key: cord-0533710-ydvbt97k
authors: Kim, Jongwon; Rhoades, Brendon
title: Lefschetz theory for exterior algebras and fermionic diagonal coinvariants
date: 2020-03-23
journal: nan
DOI: nan
sha: 94b5078e620ed4010e479ee3cac8e716a8d152e8
doc_id: 533710
cord_uid: ydvbt97k

Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $wedge (V oplus V^*)$ as well as its quotient $DR_W := wedge (V oplus V^*)/ langle wedge (V oplus V^*)^{W}_+ rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $mathfrak{S}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $wedge (V oplus V^*)$.

Let C[x 1 , . . . , x n , y 1 , . . . , y n ] be a polynomial ring in 2n variables equipped with the diagonal action of the symmetric group S n :

(1. 1) w.x i := x w(i) and w.y i := y w(i) for all w ∈ S n and 1 ≤ i ≤ n. The quotient of C[x 1 , . . . , x n , y 1 , . . . , y n ] by the ideal generated by the homogeneous S n -invariants of positive degree is the diagonal coinvariant ring; its bigraded S n -structure was calculated by Haiman [10] using algebraic geometry. In the last couple years, algebraic combinatorialists have studied variations of the diagonal coinvariants involving sets of commuting and anti-commuting variables [3, 5, 8, 14, 16, 18, 19] . In this paper we completely describe the bigraded S n -structure of the diagonal coinvariants involving two sets of anti-commuting variables (but no commuting variables). Our methods apply equally well (and uniformly) to any irreducible complex reflection group W 1 as to the symmetric group S n .

Let W be an irreducible complex reflection group of rank n acting on its reflection representation V ∼ = C n . The action of W on V induces an action of W on

• the dual space V * = Hom C (V, C),

• the direct sum V ⊕ V * of V with its dual space, and finally • the exterior algebra ∧(V ⊕ V * ) over the 2n-dimensional vector space V ⊕ V * . By placing V in bidegree (1, 0) and V * in bidegree (0, 1), this last space ∧(V ⊕ V * ) attains the structure of a doubly graded W -module.

If we let Θ n = (θ 1 , . . . , θ n ) be a basis for V and Ξ n = (ξ 1 , . . . , ξ n ) be a basis for V * , we have a natural identification (1.2) ∧ (V ⊕ V * ) = ∧{Θ n , Ξ n } of ∧(V ⊕ V * ) with the exterior algebra ∧{Θ n , Ξ n } generated by the symbols θ i and ξ i over C.

Following the terminology of physics, we refer to the θ i and ξ i as fermionic variables. In physics, such variables are used to model fermions, with relations θ 2 i = ξ 2 i = 0 corresponding to the Pauli Exclusion Principle: no two fermions can occupy the same state at the same time 2 . The model ∧{Θ n , Ξ n } for ∧(V ⊗ V * ) will be helpful in our arguments. The following quotient ring is our object of study. The ideal ∧(V ⊕ V * ) W + is W -stable and bihomogeneous, so the quotient ring DR W has the structure of a bigraded W -module. We will see (Proposition 4.1) that this ideal is principal, generated by a 'Casimir element' δ W ∈ V ⊕ V * . Our results are as follows.

• We describe the bigraded W -isomorphism type of DR W in terms of the isomorphism types of the exterior powers ∧ i V and ∧ j V * (Theorem 4.2). • We show that dim DR W = 2n+1 n whenever W has rank n and relate the dimensions of its graded pieces to Catalan and Narayana numbers (Corollaries 4.3 and 4.4).

• We describe an explicit monomial basis of DR W using a variant of Motzkin paths (Theorem 5.2) and describe the bigraded Hilbert series of DR W in terms of the combinatorics of these paths (Corollary 5.4). • When W = S n , in Section 6 we give variants of the above results as they apply to the n-dimensional permutation representation of S n (as opposed to its (n − 1)-dimensional reflection representation).

The key tool in our analysis is the realization (Theorem 3.2) of the Casimir generator δ W of the ideal defining DR W as a kind of 'W -invariant Lefschetz element' in the ring ∧(V ⊕ V * ). The ring ∧(V ⊕ V * ), similar to the cohomology ring of a compact smooth complex manifold, satisfies 'bigraded' versions of Poincaré Duality and the Hard Lefschetz Theorem. This is somewhat unusual on two counts.

