On the Nonexistence of Certain Morphisms

from Grassmannian to Grassmannian

in Characteristic 0

Ajay C. Ramadoss

Received: March 17, 2007 Revised: April 8, 2009

Communicated by Takeshi Saito

Abstract. This paper proves some properties of the big Chern classes of a vector bundle on a smooth scheme over a field of characteristic 0. These properties together with the explicit computation of the big Chern classes of universal quotient bundles of Grassmannians are used to prove the main Theorems (Theorems 1,2 and 3) of this paper.

The nonexistence certain morphisms between Grassmannians over a field of characteristic 0 follows directly from these theorems. One of our theorems, for instance, states that the higher Adams operations applied to the class of a universal quotient bundle of a Grassmannian that is not a line bundle yield elements in the K-ring of the Grassmannian that are not representable as classes of genuine vector bundles. This is not true for Grassmannians over a field of characteristic p.

2000 Mathematics Subject Classification: 14M15, 14F99, 19A99 Keywords and Phrases: Chern character, big Chern classes, Grassmannian, universal quotient bundle, Adams operations.

1 Introduction

1.1 Motivation

Problems regarding the constraints that morphisms between homogeneous spaces must satisfy have been studied by Kapil Paranjape and V. Srinivas [7], [8]. In [7], they characterize self maps of finite degree between homogeneous spaces and prove that finite surjective morphisms from Grassmannian to Grassmannian are actually isomorphisms. In [8], they prove that if S is a smooth quadric hypersurface in ℙ^{n+1}, where n = 2k + 1, and if 2^{k}|d, then there exist continuous maps f : ℙ^{n} → S so that f^{∗}(O_{S}(1)) = O_{ℙ^{n}}(d). Let G(r, n) denote the Grassmannian of r-dimensional quotient spaces of an n-dimensional vector space over a field of characteristic 0. In the same spirit, given an integer p ≥ 2, one can ask questions like whether there exists a map from a Grassmannian G(r, n) to another Grassmannian G(r, M) so that f^{∗}[Q_{G(r,M)}] = ψ^{p}[Q_{G(r,n)}] where [V] denotes the class of a vector bundle V in K-theory and Q_{G(r,n)} and Q_{G(r,M)} denote the universal quotient bundles of G(r, n) and G(r, M) respectively. Another question in the same spirit would be whether there exist morphisms f : G(r, n) → G(r − 1, M) so that f^{∗}(ch_{l}(Q)) = ch_{l}(Q). The answers to the first question is in the negative for all r ≥ 2, n ≥ 2r + 1 and the answer to the second question is in the negative for infinitely many r, with n assumed to be large enough. It may be noted that in these questions, our attention is not restricted solely to dominant/finite morphisms unlike in the results in [7] and [8]. Indeed, the results proven here are not obtainable by the methods of [7] and [8] as far I can see.

1.2 Statements of the results

The following theorems contain the answers obtained for the above questions. These theorems are proven in this paper. Before we proceed, we state that all varieties in this paper are smooth projective varieties over a field of characteristic 0. For any smooth projective variety X, let K(X) denote the K-ring of X. For any vector bundle V on X, let [V] denote the class of V in K(X)⊗ℚ.

Theorem 1. Let Q denote the universal quotient bundle of a Grassmannian G(r, n). Suppose that r ≥ 2 and that n ≥ 2r + 1. Then, for all p ≥ 2, the element ψ^{p}[Q] of K(G(r, n)) ⊗ ℚ is not equal to [V] for any genuine vector bundle V on G(r, n).

Corollary 1. If f : G(r, n) → G(r, ∞) is a morphism of schemes with r ≥ 2 and n ≥ 2r + 1, then f^{∗}[Q_{G(r,∞)}] ≠ ψ^{p}[Q_{G(r,n)}] for any p ≥ 2.

Let X be a smooth variety, and let F_{r}CH^{l}(X) ⊗ ℚ denote the subspace of CH^{l}(X) ⊗ ℚ spanned by {ch_{l}(V)|V a vector bundle of rank ≤ r}. Then, this filtration is nontrivial as a theory. Let Q_{G(r,n)} denote the universal quotient bundle of G(r, n), and let ch denotes the Chern character map, with ch_{l} denoting the degree l component of ch.

Theorem 2. Given any natural number l ≥ 2, there exist infinitely many natural numbers r > 0, and a constant C depending on l so that whenever n > Cr^{2} + r,

ch_{l}(Q_{G(r,n)}) ∈ F_{r}CH^{l}(G(r, n)) ⊗ ℚ \ F_{r−1}CH^{l}(G(r, n)) ⊗ ℚ.

Corollary 2. Given any natural number l ≥ 2, there exist infinitely many natural numbers r > 0, and a constant C depending on l so that whenever n > Cr^{2} + r, and f : G(r, n) → G(r − 1, ∞) is a morphism of varieties, then

f^{∗}(ch_{l}(Q_{G(r−1,∞)})) ≠ ch_{l}(Q_{G(r,n)}).

Corollary 3. There exist infinitely many r so that if f : G(r, n) → G(r − 1, ∞) is any morphism of schemes with n > 7r^{2} + r + 2, then

f^{∗} ch_{2}(Q_{G(r−1,∞)}) = κch_{1}(Q_{G(r,n)})^{2}

for some constant κ ∈ 𝕂 that possibly depends on r.

Theorem 3. If f : G(3, 6) → G(2, ∞) is a morphism, then

f^{∗}(ch_{2}(Q_{G(2,∞)})) = κch_{1}(Q_{G(3,6)})^{2}

for some constant κ ∈ 𝕂.

1.3 An outline of the set up of the proofs

All these results are proven using certain facts about certain characteristic classes. These characteristic classes were discovered by M. Kapranov [6] (and independently by M.V. Nori [1]) as far as I know. In this paper, I shall show that these objects are characteristic classes that commute with Adams operations (Lemma 9 and Lemma 13 of Section 4.2 in this paper). These characteristic classes are defined as follows.

Let X be a smooth projective variety and let V be a vector bundle on X. Consider the Atiyah class

θ_{V} ∈ H^{1}(X, End(V) ⊗ Ω)

of V. Denote the k -fold cup product of θ_{V} with itself by θ_{V}^{k}. Applying the composition map End(V)^{⊗k} → End(V), followed by the trace map tr : End(V) → O_{X} to θ_{V}^{k}, we obtain the characteristic class

t_{k}(V) ∈ H^{k}(X, Ω^{⊗k}).

Note that the projection Ω^{⊗k} → ∧^{k}Ω when applied to t_{k}(V) gives us k! ch_{k}(V) where ch_{k}(V) denotes the degree k part of the Chern character of V. The classes t_{k} are referred to in the paper by Kapranov [1] as the big Chern classes. These classes and their properties are discussed in greater detail in Section 4 of this paper. The big Chern classes together give a ring homomorphism ⊕ t_{k} : K(X) ⊗ ℚ → ⊕ H^{k}(X, Ω^{⊗k}) where the right hand side is equipped with a commutative product that shall be described in the Section 2. The commutative ring ⊕ H^{k}(X, Ω^{⊗k}) shall henceforth be denoted by R(X). Both this product and the usual cup product in addition to some other (λ-ring) structure on this ring are preserved under pullbacks. Moreover, the two products are distinct and the Adams gradation on R(X) is distinct from the obvious one (unlike in the case of the usual cohomology ring). These facts place serious restrictions on what pullback maps f^{∗} : R(X) → R(Y) corresponding to morphisms f : Y → X look like. An important subring of the ring R(X) will be calculated explicitly for the Grassmannian G(r, n) at the end of Section 3.

Notation: Throughout this paper, 𝕂 shall be used to denote the base field. We assume throughout this paper that the characteristic of 𝕂 is zero.

1.4 Brief outlines of the proofs

1.4.1 Outline for Theorems 2 and 3

The basic idea behind the proofs of Theorem 2, Corollary 2 and Theorem 3 is the same.

If σ ∈ S_{k} is a permutation of {1, …, k}, and if F is a vector bundle on X, then σ gives us a homomorphism σ : F^{⊗k} → F^{⊗k} of O_{X} modules. If f_{1}, …, f_{k} are sections of F over an affine open subscheme Spec(U) of X, then

σ(f_{1} ⊗ ⋯ ⊗ f_{k}) = f_{σ(1)} ⊗ ⋯ ⊗ f_{σ(k)}.

This gives us a right action of S_{k} on F^{⊗k}. If F = Ω, the cotangent bundle of X, then σ : Ω^{⊗k} → Ω^{⊗k} induces a map σ_{∗} : H^{k}(X, Ω^{⊗k}) → H^{k}(X, Ω^{⊗k}). Extending this action of S_{k} on H^{k}(X, Ω^{⊗k}) gives us an endomorphism β_{∗} of H^{k}(X, Ω^{⊗k}) corresponding to each element β of the group ring 𝕂S_{k} of S_{k}.

To prove Corollary 2, it suffices to show that for l fixed, there exist infinitely many r such that there is some natural number k with the property that there exists an element β of 𝕂S_{k} such that

β_{∗} t_{k}(α_{l}(Q_{G(r,n)})) ≠ 0

and

β_{∗} t_{k}(α_{l}(Q_{G(r−1,∞)})) = 0.

Here α_{l}(V) = ch^{−1} ch_{l}(V) for any vector bundle V. This is enough because t_{k},α_{l} and β_{∗} commute with pullbacks. If Corollary 2 were to be violated with the above situation being true, we would have something that is 0 [ in this case, β_{∗} t_{k}(α_{l}(Q_{G(r−1,∞)})) ] pulling back to something that is nonzero [ in this case, β_{∗} t_{k}(α_{l}(Q_{G(r,n)})) ]. This gives us a contradiction. A little more work is required to prove Theorem 2.

1.4.2 Outline for Theorem 1

The proof of Theorem 1 is in the same spirit, though much more complicated. We will define a functor of type (k, l) (or a functor of “Adams weight l”) to be a map ( not necessarily a ring homomorphism/abelian group homomorphism ) from K(X) ⊗ ℚ → R_{k}(X) which takes an element x ∈ K(X) ⊗ ℚ to a linear combination of expressions of the form

β_{∗} (t_{λ_{1}}(α_{l_{1}}(x)) ∪ ⋯ ∪ t_{λ_{s}}(α_{l_{s}}(x)))

where β ∈ 𝕂S_{k}. If v_{l} is a functor of type (k, l) then v_{l} commutes with pullbacks and

v_{l}(ψ^{p}x) = p^{l}v_{l}(x).

Corollary 1 will be proven by showing that there is a linear dependence relation

∑_{l} a_{l}v_{l}(Q_{G(r,n)}) = 0

for all n ≥ 2r + 1, with v_{l}(Q_{G(r,n)}) ≠ 0, where v_{l}’s are functors of type (2r, l). We will pick a linear dependence relation of this type of shortest length. If Corollary 1 is false, we will obtain yet another linear dependence relation ∑_{l} p^{l}a_{l}v_{l}(Q_{G(r,n)}) = 0, contradicting the fact that the chosen linear dependence relation is of shortest length. A little more work will give us Theorem 1.

Detailed proofs are given in Sections 6 and 7, but the previous sections are required to understand the set up for the proofs. An important ingredient required to flesh-out the proof outlined above is the explicit calculation of t_{k}(Q_{G(r,n)}). This is done in section 5.

1.5 Remarks about possible future extensions

It can be easily shown that any linear dependence relation between functors of type (k, l) applied to the universal quotient bundle of G(r, n)

∑_{l} a_{l}v_{l}(Q_{G(r,n)}) = 0

that holds for all n large enough will apply to a vector bundle of rank r on a smooth projective variety X. Thus, if we are able to prove that we have a linear dependence relation

∑_{l} a_{l}v_{l}(Q_{G(r,n)}) = 0

for all n large enough with v_{l}(V) ≠ 0 then we will be able to apply the same argument to show that in K-Theory, higher Adams operations applied to [V] give us elements not expressible as the class of any genuine vector bundle.

One can try doing this for other homogenous vector bundles in the Grassmannian, and in general, other vector bundles on a G/P space arising out of P -representations, where G is a linear reductive group and P is a parabolic subgroup. This could lead to further progress towards finding the P representations that give rise to vector bundles satisfying Theorem 1. More intricate combinatorics than was used here in this paper may be required for further progress along these lines.

At first sight, it may look that theorem 2 needs to be strengthened. Indeed, on going through the proof, one feels strongly that the filtration F_{r} of CH^{l}() ⊗ ℚ, which theorem 2 says is nontrivial as a theory, is in fact, strictly increasing as a theory. More specifically, I feel that given any l ≥ 2 fixed, and r ≥ 2, there exists some Grassmannian G = G(r, n) so that ch_{l}(Q) ∈ F_{r}CH^{l}(G) ⊗ ℚ \ F_{r−1}CH^{l}(G) ⊗ ℚ.

One approach to this question is entirely combinatorial (along the lines of the proof to theorems 2 and 3). Let V_{λ} denote the irreducible representation of S_{k} corresponding to the partition λ of k. Let |λ| denote the number of rows in the Young diagram of λ. The combinatorial approach to this question is to try to show that for some k and a particular β ∈ 𝕂S_{k} depending on l and k only, the subspace spanned by the conjugates of β_{r−1} is of strictly smaller dimension than that spanned by conjugates of β_{r}. Here, β_{i} is the image of β under the projection 𝕂S_{k} → ⊕_{|λ|≤i}End(V_{λ}). Approaching this question along these lines would indeed involve algebraic combinatorics extensively.

Acknowledgements. This work would not have been possible without the many useful discussions I had with my advisor, Prof. Madhav Nori. It is difficult to convey my gratitude to him. I also received a lot of encouragement (as well as many useful suggestions) from Prof. Spencer Bloch, to whom I am deeply grateful. I am also very grateful to Prof. Shrawan Kumar for pointing out a theorem of Bott [4] used in this work and to Prof. Victor Ginzburg for making me aware of the paper by M. Kapranov [6] where the characteristic classes used are introduced. I thank my friend and colleague Apoorva Khare for helping me LaTeX this work and Dr. Victor Protsak, Prof. Kaan Akin and Prof. Mohan Ramachandran for useful suggestions.

2 The λ-ring R(X)

We recall that a (p, q)-shuffle is a permutation σ of {1, 2, …, p + q} such that σ(1) < ⋯ < σ(p) and σ(p + 1) < ⋯ < σ(p + q). We denote the set of all (p, q)-shuffles by Sh_{p,q} throughout the rest of this work. Also, for the rest of this work, the sign of a permutation σ shall be denoted by sgn(σ).

If σ ∈ S_{k} is a permutation of {1, …, k}, and F a vector bundle on X, then σ gives us a homomorphism σ : F^{⊗k} → F^{⊗k} of O_{X} modules. If f_{1}, …, f_{k} are sections of F over an affine open subscheme Spec(U) of X, then σ(f_{1} ⊗ ⋯ ⊗ f_{k}) = f_{σ(1)} ⊗ ⋯ ⊗ f_{σ(k)}.

This gives us a right action of S_{k} on F^{⊗k}. If F = Ω, the cotangent bundle of X, then σ : Ω^{⊗k} → Ω^{⊗k} induces a map σ_{∗} : H^{k}(X, Ω^{⊗k}) → H^{k}(X, Ω^{⊗k}).

If f : Y → X is a morphism of varieties, we have a natural pullback map f̃^{∗} : H^{k}(X, Ω_{X}^{⊗k}) → H^{k}(Y, f^{∗}Ω_{X}^{⊗k}). This can be composed by the map ι^{⊗k}_{∗} : H^{k}(Y, f^{∗}Ω_{X}^{⊗k}) → H^{k}(Y, Ω_{Y} ^{⊗k}) to define the pullback f^{∗} : H^{k}(X, Ω_{X}^{⊗k}) → H^{k}(Y, Ω_{Y} ^{⊗k}), where ι : f^{∗}Ω_{X} → Ω_{Y}. We note that

f^{∗} ∘ σ_{∗} = σ_{∗} ∘ f^{∗}.

If α ∈ H^{l}(X, Ω_{X}^{⊗l}) and β ∈ H^{m}(X, Ω_{X}^{⊗m}), define

α ⊙ β := ∑_{σ∈Sh_{l,m}} sgn(σ)σ_{∗}^{−1}(α ∪ β).

⊙ gives us a product on ⊕ H^{k}(X, Ω^{⊗k}). Moreover,

Proposition 1. If α and β are as in the previous paragraph, then α⊙β = β⊙α. In other words, ⊙ equips R(X) with the structure of a commutative ring.

Proof. If γ is the permutation of {1, …, k + l} where γ(i) = l + i for 1 ≤ i ≤ k and γ(i) = i − k for k = 1 ≤ i ≤ l + k, then sgn(γ) = (−1)^{kl}. Also, σ → σ ∘ γ gives us a bijection between Sh_{l,k} and Sh_{k,l}. Thus

α ⊙ β = ∑_{σ∈Sh_{k,l}} sgn(σ)σ_{∗}^{−1}(α ∪ β) = ∑_{τ∈Sh_{l,k}} sgn(γ) sgn(τ)(τ ∘ γ)_{∗}^{−1}(α ∪ β)

= ∑_{τ∈Sh_{l,k}} sgn(τ)(γ^{−1} ∘ τ^{−1})_{∗} sgn(γ)(α∪β) = ∑_{τ∈Sh_{l,k}} sgn(τ)τ_{∗}^{−1}(sgn(γ)γ_{∗}^{−1}(α∪β))

= ∑_{τ∈Sh_{l,k}} sgn(τ)τ_{∗}^{−1}(β ∪ α) = β ⊙ α.

