Séminaire Lotharingien de Combinatoire 79 (2019), Article B79c

HALL–LITTLEWOOD EXPANSIONS OF SCHUR DELTA OPERATORS AT t = 0

JAMES HAGLUND, BRENDON RHOADES, AND MARK SHIMOZONO

Abstract. For any Schur function s_{ν}, the associated delta operator Δ′_{s_{ν}} is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When ν = (1^{n−1}) is a column of length n − 1, the symmetric function Δ′_{e_{n−1}}e_{n} appears in the Shuffle Theorem of Carlsson and Mellit. More generally, when ν = (1^{k−1}) is any column the polynomial Δ′_{e_{k−1}}e_{n} is the symmetric function side of the Delta Conjecture of Haglund, Remmel, and Wilson. We give an expansion of ωΔ′_{s_{ν}} e_{n} at t = 0 in the dual Hall–Littlewood basis for any partition ν. The Delta Conjecture at t = 0 was recently proven by Garsia, Haglund, Remmel, and Yoo; our methods give a new proof of this result. We give an algebraic interpretation of ωΔ′_{s_{ν}} e_{n} at t = 0 in terms of a Hom-space.

1. Introduction and Main Results

Let Λ = ⨁_{n≥0} Λ_{n} be the ring of symmetric functions over the ground field ℚ(q, t) in an infinite variable set x = (x_{1}, x_{2}, …). Given a partition μ, let H̃_{μ} = H̃_{μ}(x; q, t) be the associated modified Macdonald symmetric function. The collection {H̃_{μ} : μ a partition} forms a basis for the ring Λ.

If f ∈ Λ is any symmetric function, the (unprimed) delta operator Δ_{f} : Λ → Λ is the Macdonald eigenoperator given by

(1.1)  Δ_{f} : H̃_{μ} ↦ f(…, q^{i−1}t^{j−1}, …) · H̃_{μ},

where (i, j) ranges over all coordinates in the (English) Ferrers diagram of the partition μ (and all remaining variables in f are set to zero). As an example, if μ = (3, 2), we fill the Ferrers diagram of μ with monomials as

\begin{matrix}1 & q & q^{2} \\ t & qt \end{matrix}

so that Δ_{f} : H̃_{(3,2)} ↦ f(1, q, q^{2}, t, qt) · H̃_{(3,2)}.

In this paper, we will focus on a primed version Δ′_{f} : Λ → Λ of the delta operator defined by

(1.2)  Δ′_{f} : H̃_{μ} ↦ f(…, q^{i−1}t^{j−1}, …) · H̃_{μ},

where (i, j) range over all coordinates ≠ (0, 0) in the Ferrers diagram of μ. If μ = (3, 2) as above, we fill the Ferrers diagram of μ with monomials as

\begin{matrix}· & q & q^{2} \\ t & qt \end{matrix}

Key words and phrases. Hall–Littlewood function, Macdonald polynomial, delta operator.

so that Δ′_{f} : H̃_{(3,2)} ↦ f(q, q^{2}, t, qt) · H̃_{(3,2)}.

Let k ≤ n be positive integers. The Delta Conjecture of Haglund, Remmel, and Wilson [6] predicts the monomial expansion of Δ′_{e_{k−1}}e_{n} in terms of lattice paths. It reads

(1.3)  Δ′_{e_{k−1}}e_{n} = Rise_{n,k−1}(x; q, t) = Val_{n,k−1}(x; q, t),

where Rise_{n,k−1}(x; q, t) and Val_{n,k−1}(x; q, t) are certain combinatorially defined quasisymmetric functions; see [6] for their definitions.

Various special cases of the Delta Conjecture have been proven already. When k = n, the Delta Conjecture reduces to the Shuffle Theorem of Carlsson and Mellit [2]. In the specialization q = 1, Romero [13] has proven

(1.4)  Δ′_{e_{k−1}}e_{n}|_{q=1} = Δ′_{e_{k−1}}e_{n}|_{t=1,q=t} = Rise_{n,k}(x; 1, t).

Zabrocki [15] has given evidence for the Delta Conjecture at t = 1/q by showing that both sides coincide upon pairing with e_{n} under the Hall inner product. At q = 0, the following theorem summarizes work of Wilson and Rhoades.

Theorem 1.1 (Wilson [14], R. [11]). Let k ≤ n be positive integers. We have

(1.5)  Rise_{n,k−1}(x; q, 0) = Rise_{n,k−1}(x; 0, q) = Val_{n,k−1}(x; q, 0) = Val_{n,k−1}(x; 0, q).

Theorem 1.1 is proven by interpreting the four formal power series therein in terms of four statistics (called inv, maj, dinv, and minimaj) on ordered multiset partitions, and then proving the relevant equidistribution results. Let C_{n,k} = C_{n,k}(x; q) be the common symmetric function of Theorem 1.1:

(1.6) C_{n,k} := Rise_{n,k−1}(x; q, 0) = Rise_{n,k−1}(x; 0, q) = Val_{n,k−1}(x; q, 0) = Val_{n,k−1}(x; 0, q).

The authors of this paper showed [7] that the image ωC_{n,k} of C_{n,k} under the ω involution has the following expansion in the dual Hall–Littlewood basis:

(1.7)  ωC_{n,k} = ∑_{\begin{matrix}μ⊢n \\ ℓ(μ)=k \end{matrix}} q^{\overline{b}(μ)} [\begin{matrix}k \\ m(μ) \end{matrix}]_{q} · Q′_{μ}.

Here [ \begin{matrix}k \\ m(μ) \end{matrix} ]_{q} is the q-multinomial coefficient corresponding to the part multiplicities of μ, the numbers b(μ) and \overline{b}(μ) are given by

(1.8)  \begin{cases}b(μ) = ∑_{i} μ_{i}(i − 1), \\ \overline{b}(μ) = ∑_{i}(μ_{i} − 1)(i − 1), \end{cases}

and Q′_{μ} = Q′_{μ}(x; q) is the dual Hall–Littlewood symmetric function related to the Schur basis by

(1.9)  Q′_{μ} = ∑_{λ} K_{λ,μ}(q)s_{λ},

where K_{λ,μ}(q) ∈ ℤ_{≥0}[q] is the Kostka–Foulkes polynomial.

Garsia, Haglund, Remmel, and Yoo [3] recently proved the Delta Conjecture at t = 0 by using plethystic methods and Equation (1.7) to show

(1.10)  Δ′_{e_{k−1}}e_{n}|_{t=0} = Δ′_{e_{k−1}}e_{n}|_{q=0,t=q} = C_{n,k}.

HALL–LITTLEWOOD EXPANSIONS OF SCHUR DELTA OPERATORS AT t = 0 3

We give a new proof of Equation (1.10) using skewing operators e_{j}^{⊥} on the ring Λ of symmetric functions together with _{3}ϕ_{2}-hypergeometric transformations.

Finding positive Q′-basis expansions of symmetric functions is interesting for several reasons. Equation (1.9) shows that any symmetric function with a positive Q′ expansion is automatically Schur positive, and thus is the Frobenius image of some module over the symmetric group 𝔖_{n}. Even better, the function Q′_{μ} is itself (up to a twist) the Frobenius image of the action of 𝔖_{n} on the cohomology of the Springer fiber B_{μ} or on the quotient of the polynomial ring ℚ[x_{1}, …, x_{n}] by the Tanisaki ideal I_{μ}. We generalize Equation (1.10) to find the Q′-basis expansion of ωΔ′_{s_{ν}}e_{n}|_{t=0} for any partition ν.

Theorem 1.2. Let ν be a partition and let n ≥ 0. We have

(1.11)  ωΔ′_{s_{ν}}e_{n}|_{t=0} = ∑_{k=ℓ(ν)+1}^{|ν|+1} P_{ν,k−1}(q) ∑_{\begin{matrix}μ⊢n \\ ℓ(μ)=k \end{matrix}} q^{\overline{b}(μ)} · [\begin{matrix}k \\ m(μ) \end{matrix}]_{q} · Q′_{μ},

where

(1.12)  P_{ν,k−1}(q) = q^{|ν|−\binom{k}{2}} ∑_{\begin{matrix}|ρ|=|ν| \\ ℓ(ρ)=k−1 \end{matrix}} q^{b(ρ)} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} K_{ν,ρ}(q)

and K_{ν,ρ}(q) is the Kostka–Foulkes polynomial.

As operators on Λ we have the identity

Δ_{s_{ν}} = ∑_{ρ⊆ν} Δ′_{s_{ρ}},

where ρ ranges over all partitions obtainable from ν by removing a horizontal strip. Theorem 1.2 therefore also gives a positive expansion for ωΔ_{s_{ν}}e_{n}|_{t=0} in the Q′-basis, where we are using an unprimed delta operator.

