Séminaire Lotharingien de Combinatoire 63 (2010), Article B63f

TABLEAUX IN THE WHITNEY MODULE OF A MATROID

ANDREW BERGET

Abstract. The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of the matroid.

In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom (also known as Schubert, or shifted) matroids. The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a formula for the multiplicity hook shapes in this representation in terms of the no broken circuit sets.

1. Introduction and Motivation

If V is a complex vector space of dimension k we let ⋀ V denote the exterior algebra of V, T^{n}(V) the n-fold tensor product V^{⊗n} and T (V) the tensor algebra ⨂_{n≥0} T^{n}(V). For the moment we will only be concerned with the GL(V)-module structure of T (⋀ V). We begin by seeing how tableaux describe a basis for the ℂ-vector space T (⋀ V).

Recall that the irreducible polynomial representations of GL(V) are indexed by partitions λ with length at most dim V. We denote the irreducible representation with highest weight λ by 𝕊^{λ}(V). It follows from the Weyl character formula that the dimension of 𝕊^{λ}(V) is the number of column strict tableaux of shape λ with entries in [dim V] := {1, 2, …, dim V }. Using Young’s Rule we obtain the GL(V)-module decomposition of the tensor product of exterior products:

⋀^{μ_{1}}V ⊗ ⋀^{μ_{2}}V ⊗ ⋯ ⊗ ⋀^{μ_{ℓ}}V = ⨁_{λ:ℓ(λ)≤k}(𝕊^{λ}V)^{⊕K_{λ′,μ}}.

Here K_{λ′,μ} is the number of column strict tableaux of shape λ′ (the conjugate partition of λ) that contain μ_{i} i’s. From this, one easily deduces the following.

Theorem 1.1. The tensor algebra T (⋀ V) has a basis indexed by pairs of tableaux (T_{r}, T_{c}) of the same shape where T_{r} has strictly increasing rows, weakly increasing columns and entries in [dim V] and T_{c} is column strict with arbitrary entries.

The Whitney module of a matroid M, W (M), will be a quotient of a certain letterplace algebra that mimics T (⋀ V) , but takes into account the dependencies of the matroid M. Its definition is slightly more natural than the closely related Whitney algebra of a matroid, which was defined by Crapo, Rota and Schmitt in [6]. In the final section of [6] the Whitney module of a matroid is roughly described in passing. The goal of this paper is to begin to investigate how tableaux play a role in describing the structure of W (M). Our main result is that the obvious spanning set of W (M) is a basis when M is a freedom matroid (also known as Schubert, or shifted matroids). We will elaborate on and prove the following result.

Date: June 16, 2010.

Some of this work appeared in the author’s 2009 Ph.D. thesis from the University of Minnesota.

Thanks are due to Victor Reiner for advising this thesis. Some of this work was supported by NSF grant DMS-0636297. The author would like to thank Andrea Brini for sharing with the author his joint work with Regonati on the letter-place approach to the Whitney algebra.

Theorem 1.2. Let M be a freedom matroid on n elements. There is a basis for its Whitney module indexed by pairs of tableaux (T_{r}, T_{c}) where T_{r} and T_{c} have the same shape and

(1) T_{r} is row strict with entries in [n],

(2) every row of T_{r} indexes an independent set of M, and

(3) T_{c} is column strict.

We will also precisely state and prove the following result.

Theorem 1.3. In the complexified doubly multilinear submodule of the Whitney module of M, a basis for the hook shaped isotypic components are determined by the no broken circuit complex of M.

The paper is organized as follows. First we define the super algebra Super(L^{−}|P^{+}) and recall the standard basis theorem of Grosshans–Rota–Stein [9]. We then define the Whitney algebra and Whitney module of a matroid and see that the latter is spanned by certain elements indexed by pairs of tableaux, as in the standard basis theorem, with the additional condition that the rows of the row strict tableau index independent subsets of the matroid. We will then define freedom matroids and prove that the given spanning set for its Whitney module forms a basis. Following that, we will define the doubly multilinear submodule of the Whitney module and describe an action of the symmetric group on it. After complexifying this submodule, we will give a formula for the multiplicity of irreducible symmetric group modules indexed by hook shapes. The formula will be in terms of the certain no broken circuit subsets of M.

2. Letter-Place Algebras

In this section we show how to view the tensor algebra of the exterior algebra of a finite set as a letter-place superalgebra. All of the definitions come from [9] and we use the main result there, the standard basis theorem, to describe a combinatorial basis of this object in terms of pairs of tableaux. This gives a concrete explanation of our motivating result Theorem 1.1.

2.1. Exterior Algebra. Let E be a finite set and ⋀ E be the exterior algebra of the free ℤ-module with basis E. We will write decomposable elements as w = e_{i_{1}}e_{i_{2}} … e_{i_{k}}, to avoid notational clutter. This is a graded commutative algebra, which means that ⋀ E is the direct of the i-fold exterior products

⋀ E = ⨁_{0≤i} ⋀^{i}E

and if w ∈ ⋀^{i} E and w′ ∈ ⋀^{j} E then

ww′ = (−1)^{ij}w′w.

If w ∈ ⋀ E is homogeneous then we denote the degree of the piece which w is in by |w|. The tensor algebra of the graded algebra ⋀ E is the direct sum of the tensor products T^{1}(⋀ E), T^{2}(⋀ E), …. Each of the summands has its own product, called the internal product, which is induced by the rule

(w ⊗ w′) × (u ⊗ u′) = (−1)^{|w′||u|}(wu ⊗ w′u′).

The exterior algebra of E is a graded commutative Hopf algebra, with coproduct

δ : ⋀ E → ⋀ E ⊗ ⋀ E

induced by the rule δ(e) = 1 ⊗ e + e ⊗ 1. We will not need the definitions of the counit or antipode here. Given an element w ∈ ⋀ E we write its coproduct using Sweedler notation

δ(w) = ∑_{w} w_{(1)} ⊗ w_{(2)}.

The iterated coproduct δ^{(n)} : ⋀ E → T^{n}(⋀ E) is defined by the conditions that δ^{(1)} equals the identity map, δ^{(2)} = δ and δ^{(n)} = (δ ⊗ 1) ∘ δ^{(n−1)}. The iterated coproduct δ^{(n)} is the sum of its homogeneous pieces

δ^{(α)} : ⋀ E → (⋀^{α_{1}}E) ⊗ (⋀^{α_{2}}E) ⊗ ⋯ ⊗ (⋀^{α_{n}}E)

where α = (α_{1}, α_{2}, …, α_{n}) is a composition with n parts. The image w under δ^{(α)} is the α-th coproduct slice of w.