(Note that (γ^{−1} ∘ τ^{−1})_{∗} = τ_{∗}^{−1} ∘ γ_{∗}^{−1} since the action of S_{k+l} on Ω^{⊗k+l} is a right action).

We recall from Fulton and Lang [9] that a special λ-ring A is a commutative ring together with operations ψ^{p} : A → A indexed by the natural numbers so that a) ψ^{p} is a ring homomorphism for all p.

b) ψ^{p} ∘ ψ^{q} = ψ^{pq}.

c) ψ^{1} = id.

Here, we show that R(X) has a special λ-ring structure (i.e, has Adams operations). This is done in Lemma 2. It will be clear from their definition that the Adams operations commute with pullbacks. The graded tensor co-algebra T^{∗}Ω of the cotangent bundle Ω_{X} is a sheaf of graded-commutative Hopf-algebras on X. The product on T^{∗}Ω and the Adams operations on T^{∗}Ω therefore induce corresponding operations on the cohomology ring of T^{∗}Ω. Proposition 1 in fact, proves that the ring R(X) is a subring of the cohomology ring of T^{∗}Ω. It turns out that the Adams operations on the cohomology of T^{∗}Ω restrict to Adams operations on R(X) as well. The rest of this section is devoted to explaining the details of the outline we have just highlighted. We begin with a digression on Hopf-algebras.

2.1 Adams operations on commutative Hopf-algebras

We recall that a Hopf-algebra over a field 𝕂 of characteristic 0 is a vector space H together with maps μ : H ⊗ H → H (multiplication), Δ : H → H ⊗ H (comultiplication), u : 𝕂 → H (unit) and c : H → 𝕂 (counit) such that the six properties listed below are satisfied.

1. Multiplication is associative and comultiplication is coassociative.

2. Multiplication is a coalgebra homomorphism and comultiplication is an algebra homomorphism.

3. μ ∘ (u ⊗ id) = μ ∘ (id ⊗u) = id : H → H.

4. (id ⊗c) ∘ Δ = (c ⊗ id) ∘ Δ = id : H → H.

5. u is a coalgebra map and c is an algebra map.

6. c ∘ u = id : 𝕂 → 𝕂.

One can define a Hopf algebra in the category of O_{X} modules in the same spirit. It is an O_{X} module H together with maps of O_{X} modules μ : H ⊗ H → H (multiplication), Δ : H → H ⊗ H (comultiplication), u : O_{X} → H (unit) and c : H → O_{X} (counit) such that

1. Multiplication is associative and comultiplication is coassociative.

2. Multiplication is a coalgebra homomorphism and comultiplication is an algebra homomorphism.

3. μ ∘ (u ⊗ id) = μ ∘ (id ⊗u) = id : H → H.

4. (id ⊗c) ∘ Δ = (c ⊗ id) ∘ Δ = id : H → H.

5. u is a coalgebra map and c is an algebra map.

6. c ∘ u = id : O_{X} → O_{X}.

The Hopf algebra H is said to be (graded) commutative if μ ∘ τ = μ where τ is the (signed) swap map from H ⊗ H to itself. In the graded case τ(a ⊗ b) = (−1)^{|a||b|}b ⊗ a, where a and b are homogenous sections of H over an affine open subset of X. |a| and |b| denote the degrees of a and b respectively.

The following four facts are completely analogous to statements in section 4.5.1 of Loday [2]. The checks Loday [2] asks us to do to make these observations for the case of a commutative Hopf algebra over a field also go through in our case, that of a graded commutative Hopf algebra in the category of O_{X} modules. These checks are left to the reader as they are fairly simple.

Fact 1. If H is a (graded) commutative Hopf algebra in the category of O_{X} modules, we can define the convolution of two maps f, g ∈ End_{O_{X}} (H) by

f ∗ g = μ ∘ (f ⊗ g) ∘ Δ.

The convolution product ∗ is an associative product on End_{O_{X}} (H).

Fact 2. If f is an algebra morphism, then if g and h are any O_{X} linear maps,

f ∘ (g ∗ h) = (f ∘ g) ∗ (f ∘ h).

Fact 3. If H is (graded) commutative and f and g are algebra morphisms, then f ∗ g is an algebra morphism.

Fact 4. It follows from Fact 3 that

ψ^{k} := id ∗ ⋯ ∗ id ∈ End_{O_{X}} (H)

is an algebra morphism for all natural numbers k. It also follows from Fact 2 that

ψ^{p} ∘ ψ^{q} = ψ^{pq}

for all natural numbers p, q.

Further, the following proposition, which is an extension of Proposition 4.5.3 of Loday [2] to graded commutative Hopf algebras in the category of O_{X} modules, holds as well. Since the proof of Proposition 4.5.3 of [2] given by Loday [2] goes through in this case with trivial modifications, we omit the proof of the following proposition.

Proposition 2. If H = ⊕_{n≥0}H_{n} is a (graded) commutative Hopf algebra in the category of O_{X} modules, then

a) ψ^{p} maps H_{n} to itself for all p and n.

b) There exist elements e_{n}^{(i)} of End_{O_{X}} (H_{n}) such that

ψ^{k} = ∑_{i=1}^{n} k^{i}e_{n}^{(i)}.

Further,

e_{n}^{(i)} ∘ e_{n}^{(j)} = δ_{ij}e_{n}^{(i)}

where δ_{ij} is the Kronecker delta.

An immediate consequence (when k = 1) of this proposition is that

id = e_{n}^{(1)} + ⋯ + e_{n}^{(n)}.

The Hopf algebra that is relevant to us is the (graded) tensor co-algebra of a vector bundle F. Here,

T^{∗}(F)_{n} = F^{⊗n}

Δ(f_{1} ⊗ ⋯ ⊗ f_{n}) = ∑_{0≤i≤n} f_{1} ⊗ ⋯ ⊗ f_{i} ⨂ f_{i+1} ⊗ ⋯ ⊗ f_{n} ∈ T^{∗}(F) ⊗ T^{∗}(F)

(cut coproduct) and

μ(f_{1} ⊗ ⋯ ⊗ f_{p} ⨂ f_{p+1} ⊗ ⋯ ⊗ f_{p+q})

= ∑_{σ∈Sh_{p,q}} sgn(σ)f_{σ^{−1}(1)} ⊗ ⋯ ⊗ f_{σ^{−1}(p+q)}

where f_{i} is a section of F over an affine open subscheme U of X for each i.

We note that in this case,

ψ^{2}(f_{1} ⊗ ⋯ ⊗ f_{n}) = ∑_{p+q=n} ∑_{σ∈Sh_{p,q}} sgn(σ)f_{σ^{−1}(1)} ⊗ ⋯ ⊗ f_{σ^{−1}(n)}.

In this particular case, we also want to find out about the idempotents e_{n}^{(i)} ∈ End_{O_{X}} (F)^{⊗n}. The following extension of Proposition 4.5.6 from Loday [2] is what we want. Again, since the proof given in [2] extends with trivial modifications to our case. We therefore, leave the proof of the following proposition to the reader.

Lemma 1.

e_{n}^{(i)} = ∑_{j=1}^{n} a_{n}^{i,j}l_{n}^{j}

where

∑_{i=1}^{n} a_{n}^{i,j}X^{i} = \binom{X − j + n}{n}

and

l_{n}^{j} = ∑_{σ∈S_{n,j}}(sgn σ)σ_{∗}^{−1}.

Here, S_{n,j} = {σ ∈ S_{n}|card{i|σ(i) > σ(i + 1)} = j − 1}.

For example, e_{n}^{(n)} = ∑_{σ∈S_{n}} sgn(σ)σ_{∗}^{−1}.

2.2 Description of λ-ring structure on R(X)

Consider the tensor co-algebra T^{∗}Ω. Consider the Adams operations ψ^{k} on T^{∗}Ω as described in the previous subsection. Note that ψ^{k}|_{Ω^{⊗n}} induces a map ψ_{∗}^{k} : R_{n}(X) → R_{n}(X). Thus the Adams operation ψ^{k} induces a map ψ_{∗}^{k} : R(X) → R(X) that is 𝕂-linear. That ψ^{p} ∘ψ^{q} = ψ^{pq} implies that ψ_{∗}^{p} ∘ψ_{∗}^{q} = ψ_{∗}^{pq}. Define the k-th Adams operation on ⊕ H^{n}(X, Ω^{⊗n}) to be ψ_{∗}^{k}. That the Adams operations so defined are ring endomorphisms of R(X) follows from the fact that the product in R(X) is induced by the product in T^{∗}Ω. We have therefore, proven the following Lemma.

Lemma 2. R(X) is a special λ-ring with Adams operations ψ^{p} given by ψ_{∗}^{p}.

Remark. The Adams operations on R(X) are thus seen to be defined combinatorially.

3 The ring R(G(r, n))^{Gl(n)}

In this section we explicitly compute an important part of R(G(r, n)), where G(r, n) is the Grassmannian of r dimensional quotients of an n-dimensional vector space. G(r, n) is a homogenous space Gl(n)/P where P is the appropriate parabolic subgroup of Gl(n). Let N denote the unipotent normal subgroup of P.

All the vector bundles that arise during the course of stating and proving the main theorems are Gl(n)- equivariant. Thus, the big Chern classes of these vector bundles lie in the part of R(G(r, n)) fixed by Gl(n). If V is an n dimensional vector space, let S be the subspace of V preserved by P and Q the corresponding quotient. The cotangent bundle Ω of the G(r, n) is the vector bundle arising out of the P -representation Q^{∗} ⊗S on which N acts trivially.

Convention. When we refer to Ω in the category of P -representations, we shall refer to the P representation giving rise to the cotangent bundle of G(r, n).

We are now in a position to make the following four observations. Together with the step by step justifications that follow them, these observations describe the method we will use to compute R(G(r, n))^{Gl(n)} while rigorously justifying our computations at the same time. Observation 1 that follows is a serious statement. We devote the appendix of this paper to sketch its proof. Observations 2 and 3 are first stated ”proposition style” and then followed up with proofs. Observation 4 is a sequence of four computations that is crucial to the explicit description of R(G(r, n))^{Gl(n)} that we provide.

Observation 1. Let SV denote the vector bundle on G/P arising out of a P -representation V. Then, H^{k}(G/P, SV)^{G} is isomorphic to H^{k}(P, V).

Here, H^{k}(P, V) is in the category of P -modules. This statement follows from a theorem of Bott [4]. Though the base field is the field of complex numbers in [4], an extension of this result to an arbitrary base field of characteristic 0 can be shown using the method of flat descent [11] (Theorem 6 in the appendix to this paper). We sketch a proof of this fact in the appendix to this paper.

Observation 2. In the case of a Grassmannian,

H^{k}(G/P, SV)^{G} ≅ H^{k}(N, V)^{P/N}.

Proof. We have the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{pq} = H^{p}(P/N; H^{q}(N; A)) ⟹ H^{p+q}(P; A)

where A is any P -representation. In the case of a Grassmannian, P/N is isomorphic to Gl(Q)×Gl(S). The category of P/N -representations is semisimple, and all but the bottom row of the spectral sequence vanish. Thus in the case of a Grassmannian,

H^{k}(G/P, SV)^{G} ≅ H^{k}(N, V)^{P/N}.

Observation 3. From now on G = Gl(n) and P is a parabolic subgroup such that G/P is the Grassmannian G(r, n). Let N denote the category of N -representations. For any P -representations V and W on which N acts trivially,

Ext_{N}^{k} (W, V) ≅ Hom_{𝕂}(W ⊗ ∧^{k}Ω, V).

Proof. We prove the above assertion as follows.

Step 1: Note that N is a Lie group, and in our case (that of a Grassmannian) the exponential map gives a bijection between the Lie-algebra η associated to N and N itself. The category of (finite dimensional) η representations is thus equivalent to a full subcategory of N in which all our N representations lie. Note that characteristic 0 is needed to formally define the exponential map and its inverse. Also, the category of η-representations is equivalent to the category of U(η)-representations, where U(η) is the universal enveloping algebra of η. Since η is abelian, (in the case of the Grassmannian) U(η) = Sym^{∗} η. In what follows, we shall work in the category of Sym^{∗} η-modules.

Step 2: Consider the Ad action of P on η. The resulting P representation is the P -representation Q^{∗} ⊗S on which N acts trivially. Since co-tangent bundle Ω of G(r, n) arises out of this P -representation, we abuse notation and denote this P -representation by Ω. For the rest of this section as well as in Sections 5.2 and 5.3, Ω shall denote this P -representation from which the cotangent bundle of G(r, n) arises. As vector spaces, η ≃ Ω. As algebras,

U(η) ≃ Sym^{∗}(Ω).

Step 3: Note that Sym^{∗}(Ω) acts trivially on W. In other words, y.w = 0 for any w ∈ W and any y in the ideal of Sym^{∗}(Ω) generated by Ω. Therefore, a projective Sym^{∗}(Ω)-module resolution of W can be obtained by taking the Koszul complex

… → W ⊗ ∧^{k}Ω ⊗ Sym^{∗} Ω → W ⊗ ∧^{k−1}Ω ⊗ Sym^{∗} Ω → … → W ⊗ Sym^{∗} Ω → W → 0.

It follows that if V is any other Sym^{∗} Ω-module, then Ext^{k}(W, V) is just the k-th cohomology of the complex

0 → Hom(W ⊗ Sym^{∗} Ω, V) → … → … Hom(W ⊗ ∧^{k}Ω ⊗ Sym^{∗} Ω, V) → . . ..

If V is also a trivial Sym^{∗} Ω-module, then we see that

Hom(W ⊗ ∧^{k}Ω ⊗ Sym^{∗} Ω, V) = Hom_{𝕂}(W ⊗ ∧^{k}Ω, V)

and the Koszul differential in the previous complex is 0. Thus,

Ext_{N}^{k} (W, V) ≅ Hom_{𝕂}(W ⊗ ∧^{k}Ω, V).

Observation 4. R(G(r, n))^{Gl(n)} is isomorphic to a quotient of the group ring 𝕂S_{k} as a 𝕂-vector space. For the rest of this paper we identify R(G(r, n))^{Gl(n)} with this quotient via a particular isomorphism. An explicit step by step construction of this isomorphism is provided in paragraphs a).-d). below.

a). It follows from Observation 3, Observation 2 and the fact that P/N ≅ Gl(Q) × Gl(S) that

H^{k}(G(r, n), Ω^{⊗k})^{Gl(n)} ≅ Hom_{𝕂}(∧^{k}Ω, Ω^{⊗k})^{Gl(Q)×Gl(S)}.

We recall from Weyl [10] that if V is any vector space, the map

φ_{V} : 𝕂S_{k} → End_{𝕂}(V^{⊗k})^{Gl(V)}
v_{1} ⊗ ⋯ ⊗ v_{n} ↦ v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(n)}

is a surjection. It follows from this that

φ_{Q^{∗}} ⊗ φ_{S} : 𝕂S_{k} ⊗ 𝕂S_{k} → (End_{𝕂}(Q^{∗}^{⊗k}) ⊗ End_{𝕂}(S^{⊗k}))^{Gl(Q)×Gl(S)}

is surjective.

b). Let i : ∧^{k}Ω → Ω^{⊗k} denote the standard inclusion. Let p : Ω^{⊗k} → Ω^{⊗k} denote standard projection onto the image of i. Note that p = \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ω)ω. If α ∈ (End_{𝕂}(Q^{∗}^{⊗k}) ⊗ End_{𝕂}(S^{⊗k}))^{Gl(Q)×Gl(S)}, then α ∘ i = 0 iff α ∘ p = 0. Recall that Ω ≅ Q^{∗} ⊗S. Therefore, every element in Hom_{𝕂}(∧^{k}Ω, Ω^{⊗k})^{Gl(Q)×Gl(S)} is the image of a linear combination of elements of the form

(τ ⊗ σ) ∘ \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω).

Also, since we are using the right action of S_{k} × S_{k} on Q^{∗}^{⊗k} ⊗ S^{⊗k},

(τ ⊗ σ) ∘ \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω) = \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω)(τ ⊗ σ)

= \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ωσ) sgn(σ^{−1})(ωσ ⊗ ωσ)(σ^{−1}τ ⊗ id)

= \frac{1}{k!} sgn(σ) ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω)(σ^{−1}τ ⊗ id).

c). Identify End_{𝕂}(Ω^{⊗k}) with (End_{𝕂}(Q^{∗}^{⊗k})⊗End_{𝕂}(S^{⊗k})) and think of S_{k} ×S_{k} as acting on this with the left copy of S_{k} permuting the Q^{∗} and the right copy permuting the S. Then, the map p is identified with \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω). It follows from the above computation that if σ, τ ∈ S_{k} then

(σ ⊗ τ) ∘ p = sgn(σ)(σ^{−1}τ ⊗ id) ∘ p.