• Any homogeneous linear form in an exterior algebra squares to zero, and hence is not well-suited to be a (strong) Lefschetz element. [16] . The remainder of the paper is organized as follows. In Section 2 we give background material on complex reflection groups, Gröbner theory associated to exterior algebras, and the representation theory of S n . In Section 3 we prove that ∧(V ⊕ V * ) satisfies bigraded versions of the Hard Lefschetz Property and Poincaré Duality. This builds on work of Hara and Watanabe [11] showing that the incidence matrix between complementary ranks of the Boolean poset B(n) is invertible. In Section 4 we apply these Lefschetz results to determine the bigraded W -structure of DR W . In Section 5 we describe the standard monomial basis of DR W using lattice paths. In Section 6 we specialize to W = S n and translate our results to the setting of the permutation representation of S n . We close in Section 7 with some open problems.

2.1. Complex reflection groups. Let V = C n be an n-dimensional complex vector space. An element t ∈ GL(V ) = GL n (C) is a reflection if its fixed space

If W 1 and W 2 are reflection groups with reflection representations V 1 and V 2 , the direct product W 1 × W 2 is naturally a reflection group with reflection representation

2.2. Exterior Gröbner theory. Let Θ n = (θ 1 , . . . , θ n ) be a list on n anticommuting variables and let ∧{Θ n } be the exterior algebra generated by these variables over C. For any subset S = {i 1 < i 2 < · · · < i k } ⊆ {1, 2, . . . , n}, we let

where the multiplication is in increasing order of subscripts. We refer to the θ S as monomials; the set {θ S : S ⊆ {1, 2, . . . , n}} is the monomial basis of ∧{Θ n }. Given two monomials θ S and θ T , we write θ S | θ T to mean S ⊆ T . A total order < on the set {θ S : S ⊆ {1, 2, . . . , n}} is a term order if • we have 1 = θ ∅ ≤ θ S for all S and • for all subsets S, T, U with U ∩ S = U ∩ T = ∅, θ S < θ T implies θ S∪U < θ T ∪U . Given a term order <, for any nonzero element f ∈ ∧{Θ n }, let lm(f ) be the largest monomial θ S under the total order < such that θ S appears with nonzero coefficient in f . If I ⊆ ∧{Θ n } is a two-sided ideal, let The set N (I) of monomials descends to a C-basis of the quotient ∧{Θ n }/I; this is the standard monomial basis with respect to < (see for example [4] ).

2.3. Representation Theory. If V = i,j≥0 V i,j is a bigraded vector space with each piece V i,j finite-dimensional, the bigraded Hilbert series is Hilb(V ; q, t) := i,j≥0 dim V i,j · q i t j . This is a formal power series in q and t.

The irreducible representations of the symmetric group S n are in one-to-one correspondence with partitions λ ⊢ n. Given λ ⊢ n, let S λ be the corresponding S n -irreducible. For example, the trivial representation is S (n) and the sign representation is S (1 n ) .

Let Λ denote the ring of symmetric functions and let {s λ : λ a partition} denote its Schur basis. The Hall inner product on Λ declares the Schur basis to be orthonormal:

for any partitions λ and µ. Any finite-dimensional S n -module U may be expressed uniquely as a direct sum U ∼ = λ⊢n c λ S λ for some multiplicities c λ . The Frobenius image of U is the symmetric function where s λ is the Schur function. The Kronecker product of two Schur functions s λ and s µ for λ, µ ⊢ n is defined by

where S n acts diagonally on S λ ⊗ S µ .

This is a formal power series in q and t with coefficients in the ring of symmetric functions.

Let A = n i=0 A i be a commutative graded C-algebra. The algebra A satisfies Poincaré Duality (PD) if A n ∼ = C and if the multiplication map A i ⊗A n−i → A n ∼ = C is a perfect pairing for 0 ≤ i ≤ n. In particular, this implies that the Hilbert series of A is palindromic, i.e. dim A i = dim A n−i .