Haiman proved that the symmetric function Δ′_{e_{n−1}}e_{n} = Δ_{e_{n}}e_{n} (otherwise known as ∇e_{n}) is the bigraded Frobenius image of the diagonal coinvariant ring [8]. We mention that Zabrocki [16] has recently introduced a generalization of the diagonal coinvariant ring whose bigraded Frobenius image is conjecturally equal to the symmetric function Δ′_{e_{k−1}}e_{n} appearing in the Delta Conjecture.

If ν is a partition with ℓ(ν) = n, Haiman [8] gave an algebraic interpretation of Δ_{s_{ν}}e_{n} as a Schur functor applied to a vector bundle over the Hilbert scheme of n points in the plane ℂ^{2}. In particular, Haiman’s result implies that Δ_{s_{ν}}e_{n} is Schur positive when ℓ(ν) = n. Haiman conjectured that Δ_{s_{ν}}e_{n} is Schur positive for any partition ν. Haglund and Wilson have computational evidence that Δ′_{s_{ν}}e_{n} is also Schur positive for any partition ν. Theorem 1.2 gives evidence for the Schur positivity of Δ′_{s_{ν}}e_{n} (and thus also Δ_{s_{ν}}e_{n}) for arbitrary partitions ν.

In [7] the authors found an algebraic interpretation of the Delta Conjecture at t = 0. Let the symmetric group 𝔖_{n} act on the polynomial ring ℚ[x_{1}, …, x_{n}] in n variables. Following [7, Defn. 1.1], given positive integers k ≤ n we define the ideal I_{n,k} ⊆ ℚ[x_{1}, …, x_{n}] by

(1.13)  I_{n,k} := ⟨e_{n}, e_{n−1}, …, e_{n−k+1}, x_{1}^{k}, x_{2}^{k}, …, x_{n}^{k}⟩

and let

(1.14)  R_{n,k} := ℚ[x_{1}, …, x_{n}]/I_{n,k}

be the corresponding quotient. When k = n the ring R_{n,k} reduces to the classical coinvariant algebra R_{n} = ℚ[x_{1}, …, x_{n}]/⟨e_{1}, …, e_{n}⟩ obtained by modding out by symmetric polynomials in ℚ[x_{1}, …, x_{n}] with vanishing constant term. Just as algebraic properties of R_{n} are governed by combinatorial properties of permutations in 𝔖_{n}, it is shown in [7] that algebraic properties of R_{n,k} are governed by ordered set partitions of [n] := {1, 2, …, n} with k blocks.

The ring R_{n,k} has the structure of a graded 𝔖_{n}-module; in [7] it is proven that its graded Frobenius image is

(1.15)  grFrob(R_{n,k}; q) = (rev_{q} ∘ ω)C_{n,k},

where rev_{q} is the operator which reverses the coefficient sequences of polynomials in q, e.g.

rev_{q}(3s_{(2,1)}q^{2} + 2s_{(1,1,1)}q + s_{(3)}) = s_{(3)}q^{2} + 2s_{(1,1,1)}q + 3s_{(2,1)}.

Thanks to the Garsia–Haglund–Remmel–Yoo Equation (1.10) we can also express Equation (1.15) as

(1.16)  grFrob(R_{n,k}; q) = (rev_{q} ∘ ω)Δ′_{e_{k−1}}e_{n}|_{t=0}.

Informally, we think of R_{n,k} as the ‘coinvariant algebra’ attached to the operator Δ′_{e_{k−1}}e_{n}. Given Equation (1.16), one could ask for a graded 𝔖_{n}-module R_{ν,n} which satisfies

(1.17)  grFrob(R_{ν,n}; q) = (rev_{q} ∘ ω)Δ′_{s_{ν}}e_{n}|_{t=0}

for any partition ν. This would give a coinvariant algebra attached to the operator Δ′_{s_{ν}}. In [12] Rhoades and Wilson exhibited a quotient of ℚ[x_{1}, …, x_{n}] with graded Frobenius image (rev_{q} ∘ ω)Δ_{s_{ν}}e_{n}|_{t=0} when ν is a hook of the form (r, 1^{n−1}).

For general partitions ν ⊢ m, it is impossible to exhibit a module R_{n,ν} satisfying Equation (1.17) as a submodule of ℚ[x_{1}, …, x_{n}]; the graded components of the polynomial ring are not large enough for this purpose. Given Theorem 1.2, two artificial solutions to this problem are as follows.

• Very artificially, we could use the positive expansion of Q′_{μ} in the Schur basis {s_{λ}} and define R_{n,ν} as a direct sum of 𝔖_{n}-irreducibles S^{λ} with appropriate grading shifts.

• Less artificially, we could use the fact that rev_{q}(Q′_{μ}) is the graded Frobenius image of the Tanisaki quotient R_{μ} = ℚ[x_{1}, …, x_{n}]/I_{μ}, where I_{μ} is the Tanisaki ideal. Theorem 1.2 then leads to a definition of R_{n,ν} as a direct sum of R_{μ}’s with appropriate grading shifts.

The second bullet point is less artificial because the pieces R_{μ} which constitute the module R_{n,ν} are larger than the pieces S^{λ} appearing in the first bullet point.

In this paper we give a still less artificial construction for R_{n,ν} as a Hom-space. For any n, m ≥ 0 we define a graded 𝔖_{m} × 𝔖_{n}-module V_{n,m} by

(1.18)  V_{n,m} := ⨁_{k≥0}(R_{m,k−1} ⊗ R_{n,k}){−mn + km + kn − n − k(k − 1)}.

Here M{−d} denotes a graded module M with degree shifted up by d and we impose grading on tensor products by declaring

(M ⊗ N)_{d} = ⨁_{i+j=d} M_{i} ⊗ N_{j}.

If M is any 𝔖_{m}-module, the Hom-space Hom_{𝔖_{m}}(M, V_{n,m}) has the structure of a graded 𝔖_{n}-module.

Theorem 1.3. Let n ≥ 0 and let ν ⊢ m be a partition. Define the graded 𝔖_{n}-module R_{n,ν} by

(1.19)  R_{n,ν} := Hom_{𝔖_{m}}(S^{ν}, V_{n,m}){b(ν)},

where V_{n,m} is defined as in Equation (1.18). We have

(1.20)  grFrob(R_{n,ν}; q) = (rev_{q} ∘ ω)Δ′_{s_{ν}}e_{n}|_{t=0}.

The module V_{n,m} only depends on the integers n and m = |ν|, so that we may regard V_{n,m} as a universal ‘generator’ for coinvariant algebras corresponding to partitions ν ⊢ m. Since the R_{n,k} modules are ‘larger’ than the Tanisaki quotients R_{μ}, this gives a still less artificial solution to finding a ‘coinvariant algebra’ attached to the operator Δ′_{s_{ν}}. This also suggests that understanding the algebraic and geometric properties of objects related to Δ′_{s_{ν}} at t = 0 may be deduced from the case where ν is a single column.

2. Background

2.1. Symmetric functions. We adopt standard symmetric function terminology which may be found in e.g. [5, 9]. Given a partition λ, let

e_{λ} = e_{λ}(x), h_{λ} = h_{λ}(x), s_{λ} = s_{λ}(x), Q′_{λ} = Q′_{λ}(x; q), H̃_{λ} = H̃_{λ}(x; q, t)

be the associated elementary, homogeneous, Schur, dual Hall–Littlewood, and modified Macdonald symmetric function. The functions Q′_{μ} expand positively in the Schur basis. If μ ⊢ n, the transition coefficients K_{λ,μ}(q) ∈ ℤ_{≥0}[q] given by

(2.1)  Q′_{μ} = ∑_{λ⊢n} K_{λ,μ}(q)s_{λ}

are the Kostka–Foulkes polynomials. The following relationship between the modified Macdonald symmetric functions and the dual Hall–Littlewood functions is well known:

(2.2)  H̃_{λ}|_{t=0} = rev_{q}(Q′_{λ}).

If S ⊆ [n−1], we let F_{n,S} be the associated fundamental quasisymmetric function of degree n given by

(2.3)  F_{n,S} := ∑_{\begin{matrix}i_{1}≤···≤i_{n} \\ (j∈S)⇒(i_{j}<i_{j+1}) \end{matrix}} x_{i_{1}} ⋯ x_{i_{n}}.