Therefore, every element in Hom_{𝕂}(∧^{k}Ω, Ω^{⊗k})^{Gl(Q)×Gl(S)} is the image of a linear combination of elements of the form

(σ^{−1}τ ⊗ id) ∘ p.

It follows that as a 𝕂-vector space, Hom_{𝕂}(∧^{k}Ω, Ω^{⊗k})^{Gl(Q)×Gl(S)} can be identified with a quotient of the group ring 𝕂S_{k}. We shall shortly determine this quotient precisely – but not before making a final computation.

d). Identify Ω with Q^{∗} ⊗S. With this identification, if σ ∈ S_{k}, the right action of σ on Ω^{⊗k} corresponds to the right action of σ ⊗ σ on Q^{∗}^{⊗k} ⊗ S^{⊗k}. Also, if β ∈ 𝕂S_{k}, then

\frac{1}{k!} ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω)(β ⊗ id)(σ ⊗ σ)

= \frac{1}{k!} ∑_{ω∈S_{k}} sgn(ωσ) sgn(σ)(ωσ ⊗ ωσ)(σ^{−1}βσ ⊗ id)

= \frac{1}{k!} sgn(σ) ∑_{ω∈S_{k}} sgn(ω)(ω ⊗ ω)(σ^{−1}βσ ⊗ id).

The main result of this section. Henceforth B(G(r, n)) shall denote R(G(r, n))^{Gl(n)}. Observations 1-4 above enable us to conclude that B(G(r, n)) is isomorphic to a quotient of 𝕂S_{k} as a 𝕂-vector space.

We need to specify which quotient of 𝕂S_{k} gives B(G(r, n)). Recall that the irreducible representations of S_{k} over ℂ can be realized over ℚ and hence over any field of characteristic 0. We also recall that the irreducible representations of S_{k} are indexed by partitions λ of k. They are self-dual, and V_{λ} ⊗ Alt = Vλ̄, where λ̄ is the partition conjugate to λ. Note that 𝕂S_{k} is isomorphic to ⊕_{λ} End(V_{λ}).

Notation. Let |λ| denote the rank (number of summands) of the partition λ. Let P_{r} denote the projection from 𝕂S_{k} to ⊕_{|λ|≤r} End(V_{λ}) for 1 ≤ r ≤ k, and let P_{r,n} denote the projection from 𝕂S_{k} to ⊕_{|λ|≤r,|λ̄|≤n−r} End(V_{λ}). If n is large enough, P_{r,n} = P_{r}.

The main result in this section is the following.

Lemma 3. 1. As a vector space,

B(G(r, n)) ≅ ⊕_{k}P_{r,n}(𝕂S_{k}).

2. If σ ∈ S_{k} then

σ_{∗}P_{r,n}(β) = P_{r,n}(sgn(σ)σ^{−1}βσ) ∀ β ∈ 𝕂S_{k}.

3. If α ∈ S_{k} and β ∈ S_{l} then

P_{r,n}(α) ∪ P_{r,n}(β) = P_{r,n}(α × β)

where α × β is thought of as an element of S_{k+l} in the obvious fashion.

The second part of this lemma follows from the paragraph d). of Observation 4 in this subsection. The following sequence of lemmas proves the remaining parts of the above lemma.

3.1 A lemma and some corollaries

Lemma 4. Let G be a finite group and let χ : G → ℂ^{∗} be a 1-dimensional representation of G. Then, if β ∈ ℂ(G), ∑_{g∈G} χ(g)(g ⊗ g)(β ⊗ id) = 0 in ℂ(G × G) = ℂ(G) ⊗ ℂ(G) iff β = 0.

Proof. If β = 0 then clearly ∑_{g∈G} χ(g)(g ⊗ g)(β ⊗ id) = 0. For the implication in the opposite direction, let us see what ∑_{g∈G} χ(g)(g ⊗g) does to ℂ(G×G) = ⊕ End(V_{x} ⊗V_{y}) where the V_{x} are the irreducible representations of G. Let e_{i} be a basis for V_{x} and let f_{j} be a basis of V_{y}. Suppose that g(e_{i}) = ∑_{j=1}^{j=dim(V_{x})} g_{ij}^{x}e_{j}

and that g(f_{k}) = ∑_{l=1}^{l=dim(V_{y})} g_{kl}^{y}f_{l} for all i ∈ {1, …, dim(V_{x})} and for all k ∈ {1, …, dim(V_{y})}. Then,

∑_{g} χ(g)(g ⊗ g)(e_{i} ⊗ f_{j}) = ∑_{g} ∑_{k,l} g_{ik}^{x}g_{jl}^{y}χ(g)(e_{k} ⊗ f_{l})

= ∑_{k,l}(e_{k} ⊗ f_{l})(∑_{g} χ(g)g_{ik}^{x}g_{jl}^{y}) = ∑_{k,l}(e_{k} ⊗ f_{l})(∑_{g} g_{ik}^{z}g_{jl}^{y})

where V_{z} = V_{x} ⊗ χ.

Note that ∑_{g}(g ⊗ g) ∈ End(V_{z} ⊗ V_{y}) is a G-module homomorphism. In fact, G acts trivially on (∑_{g} g ⊗ g).(V_{z} ⊗ V_{y}). Thus, \frac{1}{|G|} ∑_{g}(g ⊗ g) acts as a projection to the trivial part of V_{z} ⊗ V_{y}. Note that V_{z} ⊗ V_{y} has a contains precisely ⟨χ_{z}, χ̄_{y}⟩ copies of the trivial representation of G. In particular, it contains one copy of the trivial representation of G iff V_{z} and V_{y} are dual representations. In that case, the projection to that copy of the trivial representation is given by v ⊗ w ↦ \frac{1}{d}w(v) ∑ e_{i} ⊗ f_{i} where d is the dimension of V_{z}. Here, \{e_{i}\} is a basis for V_{z} and {f_{i}} is the basis for V_{y} dual to {e_{i}}. This tells us that ∑_{g} g_{ik}^{z}g_{jl}^{y} = \frac{|G|}{d}δ_{yz̄}δ_{ij}δ_{kl}.

Therefore, in End(V_{x} ⊗ V_{y}), if V_{z} is not dual to V_{y}, then ∑_{g∈G} χ(g)(g ⊗ g) = 0. Assume that V_{z} is dual to V_{y}. Let {e_{i}} be a basis for V_{z} and let {f_{i}} be the basis of V_{y} dual to {e_{i}}. If {ẽ_{i}} is the basis of V_{x} corresponding to {e_{i}}, then with respect to the ordered basis ẽ_{1} ⊗ f_{1}, ẽ_{2} ⊗ f_{1}, …, ẽ_{d} ⊗ f_{1}, ẽ_{1} ⊗ f_{2}, …, ẽ_{d} ⊗ f_{2}, …, ẽ_{1} ⊗ f_{d}, …, ẽ_{d} ⊗ f_{d} of V_{x} ⊗ V_{y}, \frac{d}{|G|} ∑_{g∈G} χ(g)(g ⊗ g) corresponds to the matrix M such that M_{ij} = 1 if i, j ∈ {kd + k + 1|0 ≤ k ≤ d − 1} and M_{ij} = 0 otherwise. On the other hand, β ⊗ id in End(V_{x} ⊗ V_{y}) is given by a block diagonal matrix each of whose diagonal blocks is the matrix representing β in End(V_{x}). This proves the desired lemma.

In fact, in the above proof, we have also proven the following lemma.

Lemma 5. Let G be a finite group, and let χ : G → ℂ^{∗} be a 1-dimensional representation of G. Let V_{x} and V_{y} be irreducible representations of G such that V_{x} ⊗ χ is dual to V_{y}. Then, if β ∈ ℂ(G), ∑_{g∈G} χ(g)(g ⊗ g)(β ⊗ id) = 0 in End(V_{x} ⊗ V_{y}) iff β = 0 in End(V_{x}).

In our problem, the group in question is S_{k}. We note that these lemmas give us the precise description of B(G(r, n) when 𝕂 = ℂ. Let 𝕊_{λ} denote the Schurfunctor associated with the partition λ of k. In other words, if V is any vector space 𝕊_{λ}(V) = V^{⊗k} ⊗_{𝕂S_{k}} V_{λ} where V_{λ} is the irreducible representation of S_{k} corresponding to the partition λ. We know that if V is a vector space of rank m, 𝕊_{λ}(V) = 0 iff λ has more than m parts. Therefore, if Q has rank r, then 𝕊_{λ}(Q) = 0 iff |λ| > r and 𝕊λ̄(S) = 0 iff |λ̄| > n − r. Moreover, if λ and μ are two partitions of k, then V^{⊗k} ⊗ W^{⊗k} ⊗_{𝕂(S_{k}×S_{k})} V_{λ} ⊗ V_{μ} = 𝕊_{λ}(V) ⊗ 𝕊_{μ}(W). If γ ∈ 𝕂(S_{k} × S_{k}) ≠ 0 in End(V_{λ} ⊗ V_{μ}), then 𝕂(S_{k} × S_{k}).γ contains V_{λ} ⊗ V_{μ}. Therefore, V^{⊗k}⊗W^{⊗k}⊗_{𝕂(S_{k}×S_{k})}γ contains 𝕊_{λ}(V)⊗𝕊_{μ}(W). Lemma 5 therefore says the following when 𝕂 = ℂ.

Lemma 6. If the rank of Q is r and that of S is n − r, then

∑_{σ} sgn(σ)(σ ⊗ σ)(β ⊗ id) = 0

as an element of Hom_{𝕂}(Ω^{⊗k}, Ω^{⊗k}) iff β = 0 as an element of End(V_{λ}) for all partitions λ such that |λ| ≤ r and |λ̄| ≤ n − r.

Proof. Let γ = ∑_{σ} sgn(σ)(σ ⊗ σ)(β ⊗ id). Then, by Lemma 5, γ = 0 in End(V_{λ} ⊗ V_{μ}) if μ ≠ λ̄. Therefore, γ kills 𝕊_{λ}(Q^{∗}) ⊗ 𝕊_{μ}(S) whenever μ ≠ λ̄. On the other hand, if γ ≠ 0 in End(V_{λ} ⊗ Vλ̄), then Ω^{⊗k}.γ contains a copy of 𝕊_{λ}(Q^{∗}) ⊗ 𝕊λ̄(S). The desired lemma follows immediately.

Since the irreducible representations of S_{k} over ℂ can be realized over ℚ and hence over any field of characteristic 0,lemmas 4,5 and 6 thus hold for 𝕂S_{k} where 𝕂 is any field of characteristic 0. This proves the first part of Lemma 3 specifying the vector space structure of B(G(r, n)). We have so far also identified the right S_{k} module structure of B(G(r, n)). To describe the ring structure completely, we need to be able to compute cup products explicitly under this identification.

We now show how one computes the cup product of two elements X_{k} ∈ Hom_{𝕂}(∧^{k}Ω, Ω^{⊗k}) ⊂ H^{k}(G(r, n), Ω^{⊗k}) and Y_{l} ∈ Hom_{𝕂}(∧^{l}Ω, Ω^{⊗l}) ⊂ H^{l}(G(r, n), Ω^{⊗l}). Let X_{k} = (γ_{k} ⊗ id) ∘ i_{k} ∈ End_{𝕂}(Q^{∗}^{⊗k}) ⊗ End_{𝕂}(S^{⊗k}) and Y_{l} = (δ_{l} ⊗ id) ∘ i_{l} ∈ End_{𝕂}(Q^{∗}^{⊗l}) ⊗ End_{𝕂}(S^{⊗l}) where i_{k} and i_{l} are the standard inclusions ∧^{k}Ω → Ω^{⊗k} and ∧^{l}Ω → Ω^{⊗l} respectively. End_{𝕂}(Ω^{⊗∗}) is identified with End_{𝕂}(Q^{∗}^{⊗∗}) ⊗ End_{𝕂}(S^{⊗∗}) as usual. The following Lemma explicitly computes X_{k} ∪ Y_{l}.

Lemma 7.

[(γ_{k} ⊗ id) ∘ i_{k}] ∪ [(δ_{l} ⊗ id) ∘ i_{l}] = [((γ_{k} ⊗ δ_{l}) ⊗ id) ∘ i_{k+l}].

The element (γ_{k} ⊗ δ_{l}) ∈ 𝕂(S_{k} × S_{l}) ⊂ 𝕂(S_{k+l}) where S_{k} × S_{l} is embedded in S_{k+l} in the natural way.

Before proving this Lemma, we note that part 3 of Lemma 3 follows immediately from the above lemma.

Proof. Let W be any 𝕂-vector space Sym^{∗} Ω acts trivially. In other words, y.w = 0 for any w ∈ W and any y in the ideal of Sym^{∗}(Ω) generated by Ω. Let ϕ ∈ End(W). Let ϕ̄ : W ⊗ Sym^{∗}(Ω) → W denote the map ϕ ⊗ η where η : Sym^{∗} Ω → 𝕂 canonical map from Sym^{∗} Ω to its quotient by the ideal generated by Ω.

Let α_{j} : Ω^{⊗j} ⊗ Sym^{∗}(Ω) → Ω^{⊗j} ⊗ Sym^{∗}(Ω) denote the map

ω_{1} ⊗ ⋯ ⊗ ω_{j} ⨂ Y ↦ ω_{1} ⊗ ⋯ ⊗ ω_{j−1} ⨂ ω_{j}Y

for ω_{1}, …, ω_{j} ∈ Ω and Y ∈ Sym^{∗}(Ω).

Let d : ∧^{j}Ω ⊗ Sym^{∗}(Ω) → ∧^{j−1}Ω ⊗ Sym^{∗}(Ω) denote the Koszul differential.

Note that the following diagram commutes.

Ω^{⊗j} ⊗ Sym^{∗}(Ω) ⟶^{α_{j}} Ω^{⊗j−1} ⊗ Sym^{∗}(Ω)
i_{j}⊗id_{Sym^{∗}(Ω)}↓  ↓^{i_{j−1}⊗id_{Sym^{∗}(Ω)}}
∧^{j}Ω ⊗ Sym^{∗}(Ω) ⟶^{d} ∧^{j−1}Ω ⊗ Sym^{∗}(Ω)

We have the following commutative diagrams.

0 ⟶  Ω^{⊗k}  ⟶  Z_{1}  ⟶ …
γ̄_{k}↑  ^{θ_{1}}↑  ↑
… ⟶ Ω^{⊗k} ⊗ Sym^{∗} Ω ⟶^{α_{k}} Ω^{⊗k−1} ⊗ Sym^{∗} Ω ⟶ …
… ⟶ Z_{k} ⟶ 𝕂 ⟶ 0
↑  ↑  ↑^{id}
… ⟶ Sym^{∗} Ω ⟶ 𝕂 ⟶ 0
0 ⟶  Ω^{⊗l}  ⟶  W_{l}  ⟶ …
δ̄_{l}↑  ^{θ_{2}}↑  ↑
… ⟶ Ω^{⊗l} ⊗ Sym^{∗} Ω ⟶^{α_{l}} Ω^{⊗l−1} ⊗ Sym^{∗} Ω ⟶ …
… ⟶ W_{l} ⟶ 𝕂 ⟶ 0
↑  ↑  ↑^{id}
… ⟶ Sym^{∗} Ω ⟶ 𝕂 ⟶ 0

The top rows of the two commutative diagrams are exact sequences representing X_{k} and Y_{l} respectively. To compute the cup product X_{k} ∪ Y_{l} we only need to find vertical arrows making all squares in the following diagram commute.

On the Nonexistence of Certain Morphisms from … 85

0 ⟶  Ω^{⊗k+l}  ⟶  Z_{1} ⊗ Ω^{⊗l}  ⟶ …
↑  ↑  ↑
… ⟶ Ω^{⊗k+l} ⊗ Sym^{∗} Ω ⟶^{α_{k+l}} Ω^{⊗k+l−1} ⊗ Sym^{∗} Ω ⟶ …
… ⟶ Z_{k} ⊗ Ω^{⊗l} ⟶  W_{1}  ⟶ …
↑  ↑  ↑^{θ_{2}}  ↑
… ⟶ Ω^{⊗l} ⊗ Sym^{∗} Ω ⟶_{α_{l}} Ω^{⊗l−1} ⊗ Sym^{∗} Ω ⟶ …
… ⟶ W_{l} ⟶ 𝕂 ⟶ 0
↑ ↑ ↑^{id}
… ⟶ Sym^{∗} Ω ⟶ 𝕂 ⟶ 0

Note that the diagrams below commute.