If A = n i=0 A i satisifes PD, an element ℓ ∈ A 1 is called a (strong) Lefschetz element if for every 0 ≤ i ≤ n/2, the linear map

given by multiplication by ℓ n−2i is bijective. If A has a Lefschetz element, it is said to satisfy the Hard Lefschetz Property (HLP). Algebras A which satisfy PD and the HLP arise naturally in geometry as the cohomology rings (with adjusted grading) of smooth complex projective varieties. The HLP of the cohomology ring of the n-fold product P 1 × · · · × P 1 of 1-dimensional complex projective space with itself was studied combinatorially by Hara and Watanabe [11] ; we state their results below.

Recall that the Boolean poset B(n) is the partial order on all subsets S ⊆ {1, . . . , n} given by S ≤ T if and only if S ⊆ T . The poset B(n) is graded, with the i th rank given by the family B(n) i of i-element subsets of {1, . . . , n}. Hara and Watanabe proved that the incidence matrix between complementary ranks of B(n) is invertible. [11] ) Given r ≤ s ≤ n, define a n s × n r matrix M n (r, s) with rows indexed by B(n) s and columns indexed by B(n) r with entires

For example, if n = 4 and i = 1, Theorem 3.1 asserts that the 0, 1-matrix M 4 (1, 3) given by

The cohomology ring of the n-fold product P 1 × · · · × P 1 may be presented as

n where x i represents the Chern class c 1 (L i ) and L i is the dual of the tautological line bundle over the i th factor of P 1 × · · · × P 1 . Hara and Watanabe used Theorem 3.1 to give a combinatorial proof of the fact that x 1 + · · · + x n is a Lefschetz element for the ring H • (P 1 × · · · × P 1 ; C) [11] . We want to study PD and the HLP in the context of the exterior algebra ∧{Θ n , Ξ n }. This algebra satisfies a natural bigraded version of Poincaré Duality: the top bidegree ∧{Θ n , Ξ n } n,n is 1-dimensional and the multiplication map

The notion of a Lefschetz element in ∧{Θ n , Ξ n } is a bit more subtle because any linear form ℓ in the variables θ 1 , . . . , θ n , ξ 1 , . . . , ξ n satisfies ℓ 2 = 0. To get around this, we introduce the element

The following result states that δ n is a bigraded version of a Lefschetz element for the ring ∧{Θ n , Ξ n }.

given by multiplication by δ r n is bijective. Proof. The idea is to introduce strategically chosen bases of the domain and target of ϕ and show that the matrix representing ϕ with respect to these bases is invertible using Theorem 3.1.

Given two subsets A, B ⊆ {1, . . . , n}, write

The family {v(A, B) : A, B ⊆ {1, . . . , n}} is a basis of ∧{Θ n , Ξ n }. For any sets A and B, a direct computation shows

The somewhat unusual variable order in the product v(A, B) was chosen strategically so that Equation (3.8) does not contain any signs. Now suppose |A| = i and |B| = j for i + j ≤ n and set r = n − i − j. Iterating Equation (3.8) yields

We need to show that the (square) matrix of dimensions n i · n j × n n−j · n n−i defined by the system (3.9) is invertible. For any v(C, D) appearing in on the right-hand side of (3.9) we have A − B = C − D =: I and B − A = D − C =: J. The matrix representing ϕ therefore breaks up as a direct sum of smaller matrices indexed by the two sets I and J, so we only need to show that every (I, J)-submatrix is invertible.

For fixed I and J, the submatrix in the previous paragraph is determined by the system

The system (3.10) represents a linear map 

representing a linear map

Observe that

Let W be an irreducible reflection group with reflection representation V = C n . Let θ 1 , . . . , θ n be a basis of V . Given the choice of θ 1 , . . . , θ n , we let ξ 1 , . . . , ξ n be the dual basis of V * characterized by

We rename the element δ n = θ 1 ξ 1 + · · · + θ n ξ n of ∧(V ⊕ V * ) studied in the previous section as δ W :

We refer to δ W as the Casimir element of W . The full general linear group GL(V ) acts on ∧(V ⊕ V * ) and it is not difficult to check using elementary matrices and the dual basis property that δ W is invariant under this action. Equivalently, the Casimir element δ W is independent of the choice of basis θ 1 , . . . , θ n . In particular, the element