We adopt the usual q-analogs of numbers, factorials, binomial coefficients, and multinomial coefficients:

(2.4)  \begin{cases}[n]_{q} := 1 + q + q^{2} + ⋯ + q^{n−1}, & n ≥ 0, \\ [n]!_{q} := [n]_{q}[n − 1]_{q} ⋯ [1]_{q}, & n ≥ 0, \\ [ \begin{matrix}n \\ k \end{matrix}]_{q} := \frac{[n]!_{q}}{[k]!_{q}[n−k]!_{q}} , & n ≥ k ≥ 0, \\ [ \begin{matrix}n \\ a_{1},...,a_{k} \end{matrix}]_{q} := \frac{[n]!_{q}}{[a_{1}]!_{q}···[a_{k}]!_{q}} , & a_{1} + ⋯ + a_{k} = n. \end{cases}

If λ = (λ_{1}, …, λ_{k}) is a partition, we let ℓ(λ) = k be the number of parts of λ, let |λ| = λ_{1} + ⋯ + λ_{k} be the sum of the parts of λ, and set

(2.5)  b(λ) := ∑_{i=1}^{k} λ_{i} · (i − 1)

(2.6)  \overline{b}(λ) := b(λ) − \binom{ℓ(λ)}{2} = ∑_{i=1}^{k}(λ_{i} − 1)(i − 1).

We let m_{i}(λ) denote the multiplicity of i as a part of λ and adopt the q-multinomial coefficient shorthand

(2.7)  [\begin{matrix}ℓ(λ) \\ m(λ \end{matrix}]_{q} := [\begin{matrix}ℓ(λ) \\ m_{1}(λ), m_{2}(λ), … \end{matrix}]_{q} .

Let ω be the involution on Λ which interchanges e_{n} and h_{n}. We will use the following ‘twisted’ version of the polynomials C_{n,k} for k ≤ n:

(2.8)  D_{n,k} := (rev_{q} ∘ ω)C_{n,k}.

Let ⟨·, ·⟩ be the Hall inner product on Λ defined by the declaring the Schur functions to be orthonormal: ⟨s_{λ}, s_{μ}⟩ = δ_{λ,μ}. For any symmetric function f, the operator f^{⊥} : Λ → Λ is the dual operator to multiplication by f under the Hall inner product. Said differently, the operator f^{⊥} is characterized by

(2.9)  ⟨f^{⊥}g, h⟩ = ⟨g, fh⟩,

for all g, h ∈ Λ. For a proof of the following standard fact, see for example [7, Lem. 3.6].

Lemma 2.1. Let f, g ∈ Λ be symmetric functions with equal constant terms. We have f = g if and only if e_{j}^{⊥}f = e_{j}^{⊥}g for all j ≥ 1.

Lemma 2.1 will be used to form the recursions which underly our new proof of the Delta Conjecture at t = 0. The image of the D_{n,k} functions under e_{j}^{⊥} can be recursively described as follows.

Lemma 2.2 (H., R., S. [7, Lem. 3.7]). Let k ≤ n be positive integers and let j ≥ 1. We have

e_{j}^{⊥}D_{n,k} = q^{\binom{j}{2}} [\begin{matrix}k \\ j \end{matrix}]_{q} · ∑_{m=max(1,k−j)}^{min(k,n−j)} q^{(k−m)·(n−j−m)} [\begin{matrix}j \\ k − m \end{matrix}]_{q} D_{n−j,m}.

The irreducible representations of the symmetric group 𝔖_{n} over ℚ are indexed by partitions λ ⊢ n. If λ is a partition, we let S^{λ} denote the corresponding irreducible representation. If V is any finite-dimensional 𝔖_{n}-module, there exist unique integers c_{λ} ≥ 0 such that V ≅_{𝔖_{n}} ⨁_{λ⊢n} c_{λ}S^{λ}. The Frobenius image Frob(V) ∈ Λ_{n} is the symmetric function

(2.10)  Frob(V) := ∑_{λ⊢n} c_{λ}s_{λ}.

More generally, if V = ⨁_{d≥0} V_{d} is a graded 𝔖_{n}-module with each V_{d} finite-dimensional, the graded Frobenius image grFrob(V) is

(2.11)  grFrob(V) = ∑_{d≥0} Frob(V_{d}) · q^{d}.

2.2. Ordered set partitions. If π = π_{1}π_{2} ⋯ π_{n} ∈ 𝔖_{n} is a permutation (written in one-line notation), the descent set of π is

(2.12)  Des(π) := {1 ≤ i ≤ n − 1 : π_{i} > π_{i+1}}

and the inverse descent set is iDes(π) := Des(π^{−1}).

Let k ≤ n be positive integers. An ordered set partition of size n with k blocks is a sequence σ = (B_{1} | ⋯ | B_{k}) of k nonempty subsets of [n] such that we have a disjoint union decomposition [n] = B_{1} ⊎ ⋯ ⊎ B_{k}. Let OP_{n,k} be the family of ordered set partitions of size n with k blocks. As an example, we have σ = (27 | 135 | 46) ∈ OP_{7,3}. There is a natural identification OP_{n,n} = 𝔖_{n} of ordered set partitions of size n with n blocks and permutations in the symmetric group on n letters.

Let σ = (B_{1} | ⋯ | B_{k}) ∈ OP_{n,k} be an ordered set partition. An inversion in σ is a pair 1 ≤ i < j ≤ n such that

• i’s block is strictly to the right of j’s block in σ and

• i is minimal in its block.

We let inv(σ) be the number of inversions of σ. For example, if σ = (27 | 135 | 46), the inversions of σ are 12, 17, 47, and 45 so that inv(σ) = 4.

If σ = (B_{1} | ⋯ | B_{k}) ∈ OP_{n,k}, the reading word rword(σ) is the permutation π_{1}π_{2} ⋯ π_{n} ∈ 𝔖_{n} obtained by reading σ along ‘diagonals’ from left to right (where the m^{th} ‘diagonal’ is the set of elements which are m^{th} largest in their block). As an example, we have

rword(27 | 135 | 45) = 5736214.

We have (see [6] or [7, Eq. 2.20]) the following quasisymmetric expansion of C_{n,k} in terms of ordered set partitions:

(2.13)  C_{n,k} = ∑_{σ∈OP_{n,k}} q^{inv(σ)}F_{n,iDes(rword(σ))}.

2.3. Hypergeometric functions. Given a continuous parameter x and an integer k ≥ 0, the q-shifted factorial is

(2.14)  (x)_{k} = (x; q)_{k} = (1 − x)(1 − xq) ⋯ (1 − xq^{k−1}).

We adopt the abbreviation

(2.15)  ∏_{i=1}^{j}(a_{i})_{k} := (a_{1}, a_{2}, …, a_{k})_{j}.

If r, s ≥ 0 are nonnegative integers and α_{1}, …, α_{r} and β_{1}, …, β_{s} are parameters, the corresponding q-hypergeometric series is

(2.16)  _{r}ϕ_{s} (\begin{matrix}α_{1}, …, α_{r} \\ β_{1}, …, β_{s} \end{matrix}; q, z) := ∑_{n=0}^{∞} \frac{(α_{1}; q)_{n} ⋯ (α_{r}; q)_{n}}{(β_{1}; q)_{n} ⋯ (β_{s}; q)_{n}} \frac{z^{n}}{(q; q)_{n}} [(−1)^{n}q^{\binom{n}{2}}]^{1+s−r} .

In this paper we will only be concerned with the _{3}ϕ_{2}-functions. See [4] for more information on hypergeometric series.

3. Polynomial identities

In this section we will prove symmetric function and hypergeometric identities which will be used in our proof of the Delta Conjecture at t = 0, and ultimately in our proof of Theorem 1.2. The first of these is a recursive description of the image of C_{n,k} under the operator e_{j}^{⊥}.

Lemma 3.1. Let j ≥ 1 and k ≤ n. We have

e_{j}^{⊥}C_{n,k} = ∑_{r=0}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} C_{n−j,k−r}.

The proof of Lemma 3.1 should be compared with that of Lemma 2.2 (= [7, Lem. 3.7]).

Proof. We start with the quasisymmetric expansion of C_{n,k} in terms of ordered set partitions:

(3.1)  C_{n,k} = ∑_{σ∈OP_{n,k}} q^{inv(σ)}F_{n,iDes(rword(σ))}.

Let α = (α_{1}, α_{2}, …, α_{p}) be any (strict) composition of n. General facts about superization (see [5]) imply

(3.2)  ⟨C_{n,k}, e_{α_{1}}e_{α_{2}} ⋯ e_{α_{p}}⟩ = ∑_{σ∈OP_{n,k} rword(σ) is an α-shuffle} q^{inv(σ)}.

The α-shuffle condition means that the sequence rword(σ) is a shuffle of the p decreasing sequences

(α_{1}, …, 2, 1), (α_{1} + α_{2}, …, α_{1} + 2, α_{1} + 1), …, (n, n − 1, …, α_{1} + ⋯ + α_{p−1} + 1).

We are interested in the case where α_{1} = j, so that

(3.3)  ⟨C_{n,k}, e_{α_{1}}e_{α_{2}} ⋯ e_{α_{p}}⟩ = ⟨C_{n,k}, e_{j}e_{α_{2}} ⋯ e_{α_{p}}⟩ = ⟨e_{j}^{⊥}C_{n,k}, e_{α_{2}} ⋯ e_{α_{p}}⟩.