0 ⟶  Ω^{⊗k+l}  ⟶  Z_{1} ⊗ Ω^{⊗l}  ⟶ …
γ̄_{k}⊗δ_{l}↑  ^{θ_{1}⊗δ_{l}}↑  ↑
… ⟶ Ω^{⊗k+l} ⊗ Sym^{∗} Ω ⟶^{α_{k+l}} Ω^{⊗k+l−1} ⊗ Sym^{∗} Ω ⟶ …
… ⟶ Z_{k} ⊗ Ω^{⊗l}
↑ ↑
… ⟶ Ω^{⊗l} ⊗ Sym^{∗} Ω
Z_{k} ⊗ Ω^{⊗l} ⟶ Ω^{⊗l}
−⊗δ_{l}↑ ↑^{δ_{l}}
Ω^{⊗l} ⊗ Sym^{∗} Ω ⟶ Ω^{⊗l}

These diagrams prove the desired lemma. 3.2 An example. Lemma 3 tells us that if X = G(∞, ∞) = \underrightarrow{lim} G(r, ∞) then R(X) = ⊕_{k}𝕂S_{k} with σ_{∗}α = sgn(σ)σ^{−1}ασ for all σ ∈ S_{k}, α ∈ 𝕂S_{k}. Thus, by Lemma 3 and

Documenta Mathematica 14 (2009) 67–113

Proposition 1, if α ∈ S_{k}, and β ∈ S_{l}, then α ⊙ β = ∑_{σ∈Sh_{k,l}} σ(α × β)σ^{−1}. In other words, R(X) is the commutative algebra generated by symbols x_{γ} for all γ ∈ S_{k}, for all k modulo the relations x_{α}x_{β} = ∑_{σ∈Sh_{k,l}} x_{σ(α×β)σ^{−1}}. This can be seen to be larger than the usual cohomology ring of this space.

4 The big Chern Classes t_{k} and a ring homomorphism from K(X)⊗ ℚ to R(X)

Let V be a locally free coherent sheaf on a scheme X/S with X smooth over S. An algebraic connection on V is defined as an O_{S} linear sheaf homomorphism D : V → Ω_{X/S} ⨂_{O_{X}} V satisfying the Leibniz rule, i.e,

D(fv) = df ⊗ v + fDv ∀ f ∈ Γ(U, O_{X}) , v ∈ Γ(U, V),

for every U open in X. Note that a connection on V by itself is not O_{X} linear. However, if D_{1} and D_{2} are two connections on V |_{U} with U ⊆ X open, then D_{1} − D_{2} ∈ Γ(U, End(V) ⊗ Ω_{X/S}).

For each open U ⊆ X , let C_{V} (U) denote the set of connections on V |_{U}. This gives us a sheaf of sets on X on which End(V) ⊗_{O_{X}} Ω_{X/S} acts simply transitively. Consider a covering of X by open affines U_{i} such that V is trivial on U_{i}, and pick an element D_{i} ∈ C_{V} (U_{i}) ∀ i (D_{i} exists as d^{n} : O_{X}^{n} → Ω_{X}^{n} is a connection and thus gives a connection on V |_{U_{i}} ≅ O_{X}^{n}, where n is the rank of V). The D_{i} together give rise to a well defined element θ_{V} ∈ H^{1}(X, End(V) ⊗ Ω).

Lemma 8. θ_{V ⊗W} = A_{V} + B_{W}, where A_{V} and B_{W} are the elements in H^{1}(X, End(V) ⊗ End(W) ⊗ Ω) induced from θ_{V} and θ_{W} respectively by the maps End(V) → End(V) ⊗ End(W) (m ↦ m ⊗ id_{W}) and End(W) → End(V) ⊗ End(W), (m′ ↦ id_{V} ⊗m′) respectively.

Corollary 4. θ_{V ⊗V} is induced from θ_{V} by the map End(V) → End(V) ⊗ End(V), (m ↦ m ⊗ id_{V} + id_{V} ⊗m).

Proof. Since V and W are locally free, we can cover X by open sets U_{i} so that V and W are free over U_{i} for each i. Let D_{i} ∈ C_{V} (U_{i}), and E_{i} ∈ C_{W} (U_{i}) for each i. The desired result follows from the fact that id_{V} ⊗E_{i} + D_{i} ⊗ id_{W} ∈ C_{(V ⊗W)}(U_{i}).

4.1 The big Chern Classes t_{k}

Given any two locally free coherent sheaves F and G on X, one has a cup product ∪ : H^{i}(X, F) ⊗ H^{j}(X, G) → H^{i+j}(X, F ⊗ G). Hence, we can consider the cup product of θ_{V} with itself k times -

θ_{V} ∪ ⋯ ∪ θ_{V} =: θ_{V}^{k} ∈ H^{k}(X, End(V)^{⊗k} ⨂ Ω^{⊗k}).

The composition map φ : End(V)^{⊗k} → End(V) induces a map φ_{∗} : H^{k}(X, End(V)^{⊗k} ⊗ Ω^{⊗k}) → H^{k}(X, End(V) ⊗ Ω^{⊗k}). Let { \atop t_{k}(V)} := φ_{∗}θ_{V}^{k}. The trace map tr : End(V) → O_{X} is O_{X} -linear and induces

tr_{∗} : H^{k}(X, End(V) ⊗ Ω^{⊗k}) → H^{k}(X, Ω^{⊗k}). By definition, t_{k}(V) := tr_{∗}{ \atop t_{k}(V)}. The classes t_{k} are referred to in Kapranov [6] as the big Chern classes. The projection Ω^{⊗k} → ∧^{k}Ω when applied to t_{k}(V) gives us k! ch_{k}(V) where ch_{k}(V) is the degree k part of the Chern character of V. The appropriate reference for the construction of the Atiyah class and the construction of the components of the Chern character as done here is Atiyah [12].

4.2 Basic properties of the big Chern classes

Firstly, t_{k} is a characteristic class. In other words,

Lemma 9. If 0 → V′ → V → V′′ → 0 is an exact sequence of locally free coherent sheaves on X, then t_{k}(V) = t_{k}(V′) + t_{k}(V′′).

Proof. Let V,V′ and V′′ be as in the statement of this lemma. We first prove this lemma for the case when k = 1. Consider a cover of X by affine open sets U_{i} such that V and V′ are trivial over the U_{i}. On each U_{i}, choose a connection D_{i}, so that the restriction D_{i}|V′ of D_{i} to V′ is a connection on V′. In other words, D_{i}(Γ(U_{i}, V′)) ⊂ Γ(U_{i}, Ω ⊗ V′). On the other hand,for each U ⊂ X open, one can consider the 𝕂-vector space C_{V,V′}(U) of connections on V |U that give rise to a connection on V′|U. Note that the difference between any two elements of C_{V,V′}(U) is an element of Γ(U, P ⊗ Ω), which acts simply transitively on C_{V,V′}(U). Here, P is the subsheaf of sections of End(V) that preserve V′.

Let C_{V} (U_{i}) denote the space of connections on V |U_{i}. Thinking of the Π_{i}D_{i} as an element of Π_{i}C_{V} (U_{i}) we see that the Cech 1-cocycle Π_{i<j}(D_{i} − D_{j}) of Π_{i<j}Γ(U_{i} ∩ U_{j}, End(V) ⊗ Ω) yields the Atiyah class θ_{V} of V in H^{1}(X, End(V) ⊗ Ω). On the other hand, when the D_{i} are thought of as elements of C_{V,V′}(U_{i}), they similarly give rise to an element θ_{V,V′} of H^{1}(X, P ⊗ Ω). If i : P → End(V) is the natural inclusion, then clearly, (i ⊗ id)_{∗}θ_{V,V′} = θ_{V}. We shall denote (i ⊗ id) by i henceforth. Note that tr ∘ i = tr. Hence, tr_{∗}θ_{V,V′} = tr_{∗}θ_{V} = t_{1}(V). On the other hand, restriction to V′ gives us a map p_{1} : P → End(V′). Then p_{1∗}θ_{V,V′} is the cohomology class obtained by looking at D_{i}|V′ as elements of C_{V′}(Ui) which is θ_{V′}. We also have a projection p_{2} : P → End(V′′). Note that since the D_{i} are connections on V that restrict to connections on V′, they induce connections on V′′ (all restricted to U_{i}) which we will again denote by D_{i}. Note that p_{2∗}θ_{V,V′} is the cohomology class obtained by thinking of D_{i} as elements of C_{V′′}(U_{i}), i.e, θ_{V′′}. Now, tr|_{P} = tr∘p_{1}+tr∘p_{2}. This proves the lemma for k = 1.

Let θ_{V,V′}^{k} := θ_{V,V′} ∪ ⋯ ∪ θ_{V,V′} ∈ H^{k}(X, P^{⊗k} ⨂ Ω^{⊗k}). Let φ : P^{⊗k} → P denote

the composition map. Let { \atop t_{k}(V, V′)} := φ_{∗}θ^{k}(V, V′) ∈ H^{k}(X, P ⊗ Ω^{⊗k}). The following observations prove the lemma in general.

1. i_{∗}{ \atop t_{k}(V, V′)} = { \atop t_{k}(V)}. This follows from the commutativity of the following diagram.

P^{⊗k} ⟶^{i^{⊗k}} End(V)^{⊗k}
↓^{φ}  ^{φ}↓
P ⟶^{i} End(V)

2. p_{1∗}{ \atop t_{k}(V, V′)} = { \atop t_{k}(V′)} and p_{2∗}{ \atop t_{k}(V, V′)} = { \atop t_{k}(V′′)}. This is because the two

diagrams below commute.

P^{⊗k} ⟶^{p_{1}^{⊗k}} End(V′)^{⊗k}
↓^{φ}  ^{φ}↓
P ⟶^{p_{1}} End(V′)
P^{⊗k} ⟶^{p_{2}^{⊗k}} End(V′′)^{⊗k}
↓^{φ}  ^{φ}↓
P ⟶^{p_{2}} End(V′′)

From this and the additivity of trace, we see that t_{k}(V) = t_{k}(V′) + t_{k}(V′′).

Lemma 10. If f : Y → X is a morphism of varieties and V is a vector bundle on X, then t_{k}(f^{∗}V) = f^{∗}t_{k}(V).

Lemma 11. If V = V′ ⊕ V′′ as O_{X}-modules and p_{1} and p_{2} are the natural projections End(V) → End(V′) and End(V) → End(V′′) respectively, then

p_{1∗}{ \atop t_{k}(V)} = { \atop t_{k}(V′)} and p_{2∗}{ \atop t_{k}(V)} = { \atop t_{k}(V′′)}.

Lemmas 10 and 11 are fairly straightforward to verify and we shall skip their verification. Another important property that we prove here is that ⊕ t_{k} : K(X) ⊗ ℚ → R(X) is a ring homomorphism.

Lemma 12. If V and W are two locally free coherent sheaves on X, then,

t_{k}(V ⊗ W) = ∑_{l+m=k} t_{l}(V) ⊙ t_{m}(W)

where ⊙ is the product H^{l}(X, Ω^{⊗l}) ⊗ H^{m}(X, Ω^{⊗m}) → H^{k}(X, Ω^{⊗k}) appearing in Proposition 1. In other words, ⊕ t_{k} : K(X) ⊗ ℚ → R(X) is a ring homomorphism.

Proof. We know that θ_{V ⊗W} = θ_{V} ⊗ id_{W} + id_{V} ⊗θ_{W}. Therefore,

θ_{V ⊗W}^{k} = (A_{V} + B_{W}) ∪ ⋯ ∪ (A_{V} + B_{W})

where A_{V} = θ_{V} ⊗ id_{W} and B_{W} = id_{V} ⊗θ_{W}. Thus,

θ_{V ⊗W}^{k} = (A_{V} + B_{W})^{k} = ∑_{l+m=k} ∑_{σ∈Sh_{l,m}} sgn(σ)σ^{−1}_{∗}(A_{V} ^{l} ∪ B_{W}^{m}).

Here, a given permutation μ ∈ S_{k} acts on End(V ⊗ W)^{⊗k} ⨂ Ω^{⊗k} by

v_{1} ⊗ ⋯ ⊗ v_{k} ⨂ w_{1} ⊗ ⋯ ⊗ w_{k} ↦ v_{μ(1)} ⊗ ⋯ ⊗ v_{μ(k)} ⨂ w_{μ(1)} ⊗ ⋯ ⊗ w_{μ(k)}

and therefore induces a map from H^{k}(X, End(V ⊗ W)^{⊗k} ⊗ Ω^{⊗k}) to itself.

To verify that

(A_{V} + B_{W})^{k} = ∑_{l+m=k} ∑_{σ∈Sh_{l,m}} sgn(σ)σ^{−1}_{∗}(A_{V} ^{l} ∪ B_{W}^{m}),

note that in (A_{V} +B_{W})^{k}, terms having l A_{V} ’s cupped with m B_{W} ’s are in oneone correspondence with sequences b_{1} < ⋯ < b_{m}, b_{i} ∈ {1, 2, 3, …, l + m} ∀i (the b_{i}’s being the positions of the B_{W} ’s). Such sequences are in 1 − 1 correspondence with (l, m) shuffles. The sequence b := b_{1}, .., b_{m} corresponds to the (l, m)-shuffle σb such that σb(l + i) = b_{i}, 1 ≤ i ≤ m. Note that sgn(σb)σb_{∗}^{−1}A_{V} ^{l} ∪ B_{W}^{m} is exactly the term in (A_{V} + B_{W})^{k} where the B_{W} ’s are in positions b_{1}, …, b_{m}. The lemma is now proven by recognizing that tr_{∗} ∘ φ_{∗}σ_{∗}^{−1}(A_{V} ^{l} ∪ B_{W}^{m}) = σ_{∗}^{−1} t_{l}(V) ∪ t_{m}(W) if σ is any (l, m)-shuffle. This is because the inverse of an (l, m)-shuffle does not change the order of composition among the End(V)-terms and among the End(W) terms respectively.

Not only that, the ring homomorphism ⊕ t_{k} is also a homomorphism of special λ-rings. In other words, the big Chern classes commute with Adams operations. Indeed, the following lemma proves this fact. Note that in any special λ-ring A, the eigenspace corresponding to the eigenvalue p^{l} of the Adams operation ψ^{p} coincides with that corresponding to the eigenvalue 2^{l} of the operation ψ^{2} for any p ≥ 1. Therefore, to verify that ⊕ t_{k} commutes with the Adams operations, it suffices to verify that ⊕ t_{k} commutes with ψ^{2}. This is done in the lemma below.

Lemma 13. t_{k}(ψ^{2}V) = ψ^{2} t_{k}(V).

Proof. By the corollary to Lemma 8 (Corollary 4), θ_{V ⊗V} is induced from θ_{V} by the map β : End(V) → End(V) given by m → m ⊗ id_{V} + id_{V} ⊗m i.e, θ_{V ⊗V} = β_{∗}θ_{V}. Therefore,

θ_{V ⊗V} ^{k} = β_{∗}θ_{V} ∪ ⋯ ∪ β_{∗}θ_{V} = (β ⊗ ⋯ ⊗ β)_{∗}θ_{V} ^{k}.

By abuse of notation, we shall refer to β ⊗ ⋯ ⊗ β as β. Then, θ_{V ⊗V}^{k} = β_{∗}θ_{V}^{k}, where β : End(V)^{⊗k} → End(V)^{⊗k} is given by

m_{1} ⊗ ⋯ ⊗ m_{k} ↦ ⨂_{i=1}^{k}(m_{i} ⊗ id_{V} + id_{V} ⊗m_{i}).

Further, a direct computation shows that if W is a vector space over a field F, with charF ≠ 2, W ⊗W = Sym^{2} W ⊕∧^{2}W. Let p_{1} and p_{2} denote the resulting projections from End(W) ⊗ End(W) = End(W ⊗ W) onto End(Sym^{2} W) and End(∧^{2}W) respectively. If M, N ∈ End(W), then

tr(p_{1}(M ⊗ N)) − tr(p_{2}(M ⊗ N)) = tr(M ∘ N).

By this fact, and Lemma 11, we see that

t_{k}(ψ^{2}V) = t_{k}(Sym^{2} V) − t_{k}(∧^{2}V) = tr_{∗}p_{1∗}{ \atop t_{k}(V ⊗ V)} − tr_{∗}p_{2∗}{ \atop t_{k}(V ⊗ V)}

= tr_{∗}α_{∗}{ \atop t_{k}(V ⊗ V)}

where α : End(V) ⊗ End(V) → End(V) is the composition map.

Let φ : End(V ⊗ V)^{⊗k} → End(V ⊗ V) be the composition map. Observe that α ∘ φ ∘ β : End(V)^{⊗k} → End(V) is the map given by

m_{1} ⊗ ⋯ ⊗ m_{k} ↦ ∑_{p+q=k} ∑_{σ∈Sh_{p,q}} m_{σ(1)} ∘ ⋯ ∘ m_{σ(k)}

(∘ denoting the usual matrix multiplication on the right hand side of the last equation). Consider the map γ : End(V)^{⊗k} → End(V)^{⊗k} given by

m_{1} ⊗ ⋯ ⊗ m_{k} ↦ ∑_{p+q=k} ∑_{σ∈Sh_{p,q}} m_{σ(1)} ⊗ ⋯ ⊗ m_{σ(k)}.

Then, we see that

tr_{∗} ∘ φ_{∗} ∘ γ_{∗}θ_{V} ^{k} = tr_{∗} ∘ α_{∗}{ \atop t_{k}(V ⊗ V)} = t_{k}(ψ^{2}V).

Also observe that ψ^{2} t_{k}(V) = tr_{∗}φ_{∗}ψ_{∗}^{2}θ_{V} ^{k} since the following diagram commutes.