2.2. Letter-Place Algebras. The goal of this section is to view T (⋀ E) as one of the letter-place algebras of Grosshans–Rota–Stein [9]. This will be done by constructing two algebras, one of non-commutative letters the other of commutative places, and from these defining a new algebra of letter-place pairs.

We will not review the complete definition of the letter-place algebras, since this some amount of work to do precisely. Instead, we will take those pieces of the definitions suited for our needs, hinting at the form of the general definitions.

In this section we will declare the elements of E to be negatively signed and refer to them as negative letters. To emphasize that we are viewing E in this way, we will denote it by L^{−} (or sometimes L_{E}^{−} when we must emphasize the set E). Let P^{+} = {p_{1}, p_{2}, p_{3}, … } denote the infinite set of positively signed places. We associate to L^{−} and P^{+} two algebras, the exterior algebra and the divided power algebra, respectively, and associate to these the so-called letter-place algebra. Will write the exterior algebra of L^{−} as Super(L^{−}), which is constructed exactly as in the previous section.

We recall the definition of the divided power algebra. Associate to each element p ∈ P^{+} an infinite sequence of divided powers

p^{(0)}, p^{(1)}, p^{(2)}, ….

We let ℤ⟨P_{d}^{+}⟩ denote the free algebra generated by the divided powers. Let Super(P^{+}) be the quotient of ℤ⟨P_{d}^{+}⟩ by the two-sided ideal generated by the elements of the form

p^{(j)}p^{(k)} − \binom{j + k}{k}p^{(j+k)}, p^{(j)}q^{(k)} − q^{(k)}p^{(j)}, p^{(0)} − 1,

for any p, q ∈ P^{+}, p ≠ q. The elements p^{(j)} are meant to behave like p^{i}/i! in a symmetric algebra. We can endow Super(P^{+}) with a coalgebra structure by defining the coproduct of p^{(j)}, p ∈ P as

δ(p^{(j)}) = 1 ⊗ p^{(j)} + p^{(1)} ⊗ p^{(j−1)} + ⋯ + p^{(j−1)} ⊗ p^{(1)} + p^{(j)} ⊗ 1.

As before we write the coproduct of an arbitrary element of q ∈ Super(P^{+}) in Sweedler notation,

δ(q) = ∑_{q} q_{(1)} ⊗ q_{(2)}.

It is clear that Super(P^{+}) is graded and commutative in the usual sense of commutative algebra.

Finally we are in a positive to define the letter-place algebra Super(L^{−}|P^{+}). Let (L^{−}|P^{+}) denote the set of letter-place pairs:

(L^{−}|P^{+}) = {(e|p) : e ∈ L^{−}, p ∈ P^{+}}.

Since L^{−} consists of negatively signed variables and P^{+} consists of positively signed variables, we declare the letter-place pairs to be negatively signed. We define Super(L^{−}|P^{+}) to be the exterior algebra of the set (L^{−}|P^{+}).

To describe a standard basis of Super(L^{−}|P^{+}) we define a certain bilinear map

Ω : Super(L^{−}) × Super(P^{+}) → Super(L^{−}|P^{+}),

called the Laplace pairing, according to a sequence of rules.

R1. Ω(1, 1) = 1.

R2. If e ∈ L^{−} and p ∈ P^{+} then Ω(e, p^{(1)}) = (e|p).

R3. If w and p are not in the same graded piece of Super(L^{−}) and Super(P^{+}),

respectively, then Ω(w, p) = 0. R4. If δ(p) = ∑_{(p)} p_{(1)} ⊗ p_{(2)} then

Ω(ww′, p) = ∑_{(p)} Ω(w, p_{(1)})Ω(w′, p_{(2)}).

R4′. If δ(w) = ∑_{(w)} w_{(1)} ⊗ w_{(2)} then

Ω(w, pp′) = ∑_{(w)} Ω(w_{(1)}, p)Ω(w_{(2)}, p′).

That the rules R4 and R4′ are equivalent follows from a series of technical checks, which was done [9]. There they give a more general definition of the Laplace pairing was given that includes the possibility of both positively and negatively signed letters and places. In the future we will denote the Laplace pairing of w and q by (w|q), so that elements of the form (efg|p_{1}^{(2)}p_{2}) make sense.

Proposition 2.1. Viewing T (⋀ E) as a ℤ-algebra with the internal product, there is a surjection of ℤ-algebras

T (⋀ E) → Super(L^{−}, P^{+}),

that maps a tensor w_{1} ⊗ w_{2} ⊗ ⋯ ⊗ w_{n} to the product

(w_{1}|p_{1}^{|w_{1}|})(w_{2}|p_{2}^{|w_{2}|}) … (w_{n}|p_{1}^{|w_{n}|}).

Example 2.2. Suppose that e, f ∈ E = L^{−}. Then

e ⊗ e ⊗ f ↦ (e|p_{1})(e|p_{2})(f|p_{3}).

We can verify that ef + fe ↦ 0. We know that

δ(p_{1}^{(2)}) = 1 ⊗ p_{1}^{(2)} + p_{1}^{(1)} ⊗ p_{1}^{(1)} + p_{1}^{(2)} ⊗ 1,

hence

ef ↦ (e|1)(f|p_{1}^{(2)}) + (e|p_{1})(f|p_{1}) + (e|p_{1}^{(2)})(f|1) = (e|p_{1})(f|p_{1}) = −(f|p_{1})(e|p_{1}),

since Super(L^{−}|P^{+}) is an exterior algebra.

The preimage of (e|p_{2}) consists of elements of the form

1 ⊗ e ⊗ 1 ⊗ ⋯ ⊗ 1

since, e.g.,

1 ⊗ e ↦ (1|p_{1}^{(0)})(e|p_{2}) = (1|p_{1}^{(0)})(e|p_{2}) = (e|p_{2}).

Given a composition α = (α_{1}, …, α_{n}) we define

p^{(α)} := p_{1}^{(α_{1})}p_{2}^{(α_{2})} … p_{n}^{(α_{n})}

Proposition 2.3. Let w ∈ ⋀ E be a decomposable element. The image of the coproduct slice δ^{(α)}(w) in Super(L^{−}|P^{+}) is the Laplace pairing (w|p^{(α)}).