Proof. Let G be a finite group and let U, U ′ be irreducible complex representations of G. The tensor product U ⊗ U ′ is a G-module by the rule g.(u ⊗ u ′ ) := (g.u) ⊗ (g.u ′ ) for g ∈ G, u ∈ U, and u ′ ∈ U ′ . (This is the Kronecker product of the modules U and U ′ .) Character orthogonality implies

Since W is an irreducible complex reflection group, a result of Steinberg (see [12, Thm. A, §24-3, p. 250]) implies that the exterior powers ∧ 0 V, ∧ 1 V, . . . , ∧ n V are pairwise nonisomorphic irreducible representations of W . The same is true of their duals ∧ 0 V * , ∧ 1 V * , . . . , ∧ n V * . Since the (i, j)-bidegree of ∧(V ⊕ V * ) is given by ∧(V ⊕ V * ) i,j = ∧ i V ⊗ ∧ j V * , the argument of the last paragraph gives

On the other hand, we have δ W ∈ ∧(V ⊕ V * ) W 1,1 and a quick computation shows that the powers δ 0 W , δ 1 W , . . . , δ n W are nonzero. The proposition follows.

We are ready to describe the bigraded W -module structure of DR W . We state our answer in terms of the Grothendieck ring of W . Recall that this is the Z-algebra generated by the symbols 

Proof. Thanks to Proposition 4.1 we can model DR W as

If i = 0 or j = 0, the claim follows since δ W lies in bidegree (1, 1), so assume i, j > 0. If i + j ≤ n, let r = n − i − j + 1. Theorem 3.2 implies that the multiplication map

is a linear isomorphism. Whenever a composition f • g of maps is a bijection, the map g is injective and the map f is surjective. Therefore, the map

is an injection which we know to be W -equivariant. The claimed decomposition of [(DR W ) i,j ] follows from (4.8). Now suppose i + j > n. By analogous reasoning

is a W -equivariant surjection so that (DR W ) i,j = 0. dim(DR W ) k,n−k = Nar(n + 1, k + 1)

for 0 ≤ k ≤ n so that For any reflection group W , there are Catalan and Narayana numbers attached to W (see for example [2] ); the numbers appearing in Corollary 4.4 are their type A instances, and depend only on the rank of W .

Remark 4.5. The results in this section and the next apply equally well to any finite group G and any n-dimensional G-module V for which the exterior powers ∧ 0 V, ∧ 1 V, . . . , ∧ n V are pairwise nonisomorphic irreducibles. The proofs go through mutatis mutandis.

In this section we describe the standard monomial basis of DR W (with respect to a term order ≺ which we will define) in terms of a certain family of lattice paths. A Motzkin path is a lattice path in Z 2 consisting of up-steps (1, 1), down-steps (1, −1), and horizontal steps (1, 0) which starts at the origin, ends on the x-axis, and never sinks below the x-axis. We consider a variant of Motzkin paths which have decorated horizontal steps and need not end on the x-axis.

Let Π(n) be the family of n-step lattice paths σ = (s 1 , . . . , s n ) in Z 2 which start at the origin and consist of up-steps (1, 1), down-steps (1, −1), and horizontal steps (1, 0) in which each horizontal step is decorated with a θ or a ξ. We let Π(n) ≥0 ⊆ Π(n) be the family of paths which never sink below the x-axis. Two paths in Π(9) are shown in Figure 1 ; the top path lies in Π(9) ≥0 but the bottom path does not. of the steps of σ in the order in which they appear. For the paths σ and µ in Figure 1 we have (5.4) wt(σ) = θ 3 · θ 4 ξ 4 · θ 5 ξ 5 · θ 6 · θ 9 ξ 9 and wt(µ) = θ 2 ξ 2 · θ 3 ξ 3 · θ 4 ξ 4 · ξ 6 · θ 9 .

A moment's thought shows that σ → wt(σ) gives a bijection from Π(n) to the set of monomials in ∧{Θ n , Ξ n }, where monomials with differing signs are considered equivalent. We will identify paths σ with their monomials wt(σ).