As in the proof of [7, Lem. 3.7], we give a combinatorial interpretation of this expression.

Fix an index 0 ≤ r ≤ k − 1 and consider σ ∈ OP_{n−j,k−r}. Let T be a way of adding the j letters {n − j + 1, …, n} to σ (the big letters) in such a way that the resulting ordered set partition σ′ has k blocks and the big letters appear in rword(σ′) in the order n, n − 1, …, n − j + 1. An example of such a way T for n = 9, k = 5, j = 4, r = 2 is shown below, with the big letters in bold:

(4 | 15 | 23) ⇝ (4 | 159 | 78 | 23 | 6).

Notice that exactly r of the big letters are minimal in their blocks of σ′, and these minimal letters must be (from left to right)

n − j + r, …, n − j + 2, n − j + 1.

Let us consider the effect on the inv statistic of all possible ways T of producing σ′ from σ. Call the letters 1, 2, …, n − j of σ′ which are not big small. Following the notation of [7], let us call a letter i of σ′

• mins if i is small and minimal in its block,

• minb if i is big and minimal in its block,

• nmins if i is small and not minimal in its block, and

• nminb if i is big and not minimal in its block.

We observe the following, applying the standard interpretation of q-binomial coefficients as generating functions of inversions in binary strings (see for example [5, Ch. 1])

• The r letters n − j + r, …, n − j + 1 are precisely the minb letters for any way T, and they contribute amongst themselves \binom{r}{2} inversions in σ′.

• The \binom{k}{r} ways of distributing the minb letters among the blocks of σ′ generate inversions with the mins letters, contributing to a factor of [ \begin{matrix}k \\ r \end{matrix}]_{q} in the generating function for inv(σ′) when we sum over all ways T.

• The values of the nminb letters are completely determined by which block they are added to. There are j − r letters which are nminb and they may be added to any of the k blocks of σ (upon addition of the minb letters), with multiplicity. This gives \binom{k+j−r−1}{j−r} choices for distributing the nminb letters in T. The inversions contributed between the nminb letters and the mins letters generate a factor of [ \begin{matrix}k+j−r−1 \\ j−r \end{matrix} ]_{q} to the generating function for inv(σ′) when we sum over all ways T.

By the last paragraph, we have

(3.4)  ∑_{T:σ⇝σ′} q^{inv(σ′)} = q^{inv(σ)+\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} ,

where the sum is over all ways T of producing σ′ from σ. If we sum this expression over all σ with rword(σ) an (α_{2}, …, α_{p})-shuffle, and then over all r, we get the inner product

(3.5)  ⟨∑_{r=0}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} C_{n−j,k−r}, e_{α_{2}} ⋯ e_{α_{p}}⟩ ,

which is also equal to

(3.6)  ⟨e_{j}^{⊥}C_{n,k}, e_{α_{2}} ⋯ e_{α_{p}}⟩ ,

completing the proof. □

We will need the theory of hypergeometric series for our proof of Equation (1.10). In particular, we have the following transformation of the _{3}ϕ_{2} basic hypergeometric series (see [1] for background on basic hypergeometric series).

Lemma 3.2. Let j ∈ ℕ and α, x, y, z ∈ ℝ. We have

(3.7) _{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{α}, & q^{α+z} \\ & q^{α−y−j+1}, & q^{α−x−j+1} \end{matrix}; q, q)

= \frac{(q^{−y−j+1})_{j}(q^{−x−j+1})_{j}}{(q^{α−y−j+1})_{j}(q^{α−x−j+1})_{j}} \frac{(q^{−y−z−j+1})_{j}}{(q^{y})_{j}} q^{(α+x+y+z+j−1)j}

×_{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{x+y+z+j−1}, & q^{x−α} \\ & q^{x}, & q^{x+z} \end{matrix}; q, q^{1+α−y}) .

Proof. We utilize the following identities from [1, p. 525]:

(3.8) _{3}ϕ_{2} (\begin{matrix}q^{−n}, & w, & b \\ & d, & e \end{matrix}; q, q) = \frac{(e/w)_{n}w^{n}}{(e)_{n}}_{3}ϕ_{2} (\begin{matrix}q^{−n}, & w, & d/b \\ & d, & q^{1−n}w/e \end{matrix}; q, bq/e) ,
(3.9) _{3}ϕ_{2} (\begin{matrix}q^{−n}, & w, & b \\ & e, & f \end{matrix}; q, q) = \frac{(e/w)_{n}(f/w)_{n}w^{n}}{(e)_{n}(f)_{n}}_{3}ϕ_{2} (\begin{matrix}q^{−n}, & w, & wbq^{1−n}/ef \\ & q^{1−n}w/e, & q^{1−n}w/f \end{matrix}; q, q) ,

where n ∈ ℕ, and w, b, c, d, e, f are continuous parameters.

Begin by setting j = n, w = q^{α}, b = q^{α+z}, e = q^{α−y−j+1}, and f = q^{α−x−j+1} in (3.9) to get

(3.10) _{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{α}, & q^{α+z} \\ & q^{α−y−j+1}, & q^{α−x−j+1} \end{matrix}; q, q)

= \frac{(q^{−y−j+1})_{j}(q^{−x−j+1})_{j}q^{jα}}{(q^{α−y−j+1})_{j}(q^{α−x−j+1})_{j}}_{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{α}, & q^{x+y+z+j−1} \\ & q^{y}, & q^{x} \end{matrix}; q, q) .

Now apply (3.8) with n = j, w = q^{x+y+z+j−1}, b = q^{α}, d = q^{x}, e = q^{y} to the _{3}ϕ_{2} appearing in the right-hand side of (3.10) to get

(3.11) _{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{α}, & q^{α+z} \\ & q^{α−y−j+1}, & q^{α−x−j+1} \end{matrix}; q, q)

= \frac{(q^{−y−j+1})_{j}(q^{−x−j+1})_{j}}{(q^{α−y−j+1})_{j}(q^{α−x−j+1})_{j}} \frac{(q^{−x−z−j+1})_{j}}{(q^{y})_{j}} q^{(α+x+y+z+j−1)j}

×_{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{x+y+z+j−1}, & q^{x−α} \\ & q^{x}, & q^{x+z} \end{matrix}; q, q^{1+α−y}) . □

We express the hypergeometric transformation of Lemma 3.2 in a more convenient form involving q-binomials.

Lemma 3.3. Let j ≤ k ≤ n be positive integers. Let p be an integer in the range k − j ≤ p ≤ n − j. There holds the identity

(3.12) q^{\binom{k}{2}+\binom{j}{2}} ∑_{r=p}^{p+j}(−1)^{n−r} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} q^{\binom{r+1}{2}−nr} [\begin{matrix}r \\ j \end{matrix}]_{q} q^{(r−p)(n−j−p)} [\begin{matrix}j \\ r − p \end{matrix}]_{q}

= ∑_{r=k−p}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} q^{\binom{k−a}{2}}(−1)^{n−j−p} [\begin{matrix}p − 1 \\ k − r − 1 \end{matrix}]_{q} q^{\binom{p+1}{2}−(n−j)p}.

Proof. The first step is to express everything in terms of hypergeometric series. We make use of the following facts, which we refer to as the ‘simple identities’. Here u, j, a ∈ ℤ_{≥0} and p, x are continuous parameters.

(3.13)  [\begin{matrix}u + p \\ j \end{matrix}]_{q} = [\begin{matrix}p \\ j \end{matrix}]_{q} \frac{(q^{p+1})_{u}}{(q^{p−j+1})_{u}}

(3.14)  [\begin{matrix}p \\ u + a \end{matrix}]_{q} = [\begin{matrix}p \\ a \end{matrix}]_{q} \frac{(q^{p−a−u+1})_{u}}{(q^{a+1})_{u}} = [\begin{matrix}p \\ a \end{matrix}]_{q} \frac{(q^{a−p})_{u}}{(q^{a+1})_{u}} (−q^{p−a})^{u}q^{−\binom{u}{2}}

(3.15)  [\begin{matrix}p \\ j − u \end{matrix}]_{q} = [\begin{matrix}p \\ j \end{matrix}]_{q} \frac{(q^{−j})_{u}}{(q^{p−j+1})_{u}} (−q^{j})^{u}q^{−\binom{u}{2}}

(3.16)  \binom{u + a}{2} = \binom{u}{2} + \binom{a}{2} + ua.

Using the simple identities and setting u = r − p, the left-hand side of Equation (3.12) can be expressed as

(3.17)  [\begin{matrix}p \\ j \end{matrix}]_{q} [\begin{matrix}p − 1 \\ k − 1 \end{matrix}]_{q} (−1)^{n−p}q^{−np+\binom{p+1}{2}+\binom{k}{2}+\binom{j}{2}} ∑_{u=0}^{j} \frac{(q^{−j}, q^{p+1}, q^{p})_{u}}{(q, q^{p−j+1}, q^{p−k+1})_{u}} q^{u}.