Proof. The result is easy to verify if w = e ∈ L^{−}. If |w| > 1 then we may write w = w′w′′. By the homogeneity of the coproduct and induction we have

δ(w′w′′) = ∑_{β+γ=α} δ^{(β)}(w′)δ^{(γ)}(w′′) ↦ ∑_{β+γ=α}(w′|p^{(β)})(w′′|p^{(γ)}).

Since we have δ(p^{(α)}) = ∑_{β+γ=α} p^{(β)} ⊗ p^{(γ)}, we can rule R4 to write this as (w′w′′|p^{(α)}). □

2.3. The Standard Basis Theorem. From the computation of the GL(V)-module structure of T (⋀ V) in the introduction, we expect Super(L^{−}|P^{+}) to have a basis indexed by pairs of tableaux of the same shape where one is row strict and the other is column strict. This is the case, and in this section we recall how to construct this basis.

Let λ be a decreasing sequence of non-negative integers; a partition. The length of λ is the number of positive integers in the sequence. We will identify λ with its Young frame, which is a collection of boxes, north-east justified, the number of boxes in the i-th row being equal to λ_{i}. Denote the total number of boxes in the Young frame of λ by |λ|. A tableau T is a filling of the elements of A into the boxes of a partition λ. If T is a tableau we will call the partition λ the shape of T and write sh(T). For example

3 2 5 4 2 3 3 3 2 1 2 1 2

is a tableau on {1, 2, 3, 4, 5} whose shape is (5, 4, 2, 2). The content of a tableau is the number of 1’s, the number of 2’s, … that appear in the filling. We will write the content of a tableau as a composition whose i-th part is the number of i’s in the filling of the tableau. Thus the tableau above content (2, 5, 4, 1, 1, 0, 0, …). A column strict tableau is a tableau where the numbers in each row weakly increase and the numbers in each column strictly increase. A row strict tableau is a tableau where the numbers in each column weakly increase and the numbers in each row strictly increase. We will call a tableaux T a standard Young tableaux if it is both row and column strict and has entries in {1, 2, …, |sh(T)|}.

Let T and S be tableau of the same shape λ and length ℓ. Let the numbers in the i-th row of T be t_{1}, …, t_{λ_{i}}, in order. Define w_{i} to be the product in Super(L^{−}) of the elements indexed by t_{1}, …, t_{λ_{i}}, i.e., w_{i} = e_{t_{1}} … e_{t_{λ_{i}}}. Let s_{1}, …, s_{λ_{i}} be the elements in the i-th row of S, in order. Define q_{i} to be the product in Super(P^{+}) of the elements indexed by s_{1}, …, s_{λ_{i}}, where if s_{j} = ⋯ = s_{j+k−1} is a maximal string of equal entries then we take p_{s_{j}}^{(k)} instead of the product p_{s_{j}} … p_{s_{j+k−1}}. For example, if

T = 1 2 3 4 1 2 3 3 S = 3 3 4 3 5 5 5 6

then

w_{1} = w_{2} = e_{1}e_{2}e_{3}, w_{2} = e_{3}, q_{1} = p_{3}^{(2)}p_{4}^{(1)}p_{3}, q_{2} = p_{5}^{(3)}, q_{3} = p_{6}^{(1)}.

We define tab(T |S) by the formula

tab(T |S) = (w_{1}|q_{1})(w_{2}|q_{2}) … (w_{ℓ}|q_{ℓ}) ∈ Super(L^{−}|P^{+}),

which makes sense according to our definition of the Laplace pairing. We call such an element a tableaux in Super(L^{−}|P^{+}). We are finally in a position to state the main result of this section.

Theorem 2.4 (Grosshans–Rota–Stein [9]). The elements tab(T_{r}|T_{c}), where (1) T_{r} and T_{c} are tableaux of the same shape,

(2) T_{r} is row strict with entries in [n],

(3) T_{c} is column strict, form a basis for the free module Super(L^{−}|P^{+}). We will call such tableaux standard.

In the expansion of tab(T |S) as a sum of standard tableaux ∑_{i} c_{i}tab(T_{i}|S_{i}) we have that the shape of each T_{i} is larger than or equal to the shape of T, in dominance order. Further, the content of every T_{i} is equal to the content of T and the content of every S_{i} is equal to the content of S.

3. The Whitney Algebra and Module of a Matroid

We assume that the reader is familiar with the basic concepts in matroid theory (see, e.g., [11]).

In this section we define the Whitney algebra and Whitey module of matroid. We then show that if M is realizable over ℂ then the standard tableau pairs of the previous section have nonzero image in the Whitney module of M if and only if the rows of the first tableau index independent sets of M.

3.1. Definitions. Let M be a matroid on E of rank r(M). Decomposable elements of ⋀ E are given by words on E. We say that a decomposable element e_{i_{1}}e_{i_{2}} … e_{i_{k}} ∈ ⋀ E is a dependent word if {i_{1}, i_{2}, …, i_{k}} is a dependent set in M. Likewise we define independent words.

Definition 3.1 (Crapo–Schmitt [6]). The Whitney algebra of a matroid M, denoted W(M), is the quotient of T (⋀ E) by the ideal generated by the elements

δ_{α}(w)

where w is a dependent word in M and α is a composition of |w|.

The following definition was also given by Brini and Regonati (unpublished, [4]).

Definition 3.2. The Whitney module of a matroid M, denoted W (M), is defined to be the quotient of Super(L^{−}|P^{+}) by the two-sided ideal generated by the elements

(w|p^{(α)})

where w is a dependent word in M and α is a composition of |w|.

Using rule R4 it is clear that it is sufficient to take w to be the word of a circuit of M in the definition of W (M).

Proposition 2.3 implies that there is a surjective map W(M) → W (M) that takes the internal product of W(M) to the product that W (M) inherits as a quotient of an exterior algebra. One can think of W (M) as being obtained from W(M) by appending a half-infinite string of the form 1 ⊗ 1 ⊗ 1 ⊗ … to the right of every element of W(M) (see the comments at the end of [6]).

Since Super(L^{−}|P^{+}) is a graded commutative algebra and the ideal defining W (M) is homogeneous, W (M) inherits a grading. Each of the graded pieces is a finitely generated ℤ-module, and hence can be written as the direct sum of a free part and a torsion part.