End(V)^{⊗k} ⊗ Ω^{⊗k} ⟶^{id ⊗ψ^{2}} End(V)^{⊗k} ⊗ Ω^{⊗k}
tr∘(φ⊗id)↓  ↓^{tr∘(φ⊗id)}
Ω^{⊗k}  ⟶^{ψ^{2}}  Ω^{⊗k}

Here, ψ_{∗}^{2} on H^{k}(X, End(V)^{⊗k} ⊗ Ω^{⊗k}) is by definition induced on co-homology by the endomorphism id ⊗ψ^{2} of End(V)^{⊗k} ⊗ Ω^{⊗k}. Thus, the following lemma remains to be proven.

Lemma 14. γ_{∗}θ_{V} ^{k} = ψ_{∗}^{2}θ_{V} ^{k}

Proof. Note that the cup-product is anti-commutative. Therefore, if σ ∈ S_{k} , then the map given by

σ : m_{1}⊗⋯⊗m_{k} ⨂ v_{1}⊗⋯⊗v_{k} ↦ sgn(σ)m_{σ(1)}⊗⋯⊗m_{σ(k)} ⨂ v_{σ(1)}⊗⋯⊗v_{σ(k)}

preserves θ_{V} ^{k}. If σ ∈ S_{k} let σ ⊗ id denote the endomorphism of End(V)^{⊗k} ⊗ Ω^{⊗k} such that

m_{1} ⊗ ⋯ ⊗ m_{k} ⨂ v_{1} ⊗ ⋯ ⊗ v_{k} ↦ m_{σ(1)} ⊗ ⋯ ⊗ m_{σ(k)} ⨂ v_{1} ⊗ ⋯ ⊗ v_{k}.

Similarly, let id ⊗σ denote the endomorphism of End(V)^{⊗k} ⊗ Ω^{⊗k} such that

m_{1} ⊗ ⋯ ⊗ m_{k} ⨂ v_{1} ⊗ ⋯ ⊗ v_{k} ↦ m_{1} ⊗ ⋯ ⊗ m_{k} ⨂ v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(k)}.

It now suffices to note that

γ = ∑_{p+q=k} ∑_{σ∈Sh_{p,q}} σ ⊗ id = ∑_{p+q=k} ∑_{σ∈Sh_{p,q}} sgn(σ)(id ⊗σ^{−1}) ∘ (σ)

⟹ γ_{∗}θ_{V}^{k} = ∑_{p+q=k} ∑_{σ∈Sh_{p,q}} sgn(σ)(id ⊗σ^{−1})_{∗} ∘ (σ)_{∗}θ_{V}^{k}

= ∑_{p+q=k} ∑_{σ∈Sh_{p,q}} sgn(σ)(id ⊗σ^{−1})_{∗}θ_{V}^{k}

= ψ_{∗}^{2}θ_{V}^{k}.

Recalling that α_{l}(V) = ch^{−1}(ch_{l}(V)), where ch is the Chern character map, we now have the following corollary of Lemma 13 below.

Corollary 5. t_{k}(α_{l}(V)) = e_{k}_{∗}^{(l)} t_{k}(V) where e_{k}^{(l)} is the idempotent described in Lemma 1.

Proof. Note that ψ^{2} = ∑ e^{(l)}2^{l}. The fact that the e_{k}^{(l)} are mutually orthogonal idempotents adding upto id tells us that ψ^{2} ∘ e_{k}^{(l)} = 2^{l}e_{k}^{(l)}. Therefore, ψ^{2} t_{k}(V) = ∑ 2^{l}e_{k}_{∗}^{(l)} t_{k}(V) = t_{k}(ψ^{2}V) = t_{k}(∑ 2^{l}α_{l}(V)) = ∑ 2^{l} t_{k}(α_{l}(V)). Since eigenvectors corresponding to different eigenvalues of a linear operator on a finite dimensional vector space over a field of characteristic 0 are linearly independent, the desired result follows.

Remark. More conceptually, if T V is the graded tensor algebra over a vector space V, (with usual tensor product giving the multiplication, and coproduct dictated by the fact that V ⊂ T V are primitive elements), then T^{∗}V is the graded Hopf algebra dual to T V. The map ψ^{2} = μ ∘ Δ : T^{∗}V → T^{∗}V has as its dual the map μ ∘ Δ : T V → T V. The 2^{l}-eigenspace of this map is seen to be ′′ Sym^{l}(L(V))′′. Thus, the 2^{l}-eigenspace of ψ^{2} : T^{∗}V → T^{∗}V is dual to the space ′′ Sym^{l}(L(V))′′. Thus, t_{k}(α_{l}(V)) lands in k-cohomology with coefficients in a space dual to ′′ Sym^{l}(L(Ω))′′. Moreover, the last corollary explicitly describes the projector that gives t_{k}(α_{l}(V)) from t_{k}(V) as the action on t_{k}(V) of a certain idempotent in 𝕂(S_{k}). Thus, one can recover t_{k}(α_{l}(V)) from t_{k}(V) combinatorially.

5 Calculating t_{k}(Q), Q the universal quotient bundle of a Grassmannian G(r, n)

We remark that Q_{G(r,n)} is often denoted by just Q in this and subsequent sections. The Grassmannian whose universal quotient bundle we are referring to is usually clear by the context.

5.1 Alternative construction for { \atop t_{k}(V)} and t_{k}(V)

Let V be a locally free coherent sheaf on a (separated) scheme X/S. It is a fact that θ_{V} is the element in Ext^{1}(V, V ⊗ Ω) ≅ H^{1}(X, End(V) ⊗ Ω) corresponding to the exact sequence 0 → V ⊗ Ω → J_{1}(V) → V → 0 where J_{1}(V) is the first jet bundle of V. Suppose that α ∈ H^{i}(X, F) = Ext^{i}(O_{X}, F) is given by an exact sequence

0 → F → Y_{1} → … → Y_{i} → O_{X} → 0

and that β ∈ H^{j}(X, G) = Ext^{j}(O_{X}, G) is given by an exact sequence

0 → G → Z_{1} → … → Z_{j} → O_{X} → 0.

Let α∗β be the element in H^{i+j}(X, F ⊗G) = Ext^{i+j}(O_{X}, F ⊗G) defined by the exact sequence which is the tensor product of the exact sequences representing α and β respectively. We note that the product

∗ : H^{i}(X, F) ⊗ H^{j}(X, G) → H^{i+j}(X, F ⊗ G)
α ⊗ β ↦ α ∗ β

has the linearity and anticommutativity properties required of the cup product. Since all the cohomology classes we are dealing with are represented by exact sequences of O_{X}-modules, we can define the cup product to be the product ∗. With this definition of the cup product, it will follow that { \atop t_{k}(V)} ∈ Ext^{k}(V, V ⊗ Ω^{⊗k}) is given by (θ_{V} ⊗ id_{Ω}^{k−1}) ∘ ⋯ ∘ θ_{V} where ∘ denotes the Yoneda product and θ_{V} is treated as an element in Ext^{1}(V, V ⊗ Ω).

5.2 Computation of { \atop t_{1}(Q)}

Recall that Ω is identified with Q^{∗} ⊗ S. Let Δ : S → Q ⊗ Ω be the map whose dual Δ^{∗} : Q^{∗} ⊗ Q ⊗ S^{∗} → S^{∗} is ev ⊗ id_{S^{∗}}, where ev : Q^{∗} ⊗ Q → 𝕂 is the evaluation map. Also, ev ⊗ id_{S} is a map from Q ⊗ Ω to S.

Lemma 15. The element of End_{𝕂}(Q ⊗ Ω) representing θ_{Q} is Δ ∘ (ev ⊗ id_{S}).

Proof. We note that the following diagram commutes.

0 ⟶  S  ⟶^{φ} V ⟶^{ϵ} Q ⟶ 0
Δ↓  ↓  ↓^{id}
0 ⟶ Q ⊗ Q^{∗} ⊗ S ⟶ J_{1}(Q) ⟶^{γ} Q ⟶ 0

The bottom row of this diagram is the exact sequence giving θ_{V}. By the universal property of push-forwards, we see that the following diagram commutes (F denotes the pushforward V ⨿_{S} Q^{∗} ⊗ Q ⊗ S).

0 ⟶  S  ⟶^{φ} V ⟶^{ϵ} Q ⟶ 0
Δ↓  ↓  ↓^{id}
0 ⟶ Q^{∗} ⊗ Q ⊗ S ⟶ F ⟶ Q ⟶ 0
↓^{id}  ↓  ^{id}↓
0 ⟶ Q^{∗} ⊗ Q ⊗ S ⟶ J_{1}(Q) ⟶ Q ⟶ 0

Therefore, θ_{Q} can be represented by the second row of the above diagram in Ext^{1}(Q, Q ⊗ Ω). Observe, however, that every arrow in this exact sequence is a P -module homomorphism (of course, Q^{∗} ⊗ Q ⊗ S, V and therefore, F are all P -modules). Thus θ_{Q} can be represented by an exact sequence in the

category of P -representations. It follows that for all k ≥ 1, { \atop t_{k}(Q)} and t_{k}(Q) can be represented by exact sequences in the category of P -representations. Therefore, to find θ_{Q}, we need to find arrows α and β so that all squares in the following diagram commute.

0 ⟶ Q^{∗} ⊗ Q ⊗ S ⟶  F  ⟶ Q ⟶ 0
Δ↑  ↑  ↑^{id}
0 ⟶  S  ⟶^{φ}  V  ⟶^{ϵ} Q ⟶ 0
α↑  ^{β}↑  ↑^{id}
… ⟶ Q ⊗ Ω ⊗ Sym^{∗} Ω ⟶ Q ⊗ Sym^{∗} Ω ⟶ Q ⟶ 0

Observe that Ω = Hom_{𝕂}(Q, S) ⊆ End(V) (here, we have chosen a 𝕂-vector space splitting 0 → S → V ⇆ Q → 0. Choosing such a splitting describes Ω as the subspace of elements in End(V) consisting of matrices whose “upper right block” is the only nonzero block. Note that the product of two such matrices is 0. Thus, any element of Sym^{∗} Ω can be thought of as an element of Hom(Q, V) ⊂ End(V). In this scheme of things, we choose β to be the natural evaluation map, and α the restriction of β to Q ⊗ Ω ⊗ Sym^{∗} Ω. Note that β and α are Sym^{∗} Ω-module homomorphisms by construction. Note that α : Q ⊗ Ω ⊗ Sym^{∗} Ω is the Sym^{∗} Ω-module homomorphism induced by α̃ := ev ∈ Hom_{𝕂}(Q ⊗ Ω, S), where ev is the natural evaluation map. It follows that as an element in Hom_{𝕂}(Q ⊗ Ω, Q ⊗ Ω), θ_{Q} is given by Δ ∘ (ev ⊗ id_{S}).

Let {e_{i}}, 1 ≤ i ≤ r be a basis for Q. Let {f_{i}} be the basis of Q^{∗} dual to {e_{i}}. Let {u_{i}}, 1 ≤ i ≤ n − r be a basis for S, and {v_{i}} the basis for S^{∗} dual to {u_{i}}. The following is a restatement of Lemma 15.

Lemma 16. With the notation just fixed, as an element of End_{𝕂}(Q ⊗ Ω) ≏ End(Q) ⊗ End(Ω) ≏ Q^{∗} ⊗ Q ⨂ Q ⊗ S^{∗} ⨂ Q^{∗} ⊗ S,

θ_{Q} = ∑_{l_{1},m_{1},r_{1}} f_{m_{1}} ⊗ e_{l_{1}} ⨂ e_{m_{1}} ⊗ v_{r_{1}} ⨂ f_{l_{1}} ⊗ u_{r_{1}}

(l_{1}, m_{1} running from 1 to r, r_{1} running from 1 to n − r).

Proof. ev(e_{i} ⊗ f_{j} ⊗ u_{k}) = δ_{ij}u_{k} and Δ(u_{k}) = ∑_{l=1}^{r} e_{l} ⊗ f_{l} ⊗ u_{k}. Therefore, θ_{Q}(e_{i} ⊗ f_{j} ⊗ u_{k}) = δ_{ij} ∑_{l=1}^{r} e_{l} ⊗ f_{l} ⊗ u_{k}. On the other hand,

f_{m_{1}} ⊗ e_{l_{1}} ⨂ e_{m_{1}} ⊗ v_{r_{1}} ⨂ f_{l_{1}} ⊗ u_{r_{1}}(e_{i} ⊗ f_{j} ⊗ u_{k}) = δ_{im_{1}}δ_{jm_{1}}δ_{kr_{1}}e_{l_{1}} ⊗ f_{l_{1}} ⊗ u_{r_{1}}.

This is nonzero iff i = j = m_{1} and k = r_{1}. This proves the desired result.

5.3 Computing { \atop t_{k}(Q)} for k > 1

This is done inductively. The method by which Yoneda products are computed is very similar to the cup product computation in the previous section. We therefore omit the details and state the key results.

If i : ∧^{k}Ω → Ω^{⊗k} is the natural inclusion, { \atop t_{k}(Q)} is given by γ_{k} ∘ i where γ_{k} ∈ End_{𝕂}(Q ⊗ Ω^{⊗k}) is as described in the following lemma.

Lemma 17. Identifying End_{𝕂}(Q⊗Ω^{⊗k}) with End_{𝕂}(Q) ⨂ Ω^{∗}^{⊗k} ⨂ Ω^{⊗k}, we have

γ_{k} = ∑_{\begin{matrix}l_{1},...,l_{k} \\ m_{1},...,m_{k} \\ r_{1},...,r_{k} \end{matrix}} (\begin{matrix}(f_{m1}⊗e_{l1})∘···∘(f_{mk}⊗e_{lk}) ⨂(e_{m1}⊗v_{r1})⊗···⊗(e_{mk}⊗v_{rk}) \\ ⨂(f_{l1}⊗u_{r1})⊗···⊗(f_{lk}⊗u_{rk}) \end{matrix}).

Here, the l_{i}, 1 ≤ i ≤ k and the m_{i}, 1 ≤ i ≤ k run from 1 to r , while the r_{i}, 1 ≤ i ≤ k run from 1 to n − r.

Having computed { \atop t_{k}(Q)} we compute t_{k}(Q). For this, we note that t_{k}(Q) = (tr⊗id)_{∗}{ \atop t_{k}(Q)} where { \atop t_{k}(Q)} ∈ End(Q)⊗Hom_{𝕂}(∧^{k}Ω, Ω^{⊗k}) and tr : End(Q) → 𝕂 is the trace map. Calculating t_{k}(Q) is then easy. In the formula in the previous lemma, we see that

(f_{m_{1}} ⊗ e_{l_{1}}) ∘ ⋯ ∘ (f_{m_{k}} ⊗ e_{l_{k}})(e_{i}) = δ_{im_{k}}δ_{l_{k}m_{k−1}} … δ_{l_{2}m_{1}}e_{l_{1}}.

From this, we see that (f_{m_{1}} ⊗ e_{l_{1}}) ∘ ⋯ ∘ (f_{m_{k}} ⊗ e_{l_{k}}) has trace 1 iff m_{k} = l_{1}, l_{k} = m_{k−1}, …, l_{2} = m_{1} and has trace 0 otherwise. From this it follows that if i : ∧^{k}Ω → Ω^{⊗k} is the natural inclusion, t_{k}(Q) is given by μ_{k} ∘ i where μ_{k} ∈ Hom_{𝕂}(Ω^{⊗k}, Ω^{⊗k}) is as described in the following lemma.

Lemma 18. Identifying End_{𝕂}(Ω^{⊗k}) with Ω^{∗}^{⊗k} ⨂ Ω^{⊗k} we have

μ_{k} =

∑_{\begin{matrix}l_{1},...,l_{k} \\ r_{1},...,r_{k} \end{matrix}}(e_{l_{2}} ⊗ v_{r_{1}}) ⊗ ⋯ ⊗ (e_{l_{k}} ⊗ v_{r_{k−1}}) ⊗ (e_{l_{1}} ⊗ v_{r_{k}}) ⨂(f_{l_{1}} ⊗ u_{r_{1}}) ⊗ ⋯ ⊗ (f_{l_{k}} ⊗ u_{r_{k}})

= ∑_{\begin{matrix}m_{1},...,m_{k} \\ r_{1},...,r_{k} \end{matrix}} (e_{m_{1}} ⊗v_{r_{1}})⊗⋯⊗(e_{m_{k}} ⊗v_{r_{k}}) ⨂(f_{m_{k}} ⊗u_{r_{1}})⊗(f_{m_{1}} ⊗u_{r_{2}})⊗⋯⊗(f_{m_{k−1}} ⊗u_{r_{k}}).

As a consequence, the basis element f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{k}} ⨂ u_{j_{1}} ⊗ ⋯ ⊗ u_{j_{k}} of Ω^{⊗k} is mapped by t_{k}(Q) to f_{i_{k}} ⊗ f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{k−1}} ⨂ u_{j_{1}} ⊗ ⋯ ⊗ u_{j_{k}}. Therefore, if we identify End_{𝕂}(Ω^{⊗k}) with Q^{∗}^{⊗k} ⊗ S^{⊗k}, t_{k}(Q) can be thought of as (k k − 1 k − 2 .. 2 1) ⊗ id_{S^{⊗k}} where (k k − 1 k − 2 .. 2 1) is the k-cycle acting on Q^{∗}^{⊗k} by the usual action of S_{k} on V^{⊗k} for a vector space V. We denote this k-cycle by τ_{k}.