Similarly, using the simple identities and setting u = j − a, we see that the right-hand side of Equation (3.12) can be expressed as

(3.18) q^{−p(n−j)+\binom{p+1}{2}+\binom{k−j}{2}+\binom{j}{2}}(−1)^{n+j−p} [\begin{matrix}k \\ j \end{matrix}]_{q} [\begin{matrix}p − 1 \\ k − j − 1 \end{matrix}]_{q} ∑_{u=0}^{p−k+j} \frac{(q^{−j}, q^{k}, q^{−p+k−j})_{u}}{(q, q^{k−j}, q^{k−j+1})_{u}} q^{u(p+1)}.

The next step is to express (3.17) and (3.18) in terms of the hypergeometric series _{3}ϕ_{2}. The expression (3.17) is given by

(3.19)  [\begin{matrix}p \\ j \end{matrix}]_{q} [\begin{matrix}p − 1 \\ k − 1 \end{matrix}]_{q} (−1)^{n−p}q^{−np+\binom{p+1}{2}+\binom{k}{2}+\binom{j}{2}}_{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{p+1}, & q^{p} \\ & q^{p−j+1}, & q^{p−k+1} \end{matrix}; q, q)

whereas (3.18) is equal to (3.20)

q^{−p(n−j)+\binom{p+1}{2}+\binom{k−j}{2}+\binom{j}{2}}(−1)^{n+j−p} [\begin{matrix}k \\ j \end{matrix}]_{q} [\begin{matrix}p − 1 \\ k − j − 1 \end{matrix}]_{q} _{3}ϕ_{2} (\begin{matrix}q^{−j}, & q^{k}, & q^{−p+k−j} \\ & q^{k−j}, & q^{k−j+1} \end{matrix}; q, q^{p+1}) .

The fact that (3.19) = (3.20) is a consequence of Lemma 3.2. □

4. Proofs of the Main Results

Our starting point is the following expansion (see [5, Eqn. 2.72]) of e_{n} in the modified Macdonald basis:

(4.1)  e_{n} = ∑_{λ⊢n} \frac{MB_{λ}Π_{λ}H̃_{λ}}{w_{λ}} ,

where • M = (1 − q)(1 − t),

• B_{λ} = ∑_{(i,j)∈λ} q^{i−1}t^{j−1}, where the sum is over all cells (i, j) in the Ferrers diagram of λ,

• Π_{λ} = ∏_{(1,1)≠(i,j)∈λ}(1−q^{i−1}t^{j−1}), where the product is over all cells (i, j) in the Ferrers diagram of λ other than the northwest corner (1, 1), and

• w_{λ} = ∏_{c∈λ}(q^{a(c)} − t^{l(c)+1})(t^{l(c)} − q^{a(c)+1}), where the product is over all cells c in the diagram of λ and a(c), l(c) are the arm and leg lengths (see [5, p. 29]) of the cell c in λ.

If we apply the operator Δ′_{e_{k−1}} to both sides of Equation (4.1), we get

(4.2)  Δ′_{e_{k−1}}e_{n} = ∑_{λ⊢n} e_{k−1}[B_{λ} − 1]\frac{MB_{λ}Π_{λ}H̃_{λ}}{w_{λ}} .

Here we used the plethystic shorthand e_{k−1}[B_{λ} − 1] = e_{k−1}(…, q^{i−1}t^{j−1}, …) where (i, j) range over all cells ≠ (1, 1) in the Ferrers diagram of λ.

Recall that H̃_{λ}|_{t=0} = rev_{q}Q′_{λ} for any partition λ. If we evaluate both sides of Equation (4.2) at t = 0, we get

(4.3) Δ′_{e_{k−1}}e_{n}|_{t=0} = ∑_{λ⊢n}(−1)^{n−ℓ(λ)}q^{\binom{k}{2}−2b(λ)−n+∑_{i} \binom{m_{i}(λ)+1}{2}} [\begin{matrix}ℓ(λ) − 1 \\ k − 1 \end{matrix}]_{q} [\begin{matrix}ℓ(λ) \\ m(λ) \end{matrix}]_{q} · rev_{q}Q′_{λ}.

12 JAMES HAGLUND, BRENDON RHOADES, AND MARK SHIMOZONO

Here we used the evaluation

(4.4)  e_{k−1}[B_{λ} − 1] = e_{k−1}(1, q, …, q^{ℓ(λ)−1}) = q^{\binom{k}{2}} [\begin{matrix}ℓ(λ) − 1 \\ k − 1 \end{matrix}]_{q} .

Equation (4.3) can be expressed in terms of the D-functions D_{n,r}. Lemma 4.1. We have the identity

(4.5)  Δ′_{e_{k−1}}e_{n}|_{t=0} = q^{\binom{k}{2}} ∑_{r=k}^{n}(−1)^{n−r}q^{\binom{r+1}{2}−nr} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} D_{n,r}.

Proof. Starting with Equation (4.3) and grouping partitions λ ⊢ n according to their number of parts we have

Δ′_{e_{k−1}}e_{n}|_{t=0} = ∑_{λ⊢n}(−1)^{n−ℓ(μ)}q^{\binom{k}{2}−2b(μ)−n+∑_{i} \binom{m_{i}(λ)+1}{2}} [\begin{matrix}ℓ(λ) − 1 \\ k − 1 \end{matrix}]_{q} [\begin{matrix}ℓ(λ) \\ m(μ) \end{matrix}]_{q} · rev_{q}Q′_{λ}

= q^{\binom{k}{2}−n} ∑_{r=k}^{n}(−1)^{n−r} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} ∑_{\begin{matrix}λ⊢n \\ ℓ(μ)=r \end{matrix}} q^{−2b(μ)+∑_{i} \binom{m_{i}(λ)+1}{2}} [\begin{matrix}r \\ m(λ) \end{matrix}]_{q} · rev_{q}Q′_{λ}.

We focus on the internal summand. We have

∑_{\begin{matrix}μ⊢n \\ ℓ(λ)=r \end{matrix}}q^{−2b(λ)+∑_{i} \binom{m_{i}(λ)+1}{2}} [\begin{matrix}r \\ m(λ) \end{matrix}]_{q} · rev_{q}Q′_{λ}

= ∑_{\begin{matrix}λ⊢n \\ ℓ(λ)=r \end{matrix}} q^{−2b(μ)+∑_{i} \binom{m_{i}(λ)+1}{2}+∑_{i<j} m_{i}(λ)m_{j}(λ)−∑_{i<j} m_{i}(λ)m_{j}(λ)} [\begin{matrix}r \\ m(λ) \end{matrix}]_{q} · rev_{q}Q′_{λ}

= q^{\binom{r+1}{2}} ∑_{\begin{matrix}λ⊢n \\ ℓ(λ)=r \end{matrix}} q^{−2b(λ)−∑_{i<j} m_{i}(λ)m_{j}(λ)} [\begin{matrix}r \\ m(λ) \end{matrix}]_{q} · rev_{q}Q′_{λ}

= q^{\binom{r+1}{2}} ∑_{\begin{matrix}λ⊢n \\ ℓ(λ)=r \end{matrix}} q^{−b(λ)} · [q^{∑_{i<j} −m_{i}(λ)m_{j}(λ)} [\begin{matrix}r \\ m(λ) \end{matrix}]_{q}] · [q^{−b(λ)}rev_{q}Q′_{λ}(x; q)]

= q^{\binom{r+1}{2}−\binom{r}{2}} ∑_{\begin{matrix}λ⊢n \\ ℓ(λ)=r \end{matrix}} q^{−\overline{b}(λ)} · [\begin{matrix}r \\ m(λ) \end{matrix}]_{q^{−1}} · Q′_{λ}(x; q^{−1})

= q^{\binom{r+1}{2}−\binom{r}{2}}q^{−\binom{r}{2}−(n−r)(r−1)} · [q^{\binom{r}{2}+(n−r)(r−1)} · ωC_{n,r}(x; q^{−1})]

= q^{\binom{r+1}{2}−r(r−1)−(n−r)(r−1)} · D_{n,r}(x; q)

= q^{\binom{r+1}{2}−nr+n} · D_{n,r}(x; q).

The second equality used ∑_{i} \binom{m_{i}(λ)+1}{2} + ∑_{i<j} m_{i}(λ)m_{j}(λ) = \binom{ℓ(λ)+1}{2}. The fourth equality comes from the fact that the degree of the palindromic polynomial [ \begin{matrix}ℓ(λ) \\ m(λ) \end{matrix} ]_{q} is ∑_{i<j} m_{i}(λ)m_{j}(λ) and that the q-degree of Q′_{λ}(x; q) is b(λ).