Proposition 3.3. There is a direct sum decomposition

W (M) = W (M)_{free} ⊕ W (M)_{tor}

where W (M)_{free} is free and W (M)_{tor} is torsion.

It is a basic example of Crapo and Schmitt [6] that if M is not realizable over a field of characteristic zero then W (M)_{tor} can be non-zero. It is unknown if W (M)_{tor} is zero when M is realizable over a field of characteristic zero.

3.2. Tableaux in the Whitney Module. Let T be a tableau with entries in [n], and S an arbitrary tableau of the same shape. Since W (M) is a quotient of Super(L^{−}|P^{+}) we can project the elements tab(T |S) of Section 2.3 into W (M). Abusing notation, we denote the image of tab(T |S) in W (M) by tab(T |S).

Note that every standard tableaux tab(T |S) can be written as

(w_{1}|p^{(α^{1})})(w_{2}|p^{(α^{2})}) … (w_{ℓ}|p^{(α^{ℓ})}),

where α^{i} is a composition of |w_{i}| and |w_{1}| ≥ |w_{2}| ≥ ⋯ ≥ |w_{ℓ}|.

Proposition 3.4. The image of an arbitrary tableaux tab(T |S) in W (M) is zero if some row of T indexes a dependent set of M.

Proof. This follows since each tableau is a product of elements of the form (w|p^{(α)}), and we know that this element is zero in the Whitney module if w is a dependent word. □

The main theorem of this section is the following result.

Theorem 3.5. Let S be a column strict tableau. If M is realizable over ℂ, the image of image of the tableaux tab(T |S) in W (M) is non-zero if and only if the rows of T index independent sets of M.

Before we proceed with the proof we set up a nice corollary, that gives us a simple check of whether such a tableaux exists, having prescribed the content and shape of T. The content of the T determines a parallel extension of the labeled matroid M. Indeed if the content of T is μ (a composition with n parts) then the parallel extension is M_{μ}, which has the μ_{i} copies of the element e_{i}.

The rank partition of a matroid M is the sequence of numbers ρ(M) = (ρ_{1}, ρ_{2}, …) determined by the condition that

ρ_{1} + ρ_{2} + ⋯ + ρ_{k}

is the size of the largest union of k independent subsets of M. This definition was first given by Dias da Silva in [7] where he proved the following result.

Theorem 3.6 (Dias da Silva [7]). The rank partition of matroid is a partition. There is a partition of the ground set of a loopless matroid M into independent sets of size λ_{1} ≥ λ_{2} ≥ … if and only if λ ≤ ρ(M) in dominance order.

The following corollary is now immediate from the theorem.

Corollary 3.7. There is a non-zero tableaux tab(T |S) ∈ W (M) of shape λ, where S is columns strict and T has content μ, if and only if λ ≤ ρ(M_{μ}) in dominance order.

To prove Theorem 3.5 we need a lemma.

Lemma 3.8. Suppose that S is a fixed column strict tableaux of shape λ whose first row gives rise to the element p^{(α)} ∈ Super(P^{+}). Let S′ denote S with its first row removed.

Define two vector spaces: X is the subspace of Super(L^{−}|P^{+})⊗ℂ spanned by standard tableaux tab(T |S) where T has first row equal containing the numbers {1, 2, …, λ_{1}}. The second vector space X′ is the subspace of Super(L^{−}|P^{+})⊗ℂ spanned by any standard tableaux tab(T′|S′).

Then multiplication by

(e_{1}e_{2} … e_{λ_{1}}|p^{(α)})

induces an isomorphism of vector spaces X′ → X.

Proof. This follows directly from the standard basis theorem, since this map takes bases to bases. □

Proof of Theorem 3.5. Let f : E → V is a realization of M, where V is a complex n-dimensional vector space. After choosing a basis for V, we can identify ⋀ V with Super(L^{−}) and since f is a realization of M, the mapping (e|p) ↦ (f(e)|p) gives rise to a map of algebras

f : W (M) → Super(L^{−}|P^{+}) ⊗ ℂ, (w|p^{(α)}) ↦ (f(w)|p^{(α)}),

(compare Proposition 6.3 in [6]). This map will almost always fail to be surjective since M will typically not have rank n. Note that Super(L^{−}|P^{+}) comes with a left GL_{n}(ℂ) action, induced by the natural action of GL_{n}(ℂ) on Super(L^{−}) ⊗ ℂ. Taking a limit, there is a corresponding action of n × n complex matrices on Super(L^{−}|P^{+}).

Suppose that we have the tableau

tab(T |S) = (w_{1}|p^{(α^{1})})(w_{2}|p^{(α^{2})}) … (w_{ℓ}|p^{(α^{ℓ})}) ∈ W (M)

where S is column strict of shape λ, length ℓ and |w_{i}| = λ_{i}. Applying the map f from above we obtain

f(tab(T |S)) = (f(w_{1})|p^{(α^{1})})(f(w_{2})|p^{(α^{2})}) … (f(w_{ℓ})|p^{(α^{ℓ})}) ∈ Super(L^{−}|P^{+}) ⊗ ℂ.

We will prove by induction on the length of λ that f(tab(T |S)) is not zero provided that f(w_{i}) ≠ 0. Since the image of tab(T |S) is not zero, it must be that tab(T |S) ≠ 0 in W (M).

Let A be a generic matrix such that

Af(w_{1}) = e_{1}e_{2} … e_{λ_{1}}

Since A is generic, each of element A(f(w_{i})) is not zero and decomposable in Super(L^{−}) ⊗ ℂ. By Lemma 3.8, we have that

(e_{1}e_{2} … e_{λ_{1}}|p^{(α^{1})})(Af(w_{2})|p^{(α^{2})}) … (A(f(w_{ℓ}))|p^{(α^{ℓ})})

is not zero if and only if

(Af(w_{2})|p^{(α^{2})}) … (A(f(w_{ℓ}))|p^{(α^{ℓ})})

is not zero. This is not zero by induction. It only remains to check the basis step. This follows since

(e_{i_{1}} … e_{i_{k}}|p^{(α)}),

i_{1} < ⋯ < i_{k}, is a basis element of Super(L^{−}|P^{+}) ⊗ ℂ, according to the standard basis theorem. □

Remark 3.9. One cannot remove the hypothesis that S is column strict. For example if M is a boolean matroid (i.e., W (M) = Super(L^{−}|P^{+})) and

T = S = 1 2 3 1 2 3

then

tab(T |S) = (e_{1}e_{2}e_{3}|p_{1}^{(1)}p_{2}^{(1)}p_{3}^{(1)})^{2} = 0

even though (e_{1}e_{2}e_{3}|p_{1}^{(1)}p_{2}^{(1)}p_{3}^{(1)}) ≠ 0 in W (M).