The (total) degree of a path σ is This is simply the total number of exterior generators θ i and ξ i appearing in the monomial σ. We define the θ-degree deg θ (σ) and ξ-degree deg ξ (σ) analogously. Combinatorially, If σ and µ are as in Figure 1 then

We introduce the total order ≺ on paths σ ∈ Π(n), or on monomials in ∧{Θ n , Ξ n } given by

or deg(σ) = deg(σ ′ ) and d(σ) = d(σ ′ ) and σ < lex σ ′ where in the last branch < lex means the lexicographical order on the paths σ = (s 1 , . . . , s n ) and σ ′ = (s ′ 1 , . . . , s ′ n ) induced by declaring the step order (5.10)

(1, 1) < (1, 0) with θ-decoration < (1, 0) with ξ-decoration < (1, −1).

The collection of paths/monomials with a given bidegree (i, j) form a subinterval of ≺ for all 0 ≤ i, j ≤ n. In our running example of Figure 1 , we have deg(σ) = deg(µ) but d(σ) > d(µ) so that σ ≺ µ.

Lemma 5.1. The total order ≺ is a term order for ∧{Θ n , Ξ n }.

Proof. The first branch of the definition of ≺ guarantees that the monomial 1 with path consisting of a sequence of n up-steps is the minimum monomial under ≺. Checking that ≺ respects multiplication amounts to the observation that total degree, depth, and lexicographical order are all respected by multiplication.

It turns out that the set {wt(σ) : σ ∈ Π(n) ≥0 } descends to a C-basis of DR W . In fact, we prove something stronger. Proof. Let I n = δ n ⊆ ∧{Θ n , Ξ n } be the defining ideal of DR W (here we apply Proposition 4.1). Identifying paths with monomials, we want to show N (I n ) = Π(n) ≥0 with respect to ≺. We proceed by induction on n, with the base case n = 1 being immediate.

Suppose n > 1 and σ = (s 1 , . . . , s n ) ∈ Π(n) − Π(n) ≥0 . In particular, we have d(σ) < 0. The following lemma will show inductively that σ / ∈ N (I n ). Proof. (of Lemma 5.3) Let σ 0 ∈ ∧{Θ n−1 , Ξ n−1 } be the monomial σ with its last step s n removed. The proof breaks into cases depending on the step s n . Case 1: The last step s n is a horizontal step (of either decoration θ or ξ).

We assume the decoration of s n is θ; the other case is similar. In this case, we have σ 0 ∈ Π(n − 1) − Π(n − 1) ≥0 and σ = σ 0 θ n . We may inductively assume that σ 0 ∈ lm(I n−1 ) so that σ 0 = lm(f · δ n−1 ) for some polynomial f ∈ ∧{Θ n−1 , Ξ n−1 }. Since (5.11) f · δ n · θ n = f · δ n−1 θ n + f · θ n ξ n · θ n = f · δ n−1 · θ n ,

we conclude that f · δ n−1 · θ n ∈ I n . We have

completing the proof of Case 1. Case 2: The last step s n is a down-step (1, −1). If σ 0 ∈ Π(n − 1) − Π(n − 1) ≥0 sinks below the x-axis, the proof is similar to that of Case 1. One right-multiplies f · δ n by θ n ξ n instead of θ n ; we leave the details to the reader.

In this case we could have σ 0 ∈ Π(n − 1) ≥0 , but this would imply that σ 0 ends on the x-axis, so that deg(σ) = deg(σ 0 ) + 2 = (n − 1) + 2 = n + 1. Theorem 4.2 then forces σ = 0 in the quotient DR W , completing the proof of Case 2.

Case 3: The last step s n is an up-step (1, 1) . This is the most involved case. We have σ 0 = σ and σ 0 ∈ Π(n − 1) − Π(n − 1) ≥0 . By induction, we may assume that there is f ∈ ∧{Θ n−1 , Ξ n−1 } with σ 0 = lm(f · δ n−1 ). Now consider (5.13) f · δ n = f · δ n−1 + f · θ n ξ n ∈ I n .