Let P_{r,n} be as in Lemma 3. By Lemma 18 and the above paragraph,

Lemma 18’.

t_{k}(Q) = P_{r,n}(τ_{k}).

6 Proofs of Theorems 2 and 3

We recall that S_{k,j} denotes the set S_{k,j} = {σ ∈ S_{k}|card{i|σ(i) > σ(i + 1)} = j − 1}, i.e, the set of permutations of {1, …, k} with j − 1 descents. By part 2 of Lemma 3, if ∑ a_{σ} sgn(σ)σ ∈ 𝕂(S_{k}), we have

∑ a_{σ} sgn(σ)σ_{∗}(t_{k}(Q)) = P_{r,n}(∑ a_{σ}σ^{−1}τ_{k}σ).

The following lemma now follows immediately from Corollary 5.

Lemma 19.

t_{k}(α_{l}(Q)) = P_{r,n}(∑_{j=1}^{n} ∑_{σ∈S_{k,j}}(a_{k}^{l,j}στ_{k}σ^{−1}))

A remark and some notation. ∑_{j=1}^{k} ∑_{σ∈S_{k,j}} sgn(σ)a_{k}^{l,j}σ^{−1} is the operator

e_{k}^{(l)} for the graded commutative Hopf-algebra T^{∗}V. In fact, ∑_{j=1}^{k} ∑_{σ∈S_{k,j}} a_{k}^{l,j}σ

is the operator e_{k}^{(l)} for the co-commutative ordinary Hopf-algebra T V. We henceforth denote this idempotent by ẽ_{k}^{(l)}. Let ∗ denote the conjugation action of 𝕂S_{k} on itself. If a ∈ S_{k} and b ∈ 𝕂S_{k} then a ∗ b = aba^{−1} and (∑ c_{g}g) ∗ h = ∑ c_{g}ghg^{−1}, h ∈ 𝕂S_{k}. Then, Lemma 19 can be concisely restated as

t_{k}(α_{l}(Q)) = P_{r,n}(ẽ_{k}^{(l)} ∗ τ_{k}).

Note that ∗ is a left action.

6.1 Proofs of Corollary 2 and Corollary 3

Recall the definitions of the projections P_{r} and P_{r,n} from Section 3. Assume for now that n is large enough so that P_{r} = P_{r,n} for all values of k that we shall use. Let I(k, r, l) denote the annihilator in 𝕂S_{k} of t_{k}(α_{l}(Q)). By Lemma 3 and Lemma 19 this is precisely the subspace

I(k, r, l) = {∑_{g} c_{g} sgn(g)g|P_{r}((∑_{g} c_{g}g^{−1}) ∗ ẽ_{k}^{(l)} ∗ τ_{k}) = 0}.

If ⟨α⟩ denotes the subspace of 𝕂S_{k} spanned by conjugates of α by elements of S_{k} where α ∈ 𝕂S_{k}, then

dim(I(k, r, l)) = dim(⟨ẽ_{k}^{(l)} ∗ τ_{k}⟩) − dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩).

Note that since P_{r−1} factors through P_{r}, I(k, r, l) ⊆ I(k, r − 1, l). It follows that this inclusion is strict if

dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩) > dim(⟨P_{r−1}(ẽ_{k}^{(l)} ∗ τ_{k})⟩).

We will prove the following lemma.

Lemma 20. For a fixed l, there exists a constant C and infinitely many r such that there exists a k < Cr^{2} so that

dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩) > dim(⟨P_{r−1}(ẽ_{k}^{(l)} ∗ τ_{k})⟩).

Note that in such a situation, if n > Cr^{2} + r then P_{r} = P_{r,n} as projection operators on 𝕂S_{k}. We can then pick an element β in 𝕂S_{k} such that β_{∗} t_{k}(α_{l}(Q_{G(r−1,∞)})) = 0 and β_{∗} t_{k}(α_{l}(Q_{G(r,n)})) ≠ 0. If Corollary 2 were false there would be a morphism f : G(r, n) → G(r − 1, ∞) so that f^{∗} (β_{∗} t_{k}(α_{l}(Q_{G(r−1,∞)}))) = β_{∗} t_{k}(α_{l}(Q_{G(r,n)})). This gives us a contradiction. Therefore, Corollary 2 follows immediately from Lemma 20.

We will prove Lemma 20 by a simple counting argument. We however, need the following lemma.

Lemma 21.

ẽ_{k}^{(l)} ∗ τ_{k} = ẽ_{k−1}^{(l−1)} ∗ τ_{k}

where S_{k−1} ⊂ S_{k} is embedded as the subgroup fixing k.

Proof. Let α be a permutation of {1, 2, 3, …, k − 1} with j − 1 descents. Then, among the permutations α, ατ_{k}, …, ατ_{k}^{k−1}, we see that j of the permutations have j − 1 descents, while the remaining k − j have j descents. For, ατ_{k}^{i} has j descents or j − 1 descents depending on whether α(k − i) < α(k − i + 1) or not, for 2 ≤ i ≤ k − 1. For j − 1 such i, α(k − i) > α(k − i + 1) (corresponding to the descents of α). These j − 1 elements together with α have j − 1 descents. The remaining k − j permutations have j descents. As τ_{k}^{i}τ_{k}τ_{k}^{−i} = τ_{k}, the coefficient of ατ_{k}α^{−1} in ẽ_{k}^{(l)} ∗ τ_{k} is given by ja_{k}^{l,j} + (k − j)a_{k}^{l,j+1}, since among the elements α, ατ_{k}, …, ατ_{k}^{k−1}, those with j − 1 descents contribute a_{k}^{l,j} and those with j descents contribute a_{k}^{l,j+1} to the coefficient of ατ_{k}α^{−1} in ẽ_{k}^{(l)} ∗ τ_{k}. The desired lemma follows from observing that ja_{k}^{l,j} + (k − j)a_{k}^{l,j+1} = ja_{k−1}^{l−1,j}, since j\binom{X−j+k}{k} + (k − j)\binom{X−j−1−k}{k} = X\binom{X−j−1−k}{k−1}.

Proof. (Proof of Lemma 20). Suppose we have shown that there exists a constant C such that for a fixed l and r,

dim(⟨ẽ_{k}^{(l)} ∗ τ_{k}⟩) > dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩)

if k ≥ Cr^{2}. Then,

there exists s ≥ r so that dim(⟨P_{s}(ẽ_{k}^{(l)} ∗ τ_{k})⟩) < dim(⟨P_{s+1}(ẽ_{k}^{(l)} ∗ τ_{k})⟩).

Therefore, for any l and r, there exists s ≥ r so that

dim(⟨P_{s}(ẽ_{k}^{(l)} ∗ τ_{k})⟩) < dim(⟨P_{s+1}(ẽ_{k}^{(l)} ∗ τ_{k})⟩).

With l,r and s as above, pick k = Cr^{2}. Then k < C(s + 1)^{2} as well. This proves the lemma provided we actually show that there exists a constant C such that for a fixed l and r,

dim(⟨ẽ_{k}^{(l)} ∗ τ_{k}⟩) > dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩)

whenever k > Cr^{2}. This is what we will do now.

1. Observe that the stabilizer of τ_{k} under conjugation is the cyclic subgroup generated by τ_{k}. Thus, S_{k−1} acts freely on the conjugates of τ_{k} and β ∗ τ_{k} = 0 for some β ∈ 𝕂S_{k−1} iff β = 0. It follows from this remark and the Lemma 21 that dim(⟨ẽ_{k}^{(l)} ∗ τ_{k}⟩) is the dimension of the representation 𝕂S_{k−1}.ẽ_{k−1}^{(l−1)} of 𝕂S_{k−1}. By exercise 4.5 in Loday[2] that this space has dimension equal to the coefficient of q^{l−1} in q(q + 1) … (q + k − 2).

2. On the other hand, look at dim(⊕_{|λ|≤r} End(V_{λ}) for a fixed r. Note that if λ : k = λ_{1} + ⋯ + λ_{r′} is a partition of k, and if Π denotes the product of the hook lengths of the Young diagram corresponding to λ, then dim(V_{λ}) = \frac{k!}{Π} ≤ \frac{k!}{λ_{1}!λ_{2}!...λ_{r′}!}. Thus, dim(End(V_{λ})) ≤ (\frac{k!}{λ_{1}!λ_{2}!...λ_{r′}!})^{2}. Hence,

dim(⊕_{|λ|≤r} End(V_{λ})) ≤ ∑_{\begin{matrix}λ_{1}+···+λ_{r}=k \\ λ_{i}≥0 \end{matrix}} (\frac{k!}{λ_{1}!λ_{2}! … λ_{r}!})^{2}

≤ ( ∑_{\begin{matrix}λ_{1}+···+λ_{r}=k \\ λ_{i}≥0 \end{matrix}} \frac{k!}{λ_{1}!λ_{2}! … λ_{r}!})^{2} = r^{2k}.

Therefore, for a fixed r,

dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩) ≤ dim(⊕_{|λ|≤r} End(V_{λ})) ≤ r^{2k}.

On the other hand,

dim(⟨ẽ_{k}^{(l)} ∗ τ_{k}⟩) = coefficient of q^{l−1} in q(q + 1) … (q + k − 2) ≥ \frac{(k − 2)!}{(l − 2)!}.

We need to find k large enough so that \frac{(k−2)!}{(l−2)!} > r^{2k}. To see this we need to find k large enough so that

ln((k − 2)!) − ln((l − 2)!) > 2k ln r.

Note that

ln((k − 2)!) > (k − 2) ln(k − 2) − (k − 3).

We therefore, only need to find k large enough so that

(k − 2) ln(k − 2) > k − 3 + ln((l − 2)!) + (k − 2) ln(r^{2}) + 2 ln(r^{2}).

Put D = ln(r^{4}(l − 2)!). We then need k so that

(k − 2) ln(k − 2) > k − 3 + D + (k − 2) ln(r^{2}).

Certainly, there exists N ∈ ℕ so that N(k − 2) > (k − 3) + D To see this, note that we can pick N > D + 1 if k > 3 for instance. In fact, picking N > 5 + ln((l − 2)!) works as well. The latter choice of N is independent of r. If k − 2 > e^{N}r^{2}, then we see that

(k − 2) ln(k − 2) > k − 3 + D + (k − 2) ln(r^{2}).

Certainly, k > e^{N+1}r^{2} would do for our purposes.

Thus, if l and r are fixed, we have shown that there is a constant C so that when k > Cr^{2}, then

dim(⟨ẽ_{k}^{(l)} ∗ τ_{k}⟩) > dim(⟨P_{r}(ẽ_{k}^{(l)} ∗ τ_{k})⟩).

If l = 2, in particular, we need

(k − 2) ln(k − 2) > k − 3 + (k − 2) ln(r^{2}) + 2 ln(r^{2})

We see that this happens if k − 2 > 7r^{2}.

This completes the proof of Corollary 2. In addition, we have shown in Lemma 20 and hence in Corollary 2 that if l = 2, C = 7 works.

To complete the proof of Corollary 3, we make some observations.

Observation 1. By Lemma 21,

τ_{k} = ∑_{l≥2} ẽ_{k−1}^{(l−1)} ∗ τ_{k} = ∑_{l≥2} ẽ_{k}^{(l)} ∗ τ_{k}

⟹ t_{k}(Q) = ∑_{l≥2} t_{k}(α_{l}(Q)) ⟹ t_{k}(α_{1}(Q)) = 0 ∀k ≥ 2.

Observation 2. Since ⊕ t_{k} : K(X) ⊗ ℚ → ⊕ H^{k}(X, Ω^{⊗k}) is a ring homomorphism, is follows that

t_{k}(α_{1}(Q)^{2}) = 0

if k ≠ 2.

If f : G(s + 1, N) → G(s, M) is a morphism, then one sees that

f^{∗}(α_{2}(Q′)) = Aα_{1}(Q)^{2} + Bα_{2}(Q)

where Q and Q′ are the universal quotient bundles of G(s + 1, N) and G(s, M) respectively. By Observation 2,

t_{k} (f^{∗} (α_{2}(Q_{G(s,M)}))) = B t_{k}(α_{2}(Q_{G(s+1,N)})).

If B ≠ 0, one sees that I(k, s, 2) ⊆ I(k, s + 1, 2) (a contradiction). This finally proves Corollary 3.

To prove Theorem 2, we need the following lemma.

Lemma 22. X a smooth (projective) scheme. Suppose that [V] ∈ K(X) is given by [V] = ∑ a_{i}[V_{i}], where V_{i}’s are of rank ≤ r. Then, I(k, r, l) annihilates t_{k}(α_{l}([V])).

Proof. There exists N ∈ ℕ so that for each m > N there exist surjections 𝔾_{i} → V_{i}(m) where 𝔾_{i} is a free O_{X} module for each i. Let K_{i} denote the rank of 𝔾_{i}. This is equivalent to saying that for each i there exists a morphism f_{i} : X → G(rank(V_{i}), K_{i}) so that V_{i}(m) = f_{i}^{∗}Q_{i}, Q_{i} being the universal quotient bundle of G(rank(V_{i}), K_{i}). Thus for each i, I(k, r, l) kills t_{k} (α_{l}(V_{i} ⊗ O(m))) for each m > N.

To prove this lemma, it suffices to show that I(k, r, l) kills t_{k}(α_{l}(V_{i})) for each i. For this, we note that ⊕ t_{k}(O(1)) = e^{t_{1}(α_{1}(O_{1}))}, with the understanding that t_{1}(α_{1}(O_{1}))^{D+1} = 0 where D is the dimension of the ambient projective space. Thus, ⊕ t_{k}(O(m)) = e^{m t_{1}(α_{1}(O_{1}))}. Since the Vandermonde determinant Δ(N + 1, .., N + D + 1) ≠ 0, we can find a linear combination W of O(N + 1), …, O(N + D + 1) so that t_{k}(W) = 0 for every k ≥ 1 and t_{0}(W) = 1. Clearly, t_{k}(α_{l}(V_{i} ⊗ W)) = t_{k}(α_{l}(V_{i})) is killed by I(k, r, l).

Proof of Theorem 2. Lemma 20 implies that given any fixed l ≥ 2, there exists a constant C such that there exist infinitely many r such that given any n > Cr^{2} + r,

I(k, r, l) ⊊ I(k, r − 1, l).

Lemma 22 implies that I(k, r − 1, l) annihilates t_{k}(x) for any element x of F_{r−1}CH^{l}(Q_{G(r,n)}) ⊗ ℚ. Theorem 2 now follows immediately from the fact that I(k, r, l) is the annihilator of t_{k}(α_{l}(Q_{G(r,n)})) by definition.

6.2 Outline of proof of Theorem 3

Originally, the hope was for a stronger result saying that for fixed l and r, there exists a k satisfying I(k, r, l) ⊊ I(k, r − 1, l). In fact, there was the hope of being able to show that I(2r, r, l) ⊊ I(2r, r − 1, l). This would have shown that there is no morphism f : G(r, 2r) → G(r − 1, M) so that f^{∗}(α_{l}(Q′)) = α_{l}(Q). We have so far been unable to do this in general. However, we have found (by means of a computer program) that I(6, 3, 2) ⊊ I(6, 2, 2) thus proving that if f : G(3, 6) → G(2, M) is a morphism, then f^{∗}(α_{2}(Q′)) = Cα_{1}(Q)^{2}. This we do by showing that ⊕_{|λ|=3} End(V_{λ}) contains an irreducible representation V_{μ} of S_{6} not contained in ⊕_{|λ|≤2} End(V_{λ}), and that if π_{μ} denotes the projection from 𝕂S_{k} to V_{μ}, then π_{μ} ∗ ẽ_{6}^{(2)} ∗ τ_{6} ≠ 0. This is achieved using a Mathematica program.

7 Proof of Theorem 1

7.1 A certain decomposition of 𝕂S_{k}

On the Nonexistence of Certain Morphisms from … 101

Observe that 𝕂S_{k} = ⊕W_{λ} where W_{λ} is the 𝕂-span of elements of S_{k} in the conjugacy class corresponding to the partition λ. We shall break each of the spaces W_{λ} further into a direct sum of 𝕂-vector spaces in a specific manner. The significance of the new decomposition shall become clear as we proceed. First, let us decompose the conjugacy class C_{(k)} which is the conjugacy class of the cycle τ_{k}. Note that τ_{k} = ∑_{l≥2} ẽ_{k}^{(l)} ∗ τ_{k} and that ẽ_{k}^{(l)}ẽ_{k}^{(l′)} = δ_{ll′}ẽ_{k}^{(l)}. Define

operators Π_{l} on C_{(k)} by Π_{l}(βτ_{k}β^{−1}) = β ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) for β ∈ S_{k} and extend this by linearity to C_{(k)}. Note that ∑_{l≥2} Π_{l}(β ∗τ_{k}) = β ∗τ_{k}. First, we need to check that we actually have a well defined operator here. It suffices to show that if β, γ ∈ S_{k} with β ∗ τ_{k} = γ ∗ τ_{k} then Π_{l}(β ∗ τ_{k}) = Π_{l}(γ ∗ τ_{k}). In other words, we need to show that β ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) = γ ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) which is equivalent to showing that (β^{−1}γ) ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) = ẽ_{k}^{(l)} ∗ τ_{k}. But β ∗ τ_{k} = γ ∗ τ_{k} iff β^{−1}γ = τ_{k}^{s} for some s. Therefore, the fact that Π_{l} is well defined follows from the following lemma.