It is unknown if Theorem 3.5 holds for any realizable matroid.

4. Freedom Matroids

In this section we define freedom matroids and show that the obvious spanning set for their Whitney modules are bases.

4.1. Definition of a Freedom Matroid.

Definition 4.1. We will denote the direct sum of a matroid M with the one element rank one matroid on the set {e} by M ⊕ e.

Let M be a matroid of rank larger than 0. The truncation of M to rank k ≤ r(M) is the matroid whose bases are those independent sets of M with size k. The truncation of M to rank r(M) − 1 will be denoted T (M).

The principal extension of a matroid M along the improper flat is the matroid M + e obtained by truncating the direct sum M ⊕ e to the the rank of M. That is M + e = T (M ⊕ e).

We think of M + e as adding a new element generically to M without increasing its rank.

Definition 4.2. For i ∈ {0, 1} define M_{(i)} to be the rank i matroid on one element e_{1}. Let s be a binary sequence of length n > 1 and s′ be the sequence obtained by deleting its last entry s_{n}. Define a matroid M_{s} on the set {e_{1}, e_{2}, …, e_{n}} by setting

M_{s} = \begin{cases}M_{s′} ⊕ e_{n} & s_{n} = 1, \\ M_{s′} + e_{n} & s_{n} = 0. \end{cases}

A freedom matroid is a labeled matroid of the form M_{s} for some binary sequence s.

Example 4.3. The freedom matroid associated with the sequence (1, 0, 1, 0, 1, 0) is

((((M_{1} + e_{2}) ⊕ e_{3}) + e_{4}) ⊕ e_{5}) + e_{6}

where M_{1} is the one element rank one matroid with ground set {e_{1}}. It is represented linearly by the columns of the matrix

[\begin{matrix}1 & 1 & ∗ & ∗ & ∗ & ∗ \\ & & 1 & 1 & ∗ & ∗ \\ & & & & 1 & 1 \end{matrix}]

where the blank entries are zero and the ∗’d entries are generic elements of ℂ.

Freedom matroids arise in many contexts. They are the matroids associated to a generic element of a Schubert strata of a Grassmannian. They are known to be the matroids whose independence complexes are shifted. They are special cases of latticepath matroids.

4.2. The Whitney Module of a Freedom Matroid. We can now state and prove our first main theorem.

Theorem 4.4. Let M be a freedom matroid (in particular, the ground set of M is ordered). The Whitney module W (M) is free and a basis consists of those standard tableaux tab(T_{r}|T_{c}) where the rows of T_{r} index independent sets of M.

We already know that these tableaux span W (M), so it suffices to prove that W (M) is free and the stated elements are linearly independent when M is a freedom matroid.

This result does not hold in general, as we will see in Section 5. Since uniform matroids are a special case of freedom matroids (they are associated to sequences of the form (1, 1, …, 1, 0, 0, …, 0)) we see that the above theorem describes the Whitney module of uniform matroids, a result obtained in 2000 by Crapo and Schmitt [5].

Lemma 4.5. Let M and N be matroids. There is an isomorphism of graded algebras

W (M ⊕ N) ≈ W (M) ⊗ W (N),

where the ⊗ is the super tensor product of graded algebras.

Recall that the super tensor product of two graded ℤ-algebras H and L has module structure given by the ℤ-module H ⊗ L and algebra structure induced by the formula for multiplying homogeneous elements

(h ⊗ l)(h′ ⊗ l′) = (−1)^{|h′||l|}hh′ ⊗ ll′.

Proof. We use the following fact: If H and L are graded algebras and I and J are homogeneous ideals of H and L, respectively, then as graded algebras

(1)  H/I ⊗ L/J ≈ \frac{H ⊗ L}{H ⊗ J + I ⊗ L}.

In the present situation, we suppose that M is a matroid on E and N is a matroid on F. Let L_{E}^{−} denote the set of negatively signed letters E and L_{F}^{−} denote the set of negatively signed letters F. It is well known that

(2)  Super(L_{E}^{−}|P^{+}) ⊗ Super(L_{F}^{−}|P^{+}) ≈ Super(L_{E}^{−} ∪ L_{F}^{−}|P^{+}).

If I is an ideal of Super(L_{E}^{−}|P^{+}) then under this isomorphism we have

(3)  I ⊗ Super(L_{F}^{−}|P^{+}) → Super(L_{E}^{−} ∪ L_{F}^{−}|P^{+}) · I,

i.e., the ideal I ⊗Super(L_{F}^{−}|P^{+}) maps to the ideal generated by I in Super(L_{E}^{−}∪L_{F}^{−}|P^{+}).

To complete the proof all we need to note is that a circuit of M ⊕N is either a circuit of M or a circuit of N. Thus,

(4) ⟨(w|p^{α}) : w is a circuit of M ⊕ N⟩ = ⟨(w|p^{α}) : w is a circuit of M⟩ + ⟨(w|p^{α}) : w is a circuit of N⟩

where here ⟨ − ⟩ denotes taking the ideal in Super(L_{E}^{−} ∪ L_{F}^{−}|P^{+}) generated by the elements −.

Combining equations (1)—(4) with the definition of the Whitney module we have W (M) ⊗ W (N) ≈ W (M ⊕ N). □

Corollary 4.6. If W (M) and W (N) are free ℤ-modules, then so is W (M ⊕ N).

Lemma 4.7. Suppose that M is a matroid of rank larger than 0 and W (M) is free, so that W (M) has a basis B consisting of some standard tableaux. Then W (T (M)) is free and has a consisting of those standard tableaux in B whose first row has length less than r(M).