Going back to the assertion of the lemma, we have

Δ′_{e_{k−1}}e_{n}|_{t=0} = q^{\binom{k}{2}−n} ∑_{r=k}^{n}(−1)^{n−r} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} ∑_{\begin{matrix}λ⊢n \\ ℓ(λ)=r \end{matrix}} q^{−2b(μ)+∑_{i} \binom{m_{i}(λ)+1}{2}} [\begin{matrix}r \\ m(λ) \end{matrix}]_{q} · rev_{q}Q′_{λ}

= q^{\binom{k}{2}−n} ∑_{r=k}^{n}(−1)^{n−r} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} · q^{\binom{r+1}{2}−nr+n} · D_{n,r}(x; q).

Cancelation of a factor of q^{n} completes the proof. □

We are in a position to give our proof of Equation (1.10), and thus give a new proof of the Delta Conjecture at t = 0.

Theorem 4.2 (Garsia, H., Remmel, Yoo [3]). Let k ≤ n be positive integers. We have

Δ′_{e_{k−1}}e_{n}|_{t=0} = Δ′_{e_{k−1}}e_{n}|_{q=0,t=q} = C_{n,k}.

Proof. Let j ≥ 1. Given Lemma 4.1 and Lemma 2.2, the symmetric function e_{j}^{⊥}Δ′_{e_{k−1}}e_{n}|_{t=0} has the following D-function expansion, where we adopt the convention that D_{n,k} = 0 if k > n or if k < 0:

(4.6) e_{j}^{⊥}Δ′_{e_{k−1}}e_{n}|_{t=0} = q^{\binom{k}{2}+\binom{j}{2}} ∑_{r=k}^{n}(−1)^{n−r} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} q^{\binom{r+1}{2}−nr} [\begin{matrix}r \\ j \end{matrix}]_{q}

× ∑_{m=r−j}^{r} q^{(r−m)(n−j−m)} [\begin{matrix}j \\ r − m \end{matrix}]_{q} D_{n−j,m}.

If we want e_{j}^{⊥}Δ′_{e_{k−1}}e_{n}|_{t=0} to satisfy the recursion of Lemma 3.1, we must have

(4.7)  e_{j}^{⊥}Δ′_{e_{k−1}}e_{n}|_{t=0} = ∑_{r=0}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} Δ′_{e_{k−r−1}}e_{n−j}|_{t=0}.

By Lemma 4.1, we know

(4.8) ∑_{r=0}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ a \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} Δ′_{e_{k−r−1}}e_{n−j}|_{t=0}

= ∑_{r=0}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} q^{\binom{k−r}{2}} ∑_{b=k−r}^{n−j}(−1)^{n−j−b} [\begin{matrix}b − 1 \\ k − r − 1 \end{matrix}]_{q} q^{\binom{b+1}{2}−(n−j)b}D_{n−j,b}.

We want to show that the right-hand side of Equation (4.6) is equal to the right-hand side of Equation (4.8). To this end, let p be an integer in the range k − j ≤ p ≤ n − j. The coefficient of D_{n−j,p} in Equation (4.6) is

(4.9)  q^{\binom{k}{2}+\binom{j}{2}} ∑_{r=p}^{p+j}(−1)^{n−r} [\begin{matrix}r − 1 \\ k − 1 \end{matrix}]_{q} q^{\binom{r+1}{2}−nr} [\begin{matrix}r \\ j \end{matrix}]_{q} q^{(r−p)(n−j−p)} [\begin{matrix}j \\ r − p \end{matrix}]_{q} ,

whereas the coefficient of D_{n−j,p} in Equation (4.8) is

(4.10)  ∑_{r=k−p}^{j} q^{\binom{r}{2}} [\begin{matrix}k \\ r \end{matrix}]_{q} [\begin{matrix}k + j − r − 1 \\ j − r \end{matrix}]_{q} q^{\binom{k−r}{2}}(−1)^{n−j−p} [\begin{matrix}p − 1 \\ k − r − 1 \end{matrix}]_{q} q^{\binom{p+1}{2}−(n−j)p}.

Theorem 4.2 will be proven if we can only establish the equality of the expressions (4.9) and (4.10). This is Lemma 3.3. □

We use Theorem 4.2 to derive the more general Theorem 1.2. In this proof we will use the notation of plethysm; see [5].

Proof of Theorem 1.2. Let k ≤ n be positive integers. The polynomials Q′_{μ} and rev_{q}(Q′_{μ}) have Schur expansions

Q′_{μ} = ∑_{λ⊢n} K_{λ,μ}(q)s_{λ},
rev_{q}(Q′_{μ}) = ∑_{λ⊢n} q^{b(μ)}K_{λ,μ}(1/q)s_{λ}.

By Equation (1.7), Equation (4.3), and the truth of the Delta Conjecture at q = 0 (i.e., Theorem 4.2) we have the following identity:

(4.11) q^{k(k−1)} ∑_{μ⊢n}(−1)^{n−ℓ(μ)}q^{−n−b(μ)+∑_{i=1}^{n} \binom{m_{i}+1}{2}} [\begin{matrix}ℓ(μ) − 1 \\ k − 1 \end{matrix}]_{q} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} K_{λ,μ}(1/q)

= _{∑_{μ⊢n} ; ℓ(μ)=k}q^{b(μ)} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} K_{λ′,μ}(q).

Equation (4.11) is also recorded in [3, Prop. 3.2].

Using reasoning identical to that of our derivation of Equation (4.3), we see that Δ′_{s_{ν}}e_{n}|_{t=0} has the following expansion in the q-reversed Q′-basis: (4.12)

Δ′_{s_{ν}}e_{n}|_{t=0} = ∑_{μ⊢n}(−1)^{n−ℓ(μ)}s_{ν}(q, q^{2}, …, q^{ℓ(μ)−1})q^{−n−2b(μ)+∑_{i} \binom{m_{i}(μ)+1}{2}} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} · rev_{q}(Q′_{μ}).

Multiplying both sides of (4.11) by s_{λ}, summing over λ and applying ω, we get the following equivalent form of (4.11):

(4.13) q^{k(k−1)} ∑_{μ⊢n}(−1)^{n−ℓ(μ)}q^{−n−b(μ)+∑_{i=1}^{n} \binom{m_{i}+1}{2}} [\begin{matrix}ℓ(μ) − 1 \\ k − 1 \end{matrix}]_{q} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} ∑_{λ} K_{λ′,μ}(1/q)s_{λ}

= _{∑_{μ⊢n} ; ℓ(μ)=k}q^{b(μ)} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} Q′_{μ}

for all λ ⊢ n and 1 ≤ k ≤ n.

Note that the sum on the right-hand side of (4.13) also occurs on the right-hand side in Theorem 1.2. By Equation (4.12), the following equation is equivalent to Theorem 1.2:

(4.14) ∑_{μ⊢n}(−1)^{n−ℓ(μ)}q^{−b(μ)−n+∑_{i} \binom{m_{i}(μ)+1}{2}} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} ∑_{λ} s_{λ}K_{λ′,μ}(1/q)s_{ν}(q, q^{2}, …, q^{ℓ(μ)−1})

= ∑_{μ⊢n}(−1)^{n−ℓ(μ)}q^{−\binom{ℓ(μ)}{2}−\binom{ℓ(μ)}{2}+ℓ(μ)(ℓ(μ)−1)−n−b(μ)+∑_{i} \binom{m_{i}(μ)+1}{2}} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} ∑_{λ} s_{λ}K_{λ′,μ}(1/q)

×_{∑_{k,ρ} ; ℓ(ρ)=k−1, |ρ|=|ν|}q^{|ν|+b(ρ)} [\begin{matrix}ℓ(μ) − 1 \\ k − 1 \end{matrix}]_{q} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} K_{ν,ρ}(q).

If we can show that the coefficients of s_{λ}K_{λ′,μ}(1/q) in the inner sums on both sides of (4.14) are equal for any μ ⊢ n then (4.14), and hence Theorem 1.2, will follow. Replacing ℓ(μ) by j + 1, we see that this statement can be expressed as

s_{ν}(1, q, q^{2}, …, q^{j−1}) = _{∑_{k,ρ} ; ℓ(ρ)=k−1, |ρ|=|ν|}q^{b(ρ)} [\begin{matrix}j \\ k − 1 \end{matrix}]_{q} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} K_{ν,ρ}(q), (4.15)

for any nonnegative integer j.