Lemma 23.

τ_{k}^{s} ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) = ẽ_{k}^{(l)} ∗ τ_{k}

for any integer s.

Proof. This really follows from the fact that for any smooth scheme X, and for any vector bundle V on X,

sgn(τ_{k})τ_{k}_{∗} t_{k}(V) = t_{k}(V).

After all, sgn(τ_{k})τ_{k}_{∗}θ_{V} ^{k} = θ_{V} ^{k} (by the properties of the cup product). Hence,

tr_{∗}φ_{∗} sgn(τ_{k})τ_{k}_{∗}θ_{V} ^{k} = tr_{∗}φ_{∗}θ_{V} ^{k}

where φ : End(V)^{⊗k} → End(V) is k-fold composition. The right hand side of this equation is t_{k}(V) by definition. The left hand side is sgn(τ_{k})τ_{k}_{∗} t_{k}(V) since

tr ∘ φ ∘ τ_{k} = τ_{k} ∘ tr ∘ φ.

This tells us that sgn(τ_{k}^{s})τ_{k}^{s}_{∗} t_{k}(V) = t_{k}(V). To finish the proof of the lemma, we observe that by Lemma 19,

τ_{k}^{s} ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) = sgn(τ_{k}^{s})τ_{k}^{s}_{∗} t_{k}(α_{l}(Q′))

and that

ẽ_{k}^{(l)} ∗ τ_{k} = t_{k}(α_{l}(Q′))

where Q′ is the universal quotient bundle of the Grassmannian G(r′, 2r′) with r′ chosen to be greater than k.

The other detail to be verified is the fact that the operators Π_{l} are mutually orthogonal projections. For this, we see that

Π_{l}(β ∗ τ_{k}) = β ∗ (ẽ_{k}^{(l)} ∗ τ_{k}) = (βẽ_{k}^{(l)}) ∗ τ_{k} ⟹ Π_{l} ∘ Π_{m}(β ∗ τ_{k})

= (βẽ_{k}^{(m)}ẽ_{k}^{(l)}) ∗ τ_{k} = (βδ_{lm}ẽ_{k}^{(l)}) ∗ τ_{k}.

We therefore, have a direct sum decomposition W_{(k)} = ⊕_{l≥2}Π_{l}(W_{(k)}).

We now proceed to breakup W_{λ} into a direct sum of 𝕂-vector spaces in an analogous manner. Note that C_{λ} is the conjugacy class of τ_{λ} := τ_{λ_{1}}τ_{λ_{2}} … τ_{λ_{s}} where the partition λ is given by λ : k = λ_{1} + .. + λ_{s}, the λ_{i} ’s arranged in decreasing order and where τ_{λ_{i}} is the cycle (λ_{1}+⋯+λ_{i}, λ_{1}+⋯+λ_{i}−1, …, λ_{1}+⋯+λ_{i−1}) which is after all the cycle τ_{λ_{i}} embedded in S_{k} under the composition S_{λ_{i}} ⊂ S_{λ_{1}} ×⋯×S_{λ_{s}} ⊂ S_{k}. Call the map S_{λ_{1}} ×⋯×S_{λ_{s}} ⊂ S_{k} as φ. Note that φ extends to a 𝕂-algebra homomorphism φ : 𝕂(S_{λ_{1}} ×⋯×S_{λ_{s}}) → 𝕂(S_{k}). Identify 𝕂(S_{λ_{1}})⊗⋯⊗𝕂(S_{λ_{s}}) with 𝕂(S_{λ_{1}} ×⋯×S_{λ_{s}}) and consider (ẽ_{λ_{1}}^{(l_{1})}⊗⋯⊗ẽ_{λ_{s}}^{(l_{s})})∗τ_{λ}. By this we mean that we are looking at ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})} as an element of 𝕂S_{k} through the homomorphism φ. We now make the following observations that give a step by step, explicit construction of the decomposition of 𝕂S_{k} that we are interested in.

Observation 1. The elements ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})} are mutually orthogonal idempotents in 𝕂(S_{k}) adding up to id. This follows from the fact that the above statement is true in 𝕂(S_{λ_{1}} × ⋯ × S_{λ_{s}}).

Observation 2. As τ_{λ} = τ_{λ_{1}} ⊗ ⋯ ⊗ τ_{λ_{s}},

(ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}) ∗ τ_{λ} = (ẽ_{λ_{1}}^{(l_{1})} ∗ τ_{λ_{1}}) ⊗ ⋯ ⊗ (ẽ_{λ_{s}}^{(l_{s})} ∗ τ_{λ_{s}})

It follows that if for some i, λ_{i} ≥ 2 and l_{i} = 1, then

(ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}) ∗ τ_{λ} = 0.

Observation 3. Let

ẽ_{λ}^{(l)} := ∑_{l_{1}+···+l_{s}=l} ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}.

Then ẽ_{λ}^{(l)} is an idempotent with

ẽ_{λ}^{(l)}.(ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}) = (ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})})

if l_{1} + ⋯ + l_{s} = l and

ẽ_{λ}^{(l)}.(ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}) = 0

otherwise.

Let Π_{l} be defined by Π_{l}(β ∗ τ_{λ}) = (βẽ_{λ}^{(l)}) ∗ τ_{λ} for every β ∈ C_{λ} . We then have

Lemma 24. The Π_{l} are well-defined mutually orthogonal projection operators on W_{λ}.

Proof. Note that it suffices to show that if γ is a permutation in the stabilizer of τ_{λ} under conjugation, then γ ∗ (ẽ_{λ}^{(l)} ∗ τ_{λ}) = ẽ_{λ}^{(l)} ∗ τ_{λ}. Note that if γ stabilizes τ_{λ} under conjugation, then γ is of the form ζ(τ_{λ_{1}}^{r_{1}} ⊗ ⋯ ⊗ τ_{λ_{s}}^{r_{s}}) where ζ permutes blocks of equal lengths among [1, …, λ_{1}], [λ_{1} + 1, …, λ_{1} + λ_{2}], …, [λ_{1} + ⋯ + λ_{s−1} + 1, …, k] while preserving order within such blocks. Now we need to show that γ ∗ (ẽ_{λ}^{(l)} ∗ τ_{λ}) = ẽ_{λ}^{(l)} ∗ τ_{λ}. Observe that

(τ_{λ_{1}}^{r_{1}} ⊗⋯⊗τ_{λ_{s}}^{r_{s}})∗(ẽ_{λ_{1}}^{(l_{1})} ⊗⋯⊗ẽ_{λ_{s}}^{(l_{s})})∗τ_{λ} = (τ_{λ_{1}}^{r_{1}} ∗ẽ_{λ_{1}}^{(l_{1})} ∗τ_{λ_{1}})⊗⋯⊗(τ_{λ_{s}}^{r_{s}} ∗ẽ_{λ_{s}}^{(l_{s})} ∗τ_{λ_{s}})

= (ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}) ∗ τ_{λ}

(the last equality by Lemma 23). So, we only need to show that

ζ ∗ ẽ_{λ}^{(l)} ∗ τ_{λ} = ẽ_{λ}^{(l)} ∗ τ_{λ}.

But this is true since ζ induces a permutation ζ′ of 1, 2, .., s and we see that ζ.(ẽ_{λ_{1}}^{(l_{1})} ⊗ ⋯ ⊗ ẽ_{λ_{s}}^{(l_{s})}) = (ẽ_{λ_{ζ′(1)}}^{(l_{ζ′(1)})} ⊗ ⋯ ⊗ ẽ_{λ_{ζ′(s)}}^{(l_{ζ′(s)})}).

Observation 4. It now follows from this and the fact that the Π_{l} are mutually orthogonal idempotents adding upto id that

W_{λ} = ⊕Π_{l}(W_{λ}).

Also, Observation 2 tells us that Π_{1}(W_{λ}) = 0 and that Π_{2}(W_{λ}) = 0 if λ ≠ (k). Therefore, this direct sum decomposition runs over l ≥ 2. Combining this with the decomposition 𝕂S_{k} = ⊕_{λ}W_{λ}, we see that

𝕂S_{k} = ⊕_{λ} ⊕_{l≥2} Π_{l}(W_{λ}) = ⊕_{l≥2}Π_{l}(𝕂S_{k}).

7.2 Proof of Corollary 1

Definition :Define an elementary functor of type (k, l) to be a map v (not necessarily linear) from K(X) ⊗ ℚ to R_{k}(X) such that

w(x) = β_{∗} t_{λ_{1}}(α_{l_{1}}(x)) ∪ ⋯ ∪ t_{λ_{s}}(α_{l_{s}}(x))

for some β ∈ 𝕂S_{k}, some s-tuple (λ_{1}, .., λ_{s}) of non-negative integers adding up to k and some s-tuple (l_{1}, …, l_{s}) of non-negative integers adding up to l.

Define a functor of type (k, l) to be a map from K(X) ⊗ ℚ to R_{k}(X) given by a ”linear combination of elementary functors of type (k, l)”. In other words, a functor of type (k, l) is a map v from K(X) ⊗ ℚ to R_{k}(X) such that

v(x) = ∑_{j=1}^{j=p} c_{j}w_{j}(x)

where p ∈ ℕ, and w_{1}, .., w_{p} are elementary functors of type (k, l).

Define a vector of type (k, l) in P_{r,n}(𝕂S_{k}) to be an element of the form v(Q), where v is a functor of type (k, l) and Q is the universal quotient bundle of the Grassmannian G(r, n).

Note that if v is a functor of type (k, l), then

v(ψ^{p}x) = p^{l}v(x)

for any x ∈ K(X) ⊗ ℚ. Also note that functors of type (k, l) respect pullbacks.

We now try to understand what the decomposition of 𝕂S_{k} given in the Section 7.1 means. Lemma 19 together with Lemma 3 part 3 tells us that

t_{λ_{1}}(α_{l_{1}}(Q_{G(r,n)})) ∪ ⋯ ∪ t_{λ_{s}}(α_{l_{s}}(Q_{G(r,n)})) = P_{r,n}(ẽ_{λ}^{(l)} ∗ τ_{λ}).

Also, by Lemma 3 part 2

sgn(β)β_{∗}^{−1} t_{λ_{1}}(α_{l_{1}}(Q_{G(r,n)})) ∪ ⋯ ∪ t_{λ_{s}}(α_{l_{s}}(Q_{G(r,n)})) = P_{r,n}(β ∗ ẽ_{λ}^{(l)} ∗ τ_{λ}).

Let l = ∑_{i} l_{i}. Thus the space spanned by

{β_{∗} t_{λ_{1}}(α_{l_{1}}(Q_{G(r,n)})) ∪ ⋯ ∪ t_{λ_{s}}(α_{l_{s}}(Q_{G(r,n)})) | ∑_{i} l_{i} = l, ∑_{i} λ_{i} = k},

which is P_{r,n}(Π_{l}(𝕂S_{k})), is precisely the space of vectors of type (k, l).

If both r and n − r are larger than k, then P_{r,n} = id. What we did in Section 7.1 shows that in this case, 𝕂S_{k} decomposes into the direct sum of the spaces Π_{l}(𝕂S_{k}). The space Π_{l}(𝕂S_{k}) is stable under conjugation and is the space of vectors of type (k, l). However, if k is not too large, something very interesting happens primarily because the projection P_{n,r} ”behaves badly” with the projections Π_{l}. Let n ≥ 2r + 1 and let k = 2r. Then, P_{r,n} = P_{r}. Also, t_{j}(Q_{G(r,n)}) = t_{j}(Q_{G(r,M)}) for every M ≥ n and every j ≤ k. It follows that v_{l}(Q_{G(r,n)}) = v_{l}(Q_{G(r,M)}) for all M ≥ n if v_{l} is any functor of type (2r, l). Let Q denote Q_{G(r,n)}. The following claim holds in this situation.

Claim: There exists a nontrivial linear dependence relation of the form

∑_{l} v_{l}(Q) = 0

such that v_{l} is a functor of type (2r, l) for each l.

The above claim is proven in Section 7.3. This leads to Corollary 1 as follows. Choose a shortest nontrivial linear dependence relation of the form

∑_{l} v_{l}(Q) = 0

with v_{l} a functor of type (2r, l). Then, suppose that there exists a map f : G(r, n) → G(r, M) with f^{∗}([Q_{G(r,M)}]) = ψ^{p}[Q_{G(r,n)}], we can assume without loss of generality that M ≥ n. Thus,

0 = f^{∗}(∑_{l} v_{l}(Q_{G(r,M)})) = ∑_{l} v_{l}(f^{∗}Q_{G(r,M)}) = ∑_{l} v_{l}(ψ^{p}Q) = ∑_{l} p^{l}v_{l}(Q).

Since p ≥ 2, comparing this linear dependence relation with the previous one would enable us to extract a linear dependence relation of the same form but of shorter length than the one we began with. This yields a contradiction.

The proof of theorem 1 requires a little more work which we do in Section 7.4.

7.3 A linear dependence relation between functors of type (2r, l)

First, we observe that if V is a vector space with V = V_{1}⊕V_{2} and also V = ⊕W_{i}, with p_{i} being the projections to V_{i} and π_{i} being the projections to W_{i}, then

dim p_{1}(W_{1}) + ⋯ + dim p_{1}(W_{m}) ≥ dim V_{1}.

To see this, suppose that equality holds. Then,

dim p_{1}(W_{i}) = dim W_{i} − dim W_{i} ∩ V_{2}
⟹ dim W_{1} ∩ V_{2} + ⋯ + dim W_{m} ∩ V_{2} = dim V_{2}.

From this, we see that π_{i}(V_{2}) = W_{i} ∩ V_{2} for all i ∈ {1, 2, .., , }. In particular, if π_{i}(V_{2}) ≠ W_{i} ∩ V_{2}, then

dim p_{1}(W_{1}) + ⋯ + dim p_{1}(W_{m}) > dim V_{1}.

Having said this, we will prove that for V = 𝕂S_{2r} ( V = V_{1} ⊕ V_{2} where V_{1} = ⊕_{|λ|≤r} End(V_{λ}) and V_{2} = ⊕_{|λ|>r} End(V_{λ}) also V = ⊕_{l≥2}Π_{l}(V))

Π_{2}(V_{2}) ≠ Π_{2}(V) ∩ V_{2}.

This will prove that

∑_{l≥2} dim P_{r}(Π_{l}(V)) > dim V_{1}.

Observation 4 of Section 7.1 tells us that Π_{2}(V) = Π_{2}(W_{(2r)}). Any element in this space is a linear combination of conjugates of τ_{2r}. It follows that if such a linear combination is nonzero in End(V_{λ}) it is also nonzero as an element of End(Vλ̄), where λ̄ is the partition conjugate to λ. Thus Π_{(2r)}(V) ∩ V_{2} = 0. It therefore, suffices to prove that Π_{2}(V_{2}) ≠ 0.

Lemma 25. To prove that Π_{2}(V_{2}) ≠ 0, it suffices to show that

Π_{2}((1 2r) ∑_{g∈C_{μ}} g) ≠ 0

where (1 2r) is the transposition interchanging 1 with 2r and μ is some partition among {(2r − 1, 1), …, (r, r)}.

Proof. Consider the matrix M = (χ_{λ}(C_{μ})) where λ runs over all partitions of 2r that satisfy λ ≥ (r, r) (recall that there is a lexicographic ordering among the partitions,enabling one to compare them), and μ ∈ {(2r − 1, 1), …, (r, r)}. Note that if λ is such a partition and λ ≠ (r, r) then λ_{1} ≥ r + 1. We claim that M is of rank r. To prove this, it suffices to show that N is of rank r where N = (ψ_{λ}(C_{μ})), where

ψ_{λ} = Ind_{S_{λ}}^{S_{2r}}(triv) = χ_{λ} + ∑_{μ>λ} K_{μλ}χ_{μ}.

However,

ψ_{λ}(C_{μ}) = \frac{1}{|C_{μ}|}[S_{2r} : S_{λ}]|C_{μ} ∩ S_{λ}|.

Therefore, ψ_{λ}(C_{μ}) = 0 if μ > λ. This lexicographic order is a total order. Consider the restriction of N to the rows given by the partitions in {(2r − 1, 1), …, (r, r)}. This restriction of N is then a lower triangular matrix with nonzero diagonal entries if the rows are arranged in the correct order (since ψ_{λ}(C_{λ}) ≠ 0). It follows that N and therefore, M are matrices of rank r.