Proof. It is easy to convince oneself that

W (T (M)) = W (M)/⟨tab(T |S) : sh(T) = λ, λ_{1} = r(M)⟩

where ⟨ − ⟩ denotes taking the two sided ideal in W (M) generated by the elements −. Now, by the standard basis theorem we know that for non-standard tableaux tab(T |S),

tab(T |S) = ∑_{i=1}^{m} c_{i}tab(T_{i}|S_{i})

where sh(T_{i}) ≥ sh(T) in dominance order and each (T_{i}|S_{i}) ∈ B. This implies that if the first row of T has length r(M) then the first row of every T_{i} has length at least r(M). It follows that

W (T (M)) = W (M)/⟨tab(T |S) : sh(T) = λ, λ_{1} = r(M), tab(T |S) ∈ B⟩

Since W (M) is free, the claim now follows. □

Proof of Theorem 4.4. The theorem will follow by induction. We will prove that if the result holds for a matroid M on {e_{1} < e_{2} < ⋯ < e_{n}} then it also holds for M ⊕ e_{n+1} and T (M). Since freedom matroids are closed under these operations, it is sufficient to prove the result for the two one element matroids, which is trivial.

For a positive integer m let W (M)_{≤m} be the subalgebra of W (M) generated by letter place pairs (e|p) where p ∈ {p_{1}, p_{2}, …, p_{m}}. By our assumption on W (M) we see that a basis for W (M)_{≤m} consists of those tableaux tab(T_{r}|T_{c}) where T_{r} is row strict, T_{c} is column strict, every row of T_{r} indexes an independent set of M and every entry of T_{c} is at most m. This is a module of finite rank, since the shape of every tableau appearing must fit into a m-by-r(M) box.

In light of Lemma 4.5, it is straightforward to convince oneself that

W (M ⊕ e_{n+1})_{≤m} = W (M)_{≤m} ⊗ W ({e_{n+1}})_{≤m},

where {e_{n+1}} denotes the rank one element one element matroid. It follows that the the rank of W (M)_{≤m} ⊗ W ({e_{n+1}})_{≤m} as a ℤ-module is the number of pairs

((T_{r}, T_{c}) , (e_{n+1}|p_{i_{1}})(e_{n+1}|p_{i_{2}}) … (e_{n+1}|p_{i_{k}}))

where T_{r} is row strict, T_{c} is column strict, every row of T_{r} indexes an independent set of M, the entries of T_{c} are at most m, and 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ m. From this pair we produce two new tableaux T′_{r} and T′_{c} where

T′_{c} = (((T_{c} ← i_{1}) ← i_{2}) ← ⋯ ← i_{k})

is obtained by the usual Robinson–Schensted row insertion and T′_{r} is obtained from T_{r} by recording the new boxes of T′_{c} with n + 1’s.

It follows from the Super RSK Correspondence [2, 10] that this map is bijective and its image consists of pairs of tableaux (T, S) of the same shape where T is row strict with entries in [n + 1], S is column strict with entries in [m] and the rows of T index independent subsets of M ⊕ e_{n+1}. This is because e_{n+1} is in no circuit of the direct sum. This complete the proof of the induction for direct sums.

Suppose now that W (M) is free and a basis consists of those standard tableaux (T_{r}|T_{c}) such that every row of T_{r} indexes an independent set of M. Then Lemma 4.7 proves that the same statement holds for W (T (M)). □

5. The Doubly Multilinear Submodule of W (M)

The doubly multilinear submodule of W (M) is the submodule generated by elements of the form

(e_{1}|p_{σ(1)})(e_{2}|p_{σ(2)}) … (e_{n}|p_{σ(n)}) ∈ W (M)

where σ is any permutation in the symmetric group 𝔖_{n}. We denote this submodule by U(M). There is a right action of 𝔖_{n} on U(M), by permuting places. We will primarily be interested in the complexified version ℂ ⊗ U(M), where M is a matroid realizable over the complex numbers.

The module U(M) arose independently in the thesis of the author, where it was related to the smallest symmetric group (or general linear group) representation containing a fixed decomposable tensor. Recall that 𝔖_{n} acts on the right of V^{⊗n} via place permutation. If u ∈ V^{⊗n} is any tensor let 𝔖(u) be the smallest 𝔖_{n}-representation in V^{⊗n} containing u. If f : E → V is a realization of M, then it is easy to see that the map

(e_{1}|p_{1})(e_{2}|p_{2}) … (e_{n}|p_{n}) ↦ f(e_{1}) ⊗ f(e_{2}) ⊗ ⋯ ⊗ f(e_{n}) ∈ V^{⊗n}

extends to a unique surjective map of ℂ𝔖_{n}-modules

ℂ ⊗ U(M) → 𝔖(f(e_{1}) ⊗ f(e_{2}) ⊗ ⋯ ⊗ f(e_{n}))

The latter representation is a subtle projective invariant of the vector configuration f(E) ⊂ V. For example, there is no known example of two different realizations f, g : E → V of the same matroid such that

𝔖(f(e_{1}) ⊗ f(e_{2}) ⊗ ⋯ ⊗ f(e_{n})) ≉ 𝔖(g(e_{1}) ⊗ g(e_{2}) ⊗ ⋯ ⊗ g(e_{n})),

where ≈ is isomorphism of 𝔖_{n}-modules.

5.1. Which Irreducible Submodules Can Appear in ℂ ⊗ U(M). The irreducible representations of a the symmetric group 𝔖_{n} are parametrized by partitions of n. Given a partition λ and a tableaux T of shape λ with content (1, 1, …, 1) we can construct the irreducible 𝔖_{n}-representation indexed by λ by taking the left or right ideal in ℂ𝔖_{n} generated by the Young symmetrizer

c_{T} = ( ∑_{σ∈R_{T}} sign(σ)σ) ( ∑_{τ∈C_{T}} τ)

where R_{T} (respectively, C_{T}) is the subgroup of 𝔖_{n} preserving the rows (respectively the columns) of T. We will say that the partition λ appears in a representation of 𝔖_{n} if it contains a submodule isomorphic to the right ideal in ℂ𝔖_{n} generated by c_{T}. We say that a partition λ has multiplicity m in U(M) if the c_{T} ℂ𝔖_{n}-isotypic component of U(M) is isomorphic to a direct sum of exactly m copies of c_{T} ℂ𝔖_{n}.

Remark 5.1. Our indexing of the irreducible representations of 𝔖_{n} is the conjugate of the usual indexing. For example, (n) corresponds to the sign representation and (1^{n}) corresponds to the trivial representation.