To prove (4.15), multiply both sides of (4.15) by s_{ν} = s_{ν}[x] and sum over ν. Using the Cauchy identity, (4.15) is thus equivalent to

h_{n} [(1 − q^{j})\frac{x}{1 − q} ] = _{∑_{k,ρ} ; ℓ(ρ)=k−1, |ρ|=|ν|}q^{b(ρ)} [\begin{matrix}j \\ k − 1 \end{matrix}]_{q} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} ∑_{ν} s_{ν}[x]K_{ν,ρ}(q), (4.16)

for any nonnegative integer j. Using [3, Eqn. 14] this can be expressed as

h_{n} [(1 − q^{j})\frac{x}{1 − q} ] = ∑_{ρ, |ρ|=|ν|} q^{n(ρ)} [\begin{matrix}j \\ ℓ(ρ) \end{matrix}]_{q} [ℓ(ρ)]!(1 − q)^{ℓ(ρ)}P_{ρ} [\frac{x}{1 − q}; q] , (4.17)

for any nonnegative integer j. Here P_{ρ} is the Hall–Littlewood P -function. Making the transformations x ↦ x/(1 − q) and y ↦ 1 − q^{j} in [3, Eqn. 17], another expression for the left-hand side of (4.17) is

∑_{ρ, |ρ|=|ν|} P_{ρ} [\frac{x}{1 − q}; q] Q′_{ρ}[1 − q^{j}; q]. (4.18)

By [3, Lem. 3.3] we have

Q′_{ρ}[1 − q^{j}; q] = q^{b(ρ)}(1 − q^{j})(1 − q^{j−1}) ⋯ (1 − q^{j−ℓ(ρ)+1}). (4.19)

Using (4.19) in (4.18) and simplifying, we see that the right-hand side of (4.17) is the same as (4.18), which completes the proof. □

We want to prove the algebraic interpretation of Δ′_{s_{ν}}e_{n} at t = 0 given in Theorem 1.3. This interpretation is based on the q-reversal of the following symmetric function identity. We consider symmetric functions in two infinite variable sets: x and y. We let ω_{x} and ω_{y} be the ω involution acting on the x and y variables (respectively).

16 JAMES HAGLUND, BRENDON RHOADES, AND MARK SHIMOZONO

Proposition 4.3. Let n, m ≥ 0. We have

(4.20)  ∑_{ν⊢m} s_{ν}(y) · ω_{x}Δ′_{s_{ν}}e_{n}(x)|_{t=0} = ∑_{k≥0} q^{m−k+1}ω_{y}C_{m,k−1}(y; q) · ω_{x}C_{n,k}(x; q).

Proof. By Theorem 1.2,

∑_{ν⊢m}s_{ν}(y) · ω_{x}Δ′_{s_{ν}}e_{n}(x)|_{t=0}

= ∑_{ν⊢m} s_{ν}(y) · ω_{x} ∑_{k=ℓ(ν)+1}^{|ν|+1} P_{ν,k−1}(q) ∑_{\begin{matrix}μ⊢n \\ ℓ(μ)=k \end{matrix}} q^{\overline{b}(μ)} [\begin{matrix}k \\ m(μ) \end{matrix}]_{q} Q′_{μ}(x; q)

= ∑_{ν⊢m} s_{ν}(y) · ω_{x} ∑_{k=ℓ(ν)+1}^{|ν|+1} P_{ν,k−1}(q) · ω_{x}C_{n,k}(x; q)

= ∑_{ν⊢m} ∑_{k=ℓ(ν)+1}^{|ν|+1} q^{|ν|−\binom{k}{2}} ∑_{\begin{matrix}ρ⊢m \\ ℓ(ρ)=k−1 \end{matrix}} q^{b(ρ)} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} K_{ν,ρ}(q)s_{ν}(y) · ω_{x}C_{n,k}(x; q)

= ∑_{ν⊢m} ∑_{k=ℓ(ν)+1}^{|ν|+1} q^{m−k+1} ∑_{\begin{matrix}ρ⊢m \\ ℓ(ρ)=k−1 \end{matrix}} q^{\overline{b}(ρ)} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} K_{ν,ρ}(q)s_{ν}(y) · ω_{x}C_{n,k}(x; q)

= ∑_{\begin{matrix}ν⊢m \\ k≥1 \end{matrix}} q^{m−k+1} ∑_{\begin{matrix}ρ⊢m \\ ℓ(ρ)=k−1 \end{matrix}} q^{\overline{b}(ρ)} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} K_{ν,ρ}(q)s_{ν}(y) · ω_{x}C_{n,k}(x; q)

= ∑_{k≥1} q^{m−k+1} ∑_{\begin{matrix}ρ⊢m \\ ℓ(ρ)=k−1 \end{matrix}} q^{\overline{b}(ρ)} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} [∑_{ν⊢m} K_{ν,ρ}(q)s_{ν}(y)] · ω_{x}C_{n,k}(x; q)

= ∑_{k≥1} q^{m−k+1} ∑_{\begin{matrix}ρ⊢m \\ ℓ(ρ)=k−1 \end{matrix}} q^{\overline{b}(ρ)} [\begin{matrix}k − 1 \\ m(ρ) \end{matrix}]_{q} Q′_{ρ}(y; q) · ω_{x}C_{n,k}(x; q)

= ∑_{k≥1} q^{m−k+1}ω_{y}C_{m,k−1}(y; q) · ω_{x}C_{n,k}(x; q),

which is what we wanted to prove. □

We want to q-reverse the identity of Proposition 4.3. Proposition 4.4. Let n, m ≥ 0. We have (4.21)

∑_{ν⊢m} q^{b(ν)}s_{ν}(y) · (rev_{q} ∘ ω_{x})Δ′_{s_{ν}}e_{n}(x)|_{t=0} = ∑_{k≥0} q^{mn−km−kn+n+k(k−1)}D_{m,k−1}(y; q) · D_{n,k}(x; q).

Proof. We want to q-reverse both sides of Equation (4.20). We begin with the left-hand side. Claim: For any partition ν ⊢ m, the q-degree of Δ′_{s_{ν}}e_{n}|_{t=0} is (n − 1)m − b(ν).

To see why the Claim is true, let ν ⊢ m and consider Equation (4.12), recapitulated here:

Δ′_{s_{ν}}e_{n}|_{t=0} = ∑_{μ⊢n}(−1)^{n−ℓ(μ)}s_{ν}(q, q^{2}, …, q^{ℓ(μ)−1})q^{−n−2b(μ)+∑_{i} \binom{m_{i}(μ)+1}{2}} [\begin{matrix}ℓ(μ) \\ m(μ) \end{matrix}]_{q} · rev_{q}(Q′_{μ}).

We know that Q′_{μ} (and also rev_{q}(Q′_{μ})) has q-degree b(μ). If μ ⊢ n is such that ℓ(μ) > ℓ(ν) so that the μ-summand on the right-hand side of Equation (4.12) does not vanish, the q-degree of this μ-summand is therefore

(4.22)  ∑_{i=j}^{ℓ(ν)} ν_{j}(ℓ(μ) − j) − n − 2b(μ) + ∑_{i} \binom{m_{i}(μ) + 1}{2} + ∑_{i<j} m_{i}(μ)m_{j}(μ) + b(μ),

or equivalently

(4.23)  m(ℓ(μ) − 1) − b(ν) − n − b(μ) + ∑_{i} \binom{m_{i}(μ) + 1}{2} + ∑_{i<j} m_{i}(μ)m_{j}(μ).

It is not hard to see that Expression (4.23) is maximized uniquely when μ = (1^{n}), in which case it equals

(4.24)  m(n − 1) − b(ν) − n − \binom{n}{2} + \binom{n + 1}{2} = m(n − 1) − b(ν),

which completes the proof of the Claim.

Our Claim implies that the overall q-degree of the left-hand side of Equation 4.20 (and hence also the right-hand side) is m(n − 1); this corresponds to the summand ν = (m) so that b(ν) = 0. The q-reversal of the left-hand side of Equation (4.20) is therefore

(4.25)  q^{m(n−1)} ∑_{ν⊢m} q^{b(ν)−(n−1)m}s_{ν}(y) · (rev_{q} ∘ ω_{x})Δ′_{s_{ν}}e_{n}(x)|_{t=0},

which coincides with the left-hand side of Equation (4.21).

Now we q-reverse the right-hand side of Equation (4.20). Since the q-degree of C_{m,k−1}(y; q) is (k − 2)m − \binom{k−1}{2} and the q-degree of C_{n,k}(x; q) is (k − 1)n − \binom{k}{2}, the q-reversal of the right-hand side of Equation (4.20) is

(4.26)  q^{m(n−1)} ∑_{k≥0} q^{−m+k−1} · [q^{−(k−2)m+\binom{k−1}{2}}D_{m,k−1}(y; q)] · [q^{−(k−1)n+\binom{k}{2}}D_{n,k}(x; q)] ,

which is equivalent to the right-hand side of Equation (4.21). □

We are in a position to prove Theorem 1.3.

Proof of Theorem 1.3. If λ ⊢ m and μ ⊢ n are any partitions, the Frobenius image of the irreducible 𝔖_{m} ×𝔖_{n}-module S^{λ} ⊗S^{μ} is Frob(S^{λ} ⊗S^{μ}) = s_{λ}(y)·s_{μ}(x), regarded as an element of the ring Λ(x) ⊗ Λ(y) of formal power series which are separately symmetric in the x and y variables.