We further claim that if we restrict M to rows corresponding to λ > (r, r), we still get a matrix of rank r. To see this, we need to show that for some scalars a_{λ},

χ_{(r,r)}(C_{μ}) = ∑_{λ>(r,r)} a_{λ}χ_{λ}(C_{μ})

for all μ ∈ {(2r − 1, 1), …, (r, r)}. For this, it is enough to show that

ψ_{(r,r)}(C_{μ}) = ∑_{λ>(r,r)} b_{λ}ψ_{λ}(C_{μ})

for all μ ∈ {(2r − 1, 1), …, (r, r)}, for some scalars b_{λ}. In fact, we claim that there are scalars b_{i},0 ≤ i ≤ r − 1, so that

ψ_{(r,r)}(C_{μ}) = ∑_{0≤i≤r−1} b_{i}ψ_{(2r−i,i)}(C_{μ}).

Note that |C_{(2r−s,s)} ∩ S_{(2r−t,t)}| = 0 if s ≠ t and both are nonzero. Also note that ψ_{(2r)}(C_{(r,r)}) ≠ 0. Thus the vector (ψ_{(2r)}(C_{μ})), μ ∈ {(2r − 1, 1), …, (r, r)} is given by (a_{1}, .., a_{r}), where a_{r} ≠ 0. The vector ψ_{(2r−s,s)}(C_{μ}), μ ∈ {(2r − 1, 1), …, (r, r)} is given by (0, .., 0, d_{s}, …, 0), d_{s} ≠ 0 for 1 ≤ s ≤ r − 1. Thus,

ψ_{(2r)}(C_{μ}) − ∑ \frac{a_{s}}{d_{s}} ψ_{(2r−s,s)}(C_{μ}) = (0, .., 0, a_{r})

which is a nonzero multiple of ψ_{(r,r)}(C_{μ}). This shows that the matrix M = χ_{λ}(C_{μ}) where λ > (r, r) and μ ∈ {(2r − 1, 1), …, (r, r)} is of rank r. Since χλ̄ = χ_{λ}. sgn, and |λ̄| ≥ r + 1 iff λ > (r, r), the matrix M′ = χ_{λ}(C_{μ}) where |λ̄| ≥ r + 1 and μ ∈ {(2r − 1, 1), …, (r, r)} is obtained from M by multiplying some columns by −1 and is therefore of rank r.

Now suppose that Π_{2}((1 2r) ∑_{g∈C_{(2r−s,s)}} g) ≠ 0 for some 1 ≤ s ≤ r. Since M′ is of rank r, we can find a linear combination of rows of M′ that gives us the vector e_{s} i.e, ∑_{|λ|>r+1} a_{λ}χ_{λ}(C_{μ}) = 0 if μ ≠ (2r − s, s) and ∑_{|λ|>r+1} a_{λ}χ_{λ}(C_{μ}) = 1 if μ = (2r − s, s). So,

Π_{2}((1 2r)( ∑_{\begin{matrix}g∈S_{2r} \\ |λ|>r+1 \end{matrix}} a_{λ}χ_{λ}(g)g)) = Π_{2}((1 2r) ∑_{g∈C_{(2r−s,s)}} g) ≠ 0.

The first equality is because only the 2r cycles contribute to Π_{2}(V). Note that since ∑ χ_{λ}(g)g ∈ End(V_{λ}) it follows that

( ∑_{\begin{matrix}g∈S_{2r} \\ |λ|>r+1 \end{matrix}} a_{λ}χ_{λ}(g)g) ∈ V_{2}

and hence

(1 2r)( ∑_{\begin{matrix}g∈S_{2r} \\ |λ|>r+1 \end{matrix}} a_{λ}χ_{λ}(g)g) ∈ V_{2}.

It follows that Π_{2}(V_{2}) ≠ 0.

Lemma 26. For some s, 1 ≤ s ≤ r, we have Π_{2}((1 2r) ∑_{g∈C_{(2r−s,s)}} g) ≠ 0.

Proof. Every 2r cycle that arises in (1 2r) ∑_{g∈C_{(2r−s,s)}} g arises with coefficient 1. We therefore need to identify the 2r cycles that do arise. They are those of the form (1 a_{2} .. a_{s} 2r a_{s+2} …) or (1 a_{2} … a_{2r−s} 2r …). For this proof, denote the subgroup of S_{2r} fixing the elements i and j by S(i, j) for any 1 ≤ i < j ≤ 2r. We note that

(1 2r) ∑_{g∈C_{(2r−s,s)}} g

= ∑_{α∈S(1,2r)} α ∗ (2r 2r − s 2r − s − 1....1 2r − 1 2r − 2 … 2r − s + 1)

+α ∗ (2r s s − 1 … 1 2r − 1.. s + 1)

= ∑_{α∈S(1,2r)} α ∗ (τ_{2r−1}^{s−1} + τ_{2r−1}^{2r−s−1}) ∗ τ_{2r}
= (τ_{2r−1}^{−1}( ∑_{β∈S(2r−1,2r)} β)τ_{2r−1}) ∗ (τ_{2r−1}^{s−1} + τ_{2r−1}^{2r−s−1}) ∗ τ_{2r}
= (τ_{2r−1}^{−1} ∑_{β∈S(2r−1,2r)} β) ∗ (τ_{2r−1}^{s} + τ_{2r−1}^{2r−s}) ∗ τ_{2r}.

Therefore,

Π_{2}((1 2r) ∑_{g∈C_{(2r−s,s)}} g) = (τ_{2r−1}^{−1} ∑_{β∈S(2r−1,2r)} β) ∗ (τ_{2r−1}^{s} + τ_{2r−1}^{2r−s}) ∗ (ẽ_{2r}^{(2)} ∗ τ_{2r})

= (τ_{2r−1}^{−1} ∑_{β∈S(2r−1,2r)} β) ∗ (τ_{2r−1}^{s} + τ_{2r−1}^{2r−s}) ∗ (ẽ_{2r−1}^{(1)} ∗ τ_{2r}),

the last equality following from Lemma 21.

For this proof, denote the subgroup of S_{2r−1} fixing the element i by S(i) for any 1 ≤ i ≤ 2r − 1. It therefore, suffices to show that

(τ_{2r−1}^{−1} ∑_{β∈S(2r−1)} β)(τ_{2r−1}^{s} + τ_{2r−1}^{2r−s})(ẽ_{2r−1}^{(1)}) ≠ 0

for some s, 1 ≤ s ≤ r. It therefore, suffices to show that

W_{s} := ( ∑_{β∈S(2r−1)} β)(τ_{2r−1}^{s} + τ_{2r−1}^{2r−s})(ẽ_{2r−1}^{(1)}) ≠ 0

for some s, 1 ≤ s ≤ r. Consider a vector space V of finite dimension, and let u and v be two basis vectors of V. We will show that the right action of W_{s} on u^{⊗2r−2} ⊗ v is nonzero. Note that

\frac{1}{(2r − 2)!}(u^{⊗2r−2} ⊗ v)W_{s} = (u^{⊗2r−2} ⊗ v)(τ_{2r−1}^{s} + τ_{2r−1}^{2r−s})ẽ_{2r−1}^{(1)}

= (u^{⊗s−1} ⊗ v ⊗ u^{⊗2r−1−s} + u^{⊗2r−1−s} ⊗ v ⊗ u^{⊗s−1})ẽ_{2r−1}^{(1)}.

Therefore, it is enough to show that

(u^{⊗s−1} ⊗ v ⊗ u^{⊗2r−1−s} + u^{⊗2r−1−s} ⊗ v ⊗ u^{⊗s−1})ẽ_{2r−1}^{(1)} ≠ 0

for some s, 1 ≤ s ≤ r. For this, we note that

0 ≠ ad(u)^{2r−2}(v) = (l_{u} − r_{u})^{2r−2}(v) = ∑_{i} \binom{2r − 2}{i}u^{⊗i} ⊗ v ⊗ u^{2r−2−i}.

Now, ad(u)^{2r−2}(v) is an element of the free Lie algebra generated by V. The idempotent ẽ_{2r−1}^{(1)} therefore acts as the identity on this vector, which is a linear combination of (u^{⊗s−1} ⊗ v ⊗ u^{⊗2r−1−s} + u^{⊗2r−1−s} ⊗ v ⊗ u^{⊗s−1}) where s runs from 1 to r.

7.4 Final step to the proof of Theorem 1

Suppose that [ψ^{p}Q] = [Y] for some genuine vector bundle Y. Then Y is of rank r, and for all sufficiently large m, Y ⊗ O(m) is a quotient of O_{G}^{s} for some s. It follows that Y ⊗ O(m) = f^{∗}Q′ for some morphism f : G(r, n) → G(r, n′), where Q′ is the universal quotient bundle of G(r, n′). Without loss of generality we may assume that n′ ≥ 2r + 1. Let Q denote the universal quotient bundle of G(r, n). As in Section 7.2, choose a shortest linear dependence relation of the form

∑_{l} v_{l}(Q) = 0

where v_{l} is a functor of type (2r, l).

Then, ∑_{l} v_{l}(Q′) = 0. Since the v_{l}’s respect pullbacks,

∑_{l} v_{l}(Y ⊗ O(m)) = 0

for all sufficiently large m. Note that ⊕ t_{k}(O(m)) = exp(t_{1}(α_{1}(O(1)))). Therefore,

t_{λ_{i}}(α_{l_{i}}(Y ⊗ O(m))) = t_{λ_{s}}(α_{l_{s}}(Y) + mα_{l_{s}−1}(Y)α_{1}(O(1)) + …).

Therefore,

v_{l}(Y ⊗ O(m)) = v_{l}(Y) + m.A_{1}(Y) + ⋯ + m^{s}A_{s}(Y)

for all l with A_{i}(Y) ∈ R(G(r, n)). In other words, v_{l}(Y ⊗O(m)) is a polynomial in m with coefficients in R(G(r, n)) whose constant term is v_{l}(Y). It follows that ∑_{l} v_{l}(Y ⊗ O(m)) is a polynomial in m with coefficients in R(G(r, n)) whose constant term is ∑_{l} v_{l}(Y). The fact that ∑_{l} v_{l}(Y ⊗ O(m)) vanishes for all sufficiently large m implies that ∑_{l} v_{l}(Y) = 0. Thus,

∑_{l} v_{l}(ψ^{p}Q) = ∑_{l} p^{l}v_{l}(Q) = 0

as well. As in Section 7.2, since p ≥ 2, this together with the linear dependence relation ∑_{l} v_{l}(Q) = 0 yields a linear dependence relation of the same form but of shorter length, thereby giving a contradiction. This finally proves Theorem 1.

Appendix

This appendix if for sketching a proof of Observation 1 of Section 3. This material is by and large reproduced from notes by Jinhyun Park [13] of a course taught by Madhav Nori at the University of Chicago in Fall 2004.

Recall that given a morphism f : Y → X of schemes, a sheaf F on Y is said to have descent data if it satisfies the following three properties.

[D_{1}]. Given any two morphisms g_{1}, g_{2} : Z → Y such that f ∘ g_{1} = f ∘ g_{2}, there is an isomorphism c(g_{1}, g_{2}) : g_{1}^{∗}F ≅ g_{2}^{∗}F. [D_{2}]. (Functoriality). Given any morphisms h : W → Z and g_{1}, g_{2} : Z → Y such that f ∘ g_{1} = f ∘ g_{2}, the following diagram commutes.

h^{∗} ∘ g_{1}^{∗}F ⟶^{h^{∗}c(g_{1},g_{2})} h^{∗} ∘ g_{2}^{∗}F
≃↓  ↓^{≃}
(g_{1} ∘ h)^{∗}F ⟶^{c(g_{1}∘h,g_{2}∘h)} (g_{2} ∘ h)^{∗}F

[D_{3}]. Given any three morphisms g_{1}, g_{2}, g_{3} : Z → Y such that f ∘ g_{1} = f ∘ g_{2} = f ∘ g_{3} the following diagram commutes.

g_{1}^{∗}F ⟶^{c(g_{1},g_{2})} g_{2}^{∗}F
c(g_{1},g_{3})↓  ↓^{c(g_{2},g_{3})}
g_{3}^{∗}F ⟶^{id} g_{3}^{∗}F

We now recall a theorem of Grothendieck [15].

Theorem 4. Let f : Y → X be a flat surjective morphism of schemes. There is an equivalence of categories

{Quasicoherent sheaves on X} ⟷ {quasicoherent sheaves on Y with descent data}

G ↦ f^{∗}G.

The following construction due to Grothendieck [15] gives the inverse to the above equivalence of categories.

Construction 1. Let F be a quasicoherent sheaf on Y with descent data. Note that for every open U ⊂ X, F|_{f^{−1}(U)} is a quasicoherent sheaf with descent data for the morphism f|_{f^{−1}(U)} : f^{−1}(U) → U. Let \overline{F} denote the sheafification of the presheaf

U ↦ {s ∈ Γ(f^{−1}(U), F) | c(g_{1}, g_{2})g_{1}^{∗}s = g_{2}^{∗}s for all g_{1}, g_{2} : Z → f^{−1}(U)}.

The inverse to the equivalence of categories in Theorem 4 is given by F ↦ \overline{F}. For example, \overline{O_{Y}} = O_{X}. Let P be an affine group scheme over 𝕂. Let f : Y → X be a principal P - bundle on X. Then, descent data for f on a sheaf F is indeed equivalent to a P -action on F. Theorem 4 therefore implies the following theorem.

Theorem 5. Let f : Y → X be a principal P -bundle. There is an equivalence of categories

{ Quasicoherent sheaves on X} ⟷ {Quasicoherent sheaves on Y with P action}

G ↦ f^{∗}G.

Corollary 6. The functor

F : { P-representations} ⟶ {locally free Quasicoherent sheaves on X}

F (V) = \overline{O_{Y} ⊗_{𝕂} V}

is an exact functor commuting with ⊗.

Proof. O_{Y} is naturally a P -sheaf on Y. A representation V of P is a H-sheaf on Spec 𝕂. Therefore, O_{Y} ⊗_{𝕂} V is a P -sheaf on Y = Y ×_{Spec 𝕂} Spec 𝕂. By Theorem 5, F (V) is a quasicoherent sheaf on X. Clearly, F (V) is locally free. It can also be verified without difficulty that V ↦ O_{Y} ⊗_{𝕂} V is an exact functor commuting with ⊗. Since the functor from Theorem 5 is an equivalence of categories, the desired corollary follows.

We can now sketch the proof of the following theorem. Let G be a affine algebraic group and let P be a closed subgroup of G. Let P denote the category of P -representations. With these assumptions, we have the following theorem of Bott [4]. This theorem has been referred to in Section 3 as Observation 1.

Theorem 6. Let G be reductive. If 𝕂 is regarded as the trivial P - representation,

H^{i}(G/P, F (V))^{G} ≃ Ext_{P}^{i}(𝕂, V).

Proof. For any V ∈ P, let T^{i}(V) = H^{i}(G/P, F (V))^{G}. We shall show that in the language of Grothendieck [14], T^{0}(V) = Hom_{P}(𝕂, V) and T^{i}(V) = R^{i}T^{0}(V). This will prove the desired theorem. To do this, we need to verify the following list of properties.

(a) T^{i} : P → 𝕂 − vector spaces is a functor.

(b) Given a short exact sequence 0 → V′ → V → V′′ → 0 in P, there is a long exact sequence

… ⟶^{δ} T^{i}(V′) ⟶ T^{i}(V) ⟶ T^{i}(V′′) ⟶^{δ} T^{i+1}V′ ⟶ …

The given short exact sequence gives a long exact sequence H^{i}(G/P, −). Now, for any exact sequence W′ → W → W′′ of G-representations, the sequence W′^{G} → W^{G} → W′′^{G} is exact. This verifies (b).

(c) The data in (b) is functorial.

(d) T^{0}(V) = V^{P}.

(e) (effaceability) For all i > 0, for all α ∈ T^{i}(V), there is a monomorphism j : V → W in P such that T^{i}(j)(α) = 0.

We check (e), the only nontrivial assertion above. Put W = Γ(P, O_{P}). Then, F (W) = f_{∗}O_{G} where f : G → G/P is the natural morphism. Note that G is affine and f ia an affine morphism. Therefore, for any quasicoherent sheaf F on G, H^{i}(G, F) = 0 for every i > 0 and R^{i}f_{∗}F = 0 for every i > 0. The Leray spectral sequence then tell us that H^{i}(G/P, f_{∗}F) ≃ H^{i}(G, F) = 0 for all i > 0. In particular, H^{i}(G/P, F (W)) = 0 for every i > 0. Let V be any P -representation. We have an isomorphism

Hom_{P}(V, Γ(P, O_{P})) ≃ V^{∗}  (1)
L ↦ ev_{id} ∘ L.

Here, ev_{id} ∘ L is the composite

V ⟶^{L} Γ(P, O_{P}) ⟶^{ev_{id}} 𝕂.

Denote the inverse of the isomorphism (1) by S. Choose linear functionals u_{1}, …, u_{i}, … on V such that ∩Ker(u_{i}) = 0. Then, S(u_{i}) ∈ Hom_{P}(V, Γ(P, O_{P})). Clearly, the morphism ⊕_{i}S(u_{i}) : V → ⊕_{i}Γ(P, O_{P}) is a monomorphism in P. Further, T^{p}(⊕_{i}Γ(P, O_{P})) = 0 whenever p > 0 since we just showed that T^{p}(Γ(P, O_{P})) = 0 whenever p > 0. This completes the verification of (e) and therefore, the proof of the desired theorem.

References

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