It follows from the our discussion above that if f : E → V is realization of M in a complex vector space V, and λ appears in

𝔖(f(e_{1}) ⊗ f(e_{2}) ⊗ ⋯ ⊗ f(e_{n}))

then λ appears in ℂ ⊗ U(M) with positive multiplicity.

The following result is equivalent to Gamas’ Theorem on the vanishing of symmetrized tensors (see [1]).

Theorem 5.2 (Berget [1]). Let f : E → V be a realization of a matroid M in a complex vector space V. A partition λ appears in

𝔖(f(e_{1}) ⊗ f(e_{2}) ⊗ ⋯ ⊗ f(e_{n}))

if and only if there is a set partition of E into independent sets whose sizes are the part sizes of λ.

Corollary 5.3. Let M be a matroid realizable over ℂ. The partition λ appears in U(M) if and only if λ ≤ ρ(M) in dominance order.

Proof. One direction follows immediately from the previous theorem and Dias da Silva’s Theorem 3.6. It remains to prove the converse. The image of an antisymmetrizer

∑_{σ∈𝔖_{k}} sign σ ∈ ℂ𝔖_{n},

where k ≤ n, on (e_{1}|p_{1})(e_{2}|p_{2}) … (e_{n}|p_{n}) is

(e_{1}e_{2} … e_{k}|p_{1}p_{2} … p_{k})(e_{k+1}|p_{k+1}) … (e_{n}|p_{n}).

It follows that if some row of T indexes a dependent set of M then c_{T} applied to (e_{1}|p_{1})(e_{2}|p_{2}) … (e_{n}|p_{n}) is zero. Since the projector of an 𝔖_{n}-module to its λ-th isotypic component is, up to a scalar,

∑_{σ∈𝔖_{n}} σc_{T} σ^{−1}

we conclude that if every tableaux of shape λ has a row indexing a dependent set of M then λ cannot appear in ℂ ⊗ U(M). □

5.2. Multiplicities of Hook Shapes. A hook is a partition with at most one part not equal to one. Let λ^{k} denote the hook whose first part is k, and all other parts are equal to one. In this subsection, we show how the multiplicities of the irreducible 𝔖_{n} representations indexed by hook shapes are related to the no broken circuit complex of M. The results of this subsection hold for any matroid, regardless of realizability.

To ease notation for the rest of this section, we assume that the ground set of M is E = {1, 2, …, n}. We define a broken circuit of M as a circuit with its smallest element deleted. A subset of the ground set of M is said to be nbc if it contains no broken circuits of M. The collection of nbc sets of M is a simplicial complex called the nbc complex of M.

Theorem 5.4. The multiplicity λ^{k} in U(M) is the number of nbc sets of M of size k which contain the ground set element 1.

Example 5.5. Let M be the matroid realizable over ℂ be the columns of the matrix

[\begin{matrix}1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{matrix}]

Label the columns 1, 2, …, 6, left to right. The circuits of size three of M are 124, 136, 235, where ijk denotes {i, j, k}, hence the broken circuits of M are 24, 36, 35. It follows that the no broken circuit sets of size 3 are

123, 125, 126, 134, 145, 146, 156

and so the multiplicity of λ^{3} in ℂ ⊗ U(M) is 7. For any ordering of the ground set of M, the smallest element is in at most two dependent sets of size 3. It follows that for any ordering of the ground set, the number of standard Young tableaux of shape λ^{3} whose first row indexes an independent set of M is at least 8.

Remark 5.6. Even for matroids realizable over ℂ the standard basis theorem of Theorem 4.4 does not hold. Indeed the previous example proves that the obvious spanning set of hook shaped tableaux tab(T_{r}|T_{c}) ∈ W (M) where T_{r} has independent rows cannot be linearly independent.

The nbc sets of M with size k that contain 1 are precisely the nbc bases of the truncation of M to rank k. By the proof of Lemma 4.7 it is sufficient to prove the result when k is equal to the rank of M.

Definition 5.7. If D is a subset of [n] we let b_{D} ∈ ℂ𝔖_{n} denote the antisymmetrizer of the set D, i.e., b_{D} = ∑_{σ∈𝔖_{D}} sign(σ)σ.

For a given set B ⊂ [n] we let c_{B} denote the Young symmetrizer of the tableaux of shape λ^{|B|} that has the elements of B in its first row and the remaining elements [n]−B in the rows rows.

It follows directly from the definition of the Young symmetrizer that b_{B} is a left factor of c_{B}.

Proposition 5.8. Let ⟨ − ⟩ denote taking the right ideal in ℂ𝔖_{n} generated by the elements −. There is an isomorphism of ℂ𝔖_{n}-modules

U(M) ≈ ℂ𝔖_{n}/⟨b_{D} : D indexes a dependent set of M⟩

induced by the map that sends (e_{1}|p_{1})(e_{2}|p_{2}) … (e_{n}|p_{n}) to the image of 1 in the quotient.

Proof. The map that sends 1 ∈ ℂ𝔖_{n} to (e_{1}|p_{1})(e_{2}|p_{2}) … (e_{n}|p_{n}) ∈ Super(L^{−}|P^{+}) is an isomorphism onto its image. One then verifies that, up to a sign, the image of the antisymmetrizer b_{D} is the tableaux tab(T |S) of hook shape where the first row of T and S consists of the numbers in D. □

By taking the λ^{r(M)}-th isotypic component of the quotient in the above proposition we immediately have the following result.

Corollary 5.9. Let ⟨−⟩ denote taking the right ideal in ℂ𝔖_{n} generated by the elements −. There is an ℂ𝔖_{n}-isomorphism between the λ^{r(M)}-th isotypic component of ℂ⊗U(M) and the quotient

⟨c_{B} : B is any set of size r(M)⟩/⟨c_{D} : D is a dependent set of M⟩

Lemma 5.10. Let S be any set of [n], from which we form the Young symmetrizer of the tableaux of shape λ^{|S|} that has the elements of S in its first row. In ℂ𝔖_{n} we have the equality

c_{S} = ∑_{f∈S} c_{S∪e−f}(ef)

where e is any element not in S.