More generally, if V is any finite-dimensional 𝔖_{m} ×𝔖_{n}-module, there exist unique integers c_{λ,μ} ≥ 0 such that

(4.27)  V ≅_{𝔖_{m}×𝔖_{n}} ⨁_{λ,μ} c_{λ,μ}S^{λ} ⊗ S^{μ}.

We then set

(4.28)  Frob(V) := ∑_{λ,μ} c_{λ,μ}s_{λ}(y)s_{μ}(x).

Finally, if V = ⨁_{d≥0} V_{d} is a graded 𝔖_{m} × 𝔖_{n}-module with each graded piece V_{d} finitedimensional, we set

(4.29)  grFrob(V; q) := ∑_{d≥0} Frob(V) · q^{d}.

If U is a graded 𝔖_{m}-module and W is a graded 𝔖_{n}-module, we have

(4.30)  grFrob(U ⊗ W; q) = grFrob(U; q) · grFrob(W; q).

Recall that the 𝔖_{m} × 𝔖_{n}-module V_{n,m} is defined by

(4.31)  V_{n,m} = ⨁_{k≥0}(R_{m,k−1} ⊗ R_{n,k}){−mn + km + kn − n − k(k − 1)}.

Applying Equation (1.15), we see that the right-hand side of Equation (4.21) may be expressed as

(4.32)  ∑_{k≥0} q^{mn−km−kn+n+k(k−1)}D_{m,k−1}(y; q) · D_{n,k}(x; q) = grFrob(V_{n,m}; q).

On the other hand, for any graded 𝔖_{m} × 𝔖_{n}-module V and any partition ν ⊢ m, we have

(4.33)  coefficient of s_{ν}(y) in grFrob(V; q) = grFrob(Hom_{𝔖_{m}}(S^{ν}, V); q).

Therefore, we have

(4.34)  (rev_{q} ∘ ω_{x})Δ′_{s_{ν}}e_{n}(x)|_{t=0} = q^{−b(ν)} · grFrob(Hom_{𝔖_{m}}(S^{ν}, V_{n,m}); q),

which is what we wanted to prove. □

5. Closing remarks

In this paper we found an expansion of ωΔ′_{s_{ν}}e_{n}|_{t=0} in the dual Hall–Littlewood basis for any partition ν ⊢ m. This led to the algebraic interpretation of (rev_{q} ∘ω)Δ′_{s_{ν}}e_{n}|_{t=0} presented in Theorem 1.3 involving tensor products of R_{n,k} modules. It may be interesting to find a simpler module whose graded Frobenius image is (rev_{q} ∘ ω)Δ′_{s_{ν}}e_{n}|_{t=0}.

Let k ≤ n be positive integers. The ring R_{n,k} has the following geometric interpretation. Denote by ℙ^{k−1} the (k − 1)-dimensional complex projective space of lines through the origin in ℂ^{k} and let (ℙ^{k−1})^{n} denote the n-fold Cartesian product of ℙ^{k−1} with itself. In joint work with Pawlowski [10], the second author defined the open subvariety X_{n,k} ⊆ (ℙ^{k−1})^{n} given by

(5.1)  X_{n,k} := {(ℓ_{1}, …, ℓ_{n}) ∈ (ℙ^{k−1})^{n} : ℓ_{1} + ⋯ + ℓ_{n} = ℂ^{k}}.

A typical point in X_{n,k} is an n-tuple (ℓ_{1}, …, ℓ_{n}) of one-dimensional subspaces of ℂ^{k} which together span ℂ^{k}.

The symmetric group 𝔖_{n} acts on X_{n,k} by the rule π.(ℓ_{1}, …, ℓ_{n}) := (ℓ_{π_{1}}, …, ℓ_{π_{n}}) for any permutation π = π_{1} … π_{n} ∈ 𝔖_{n}. If H^{•}(X_{n,k}) denotes the singular cohomology of X_{n,k} with integer coefficients, this gives rise to an action of 𝔖_{n} on H^{•}(X_{n,k}). Pawlowski and the second author prove [10] that

(5.2)  H^{•}(X_{n,k}) = ℤ[x_{1}, …, x_{n}]/⟨e_{n}, e_{n−1}, …, e_{n−k+1}, x_{1}^{k}, x_{2}^{k}, …, x_{n}^{k}⟩.

The identification (5.2) may be regarded as both an isomorphism of graded rings and an isomorphism of graded ℤ[𝔖_{n}]-modules. The variable x_{i} represents the Chern class c_{1}(ℓ_{i}^{∗}) of the dual to the i^{th} tautological line bundle ℓ_{i} ↠ X_{n,k}. In particular, we have R_{n,k} = ℚ ⊗_{ℤ} H^{•}(X_{n,k}).

The isomorphism (5.2), together with the fact that X_{n,n} is homotopy equivalent to the manifold Fℓ(n) of complete flags in ℂ^{n}, justify the statement that X_{n,k} is the flag variety attached to the Delta Conjecture (i.e., the Macdonald eigenoperator Δ′_{e_{k−1}}). Given an arbitrary partition ν, it would be interesting to find an analogous variety X_{n,ν} with an action of 𝔖_{n} which would play the corresponding role for the operator Δ′_{s_{ν}}. That is, the cohomology ring H^{•}(X_{n,ν}) should carry an action of 𝔖_{n} such that (upon tensoring with ℚ), the graded Frobenius image of this action is (rev_{q} ∘ ω)Δ′_{s_{ν}}e_{n}|_{t=0}. The space X_{n,k} solves this problem when ν = (1^{k−1}). Theorem 1.3 might be helpful in constructing such a space X_{n,ν} in general.

We close by giving a geometric interpretation of Equation (4.20), recapitulated here:

∑_{ν⊢m} s_{ν}(y) · ω_{x}Δ′_{s_{ν}}e_{n}(x)|_{t=0} = ∑_{k≥0} q^{m−k+1}ω_{y}C_{m,k−1}(y; q) · ω_{x}C_{n,k}(x; q).

If M = M_{0}⊕M_{1}⊕⋯⊕M_{d} is any graded vector space with M_{d} ≠ 0, let M̃ = M̃_{0}⊕M̃_{1}⊕⋯⊕M̃_{d} be the reversed graded vector space with components

(5.3)  M̃_{i} := M_{d−i}, 0 ≤ i ≤ d.

In terms of reversals of R-modules, Equation (4.20) reads

(5.4)  ωΔ′_{s_{ν}}e_{n}|_{t=0} = grFrob(Hom_{𝔖_{n}}(S^{ν}, W_{n,m})),

where

(5.5)  W_{n,m} := ⨁_{k≥0}(R̃_{m,k−1} ⊗ R̃_{n,k}){−m + k − 1}.

Taking the reversal R̃_{n,k} of the quotient R_{n,k} := ℚ[x_{1}, …, x_{n}]/I_{n,k} is not a natural ringtheoretic operation, but it has a geometric interpretation in terms of the variety X_{n,k}.

Let X_{n,k}^{+} = X_{n,k} ∪ {∞} denote the one-point compactification of X_{n,k}, where ∞ is the adjoined point. The Borel–Moore homology H̄_{•}(X_{n,k}) of X_{n,k} is the (singular) homology of the pair (X_{n,k}^{+}, {∞}):

(5.6)  H̄_{•}(X_{n,k}) := H_{•}(X_{n,k}^{+}, {∞}).

The action of 𝔖_{n} on R_{n,k} is both continuous and proper, and so induces a (graded) action of 𝔖_{n} on the Borel–Moore homology H̄_{•}(X_{n,k}). By Poincaré duality and Equation (5.2), we have the isomorphism of 𝔖_{n}-modules

(5.7)  R̃_{n,k} ≅_{𝔖_{n}} ℚ ⊗_{ℤ} H̄_{•}(X_{n,k}).

This yields a geometric interpretation of Equation (4.20) in terms of Borel–Moore homology.

6. Acknowledgements

The authors thank an anonymous referee for their careful reading of this paper. J. Haglund was partially supported by NSF Grant DMS-1600670. B. Rhoades was partially supported by NSF Grant DMS-1500838. M. Shimozono was partially supported by NSF Grant DMS-1600653.

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Department of Mathematics University of Pennsylvania Philadelphia, PA, 19104-6395, USA

E-mail address: jhaglund@math.upenn.edu

Department of Mathematics University of California, San Diego La Jolla, CA, 92093-0112, USA

E-mail address: bprhoades@math.ucsd.edu

Department of Mathematics 460 McBryde Hall, Virginia Tech 255 Stanger St. Blacksburg, VA, 24601, USA

E-mail address: mshimo@math.vt.edu