Proof. We have c_{S∪e−f}(ef) = (ef)c_{S} in ℂ𝔖_{n}. It follows that we can write this statement as

(1 − ∑_{e∈S}(ef)) c_{S} = 0

We can write this as

b_{S∪e}c_{S}/|S|!

where b_{S∪e} is the antisymmetrizer of the set S ∪ e, since the antisymmetrizer is near idempotent. It is well known that every irreducible that appears in the left or right ideal generated by b_{S∪e} is larger than λ^{|S|+1} in dominance order. Since c_{S} generates an irreducible of shape λ^{|S|} < λ^{|S|+1}, the product of the two elements must be zero. □

Corollary 5.11. The quotient

⟨c_{B} : B is any set of size r(M)⟩/⟨c_{D} : D is a dependent set of M⟩

is generated by the Young symmetrizers of the nbc bases of M.

Proof. Inducting on the number broken circuits contained in a given base, the proof follows at once from the lemma. □

To prove the remainder of the theorem, we will show that the ideal generated by the Young symmetrizers of nbc bases does not meet the ideal generated by the Young symmetrizers of dependent sets. That is, we prove that

⟨c_{B} : B is an nbc basis of M⟩ ∩ ⟨c_{D} : D is a dependent set of M⟩ = 0.

This will be done by straightening the latter Young symmetrizers into sums of Young symmetrizers of standard Young tableaux, which in turn is accomplished by a series of somewhat tedious reductions. In the end, the proof comes down to the well known fact that the right ideals generated by Young symmetrizers of standard Young tableaux have intersection equal to zero.

For the rest of this section D will denote a dependent set of M.

Claim 5.12. We have

⟨c_{D} : D contains two circuits⟩ ⊂ ⟨c_{D} : 1 ∈ D⟩.

Proof. If 1 ∉ D and D contains more than one circuit then each of the sets D − e ∪ 1 is dependent since D − e is. We have c_{D} = ∑_{e∈D} c_{D−e∪1}(1e) which proves the result. □

The remainder of the proof is adapted from Las Vergnas and Forge [8]. We call a set unicyclic if it contains a unique circuit. Using the circuit elimination axioms it can be shown that D is is unicyclic if and only if it contains an element e such that D − e is independent. The proof of the following claim is exactly the same as the proof of the previous one.

Claim 5.13. Let cl(D) denote the closure of D in M. We have the inclusion,

⟨c_{D} : D unicyclic, 1 ∈ cl(D)⟩ ⊂ ⟨c_{D} : 1 ∈ D dependent⟩.

For the unicyclic sets where 1 ∉ cl(D) note that we can write D = I ∪ e, where I is independent and e is the smallest element of the unique circuit of D. For a general independent set I, though, it is possible to choose many elements e such that e is the minimum element of a circuit of I ∪ e. The external activity of an independent set I, denoted ex(I), is the number of elements e such that I ∪ e contains a unique circuit and e is the minimum element of that circuit. Let Ex(I) denote the set of elements e such that e is the minimum element of a circuit of I ∪ e.

Claim 5.14. We have the inclusion,

⟨c_{D} : 1 ∉ cl(D), D unicyclic⟩ ⊂ ⟨c_{D} : 1 ∈ D dependent⟩ + ⟨c_{I∪Ex(I)} : ex(I) = 1, 1 ∉ cl(I)⟩ Proof. Let I ∪ e be a unicyclic set that has ex(I) > 1 but 1 ∉ Ex(I). Then there is an element f ∈ Ex(I) − e. We have

c_{I∪e} = ∑_{g∈I} c_{(I−g∪f)∪e}(fg)

We see that for all g ∈ I, (I − g ∪ f) ∪ e is unicyclic, does not contain 1 in its closure and is lexicographically smaller than I ∪ e. Assuming inductively that c_{(I−g∪f)∪e} is in the ideal

⟨c_{D} : 1 ∈ D dependent⟩ + ⟨c_{I∪Ex(I)} : ex(I) = 1, 1 ∉ cl(I)⟩

we have that c_{I∪e} is in this ideal too. □

We now straighten the generators of the ideal

⟨c_{I∪Ex(I)} : ex(I) = 1, 1 ∉ cl(I)⟩.

If I is independent of rank r(M) − 1, has external activity equal to one but does not contain 1 in its closure then I ∪ 1 is a broken circuit base. However, for all elements g ∈ I, (I − g ∪ 1) ∪ Ex(I) is a no broken circuit base of M. Since a Young symmetrizer c_{S}, |S| = r(M) is that of a standard Young tableau if and only if 1 ∈ S, we have proved

⟨c_{B} : B an nbc base⟩ ∩ ⟨c_{D} : D dependent⟩ = ⟨c_{B} : B an nbc base⟩ ∩ ⟨c_{I∪ex(I)} : ex(I) = 1, 1 ∉ cl(I)⟩.

Finally, every Young symmetrizer on the last ideal has support on a unique broken circuit base containing 1, so this intersection must be zero. This completes the proof of the theorem.

References

[1] Andrew Berget. A short proof of Gamas’s theorem. Linear Algebra Appl., 430(2-3):791–794, 2009.

[2] F. Bonetti, D. Senato, and A. Venezia. The Robinson-Schensted correspondence for the fourfold algebra. Boll. Un. Mat. Ital. B (7), 2(3):541–554, 1988.

[3] A. Brini. Combinatorics, superalgebras, invariant theory and representation theory. Sém. Lothar. Combin., 55:Art. B55g, 117, 2005/07.

[4] Andrea Brini. Private communication, 2009.

[5] Henry Crapo. Private communication, 2009.

[6] Henry Crapo and William Schmitt. The Whitney algebra of a matroid. J. Combin. Theory Ser. A, 91(1-2):215–263, 2000. In memory of Gian-Carlo Rota.

[7] J. A. Dias da Silva. On the μ-colorings of a matroid. Linear and Multilinear Algebra, 27(1):25–32, 1990.

[8] David Forge and Michel Las Vergnas. Orlik-Solomon type algebras. European J. Combin., 22(5):699–704, 2001. Combinatorial geometries (Luminy, 1999).

[9] Frank D. Grosshans, Gian-Carlo Rota, and Joel A. Stein. Invariant theory and superalgebras, volume 69 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1987.

[10] Roberto La Scala, Vincenzo Nardozza, and Domenico Senato. Super RSK-algorithms and super plactic monoid. Internat. J. Algebra Comput., 16(2):377–396, 2006.

[11] Neil White, editor. Theory of matroids, volume 26 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1986.

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 E-mail address: berget@math.ucdavis.edu