By discarding redundant terms if necessary, we may assume that f is bi-homogeneous. The monomial σ = σ 0 is the ≺-largest monomial appearing in f · δ n−1 . Since σ does not involve θ n or ξ n , it does not appear in f · θ n ξ n . We will have σ = lm(f · δ n ) unless some monomial µ appearing in f · θ n ξ n satisfies µ ≻ σ.

Let µ be the ≺-largest element of f · θ n ξ n and assume σ ≺ µ. Let µ 0 ∈ Π(n − 1) be the path obtained from µ by removing its last step (which is necessarily a down-step since µ appears in f · θ n ξ n ). Since σ ≺ µ, the bihomogeneity of f forces d(µ) ≤ d(σ) < 0. Subcase 3.1: We have µ 0 ∈ Π(n − 1) ≥0 , or equivalently d(µ 0 ) ≥ 0.

Since d(µ) = d(µ 0 ) + 1 < 0, this can only happen if d(µ 0 ) = 0 and µ 0 ends at the lattice points (n − 1, 0). This implies that deg(µ) = deg(µ 0 ) + 2 = (n − 1) + 2 = n + 1 and Theorem 4.2 forces µ ∈ I n . We may therefore discard the term involving µ from (5.13) and still have an element of I n involving σ.

In this case, we induct on n to obtain some polynomial g ∈ ∧{Θ n−1 , Ξ n−1 } whose leading monomial is µ 0 = lm(g · δ n−1 ). We calculate (5.14) lm(g · δ n · θ n ξ n ) = lm(g · δ n−1 · θ n ξ n ) = lm(g · δ n−1 ) · θ n ξ n = µ 0 · θ n ξ n = µ where the second equality used the fact that g · δ n−1 does not involve θ n or ξ n . Since σ does not involve θ n or ξ n , it does not appear in g · δ n · θ n ξ n . We may therefore replace (5.13) by (5.15) f · δ n−1 + (f − g · δ n−1 ) · θ n ξ n ∈ I n to obtain another element of I n which involves σ only in its first term, still satisfies σ = lm(f ·δ n−1 ), but now only involves monomials ≺ µ.

Iterating the arguments of Subcases 3.1 and 3.2, we see that σ ∈ lm(I n ), proving both Case 3 and the lemma.

We complete the proof of Theorem 5.2 using Lemma 5.3. Lemma 5.3 implies N (I n ) ⊆ Π(n) ≥0 , and to force equality it suffices to verify

In fact, we verify the equality of bigraded Hilbert series

If we let Π(n) =0 ⊆ Π(n) ≥0 be the subset of paths that end on the x-axis and let

considering the addition of one more step to a path yields (5.19) P n+1 (q, t) = (1 + q + t + qt) · P n (q, t) − (qt) · P ′ n (q, t).

On the other hand (adopting the notation DR W (n) for DR W whenever W has rank n) Theorem 4.2 yields

It can be shown using the Pascal identity and Equation (5.20) that

which matches the combinatorial recursion in Equation (5.19 ).

In the course of proving Theorem 5.2, we derived the following combinatorial expression for the bigraded Hilbert series of DR W .

Corollary 5.4. If W has rank n, we have

In the coinvariant theory of the symmetric group S n , it is more common to consider its ndimensional permutation representation U as opposed to its (n − 1)-dimensional reflection representation V In this section we describe how to translate our results into this setting.

The following decompositions of U and U * into S n -irreducibles are well-known:

It follows that

Modding out by ideals generated by S n -invariants with vanishing constant term, we see that

Let S n act on ∧{Θ n , Ξ n } diagonally, viz. w.θ i := θ w(i) and w.ξ i := ξ w(i) . Expressing the left-hand side of (6.5) in terms of coordinates, we have the following translation of Theorem 4.2, Corollary 4.3, and Corollary 4.3.

Theorem 6.1. Let DR n be the bigraded S n -module (6.6) DR n := ∧{Θ n , Ξ n }/ ∧{Θ n , Ξ n } Sn + . We have (DR n ) i,j = 0 whenever i + j ≥ n. If i + j < n, we have

where * denotes Kronecker product. Here we interpret s (n+1,−1) = 0. We have (6.8) dim DR n = 2n − 1 n and, for 1 ≤ k ≤ n, we have (6.9) dim(DR n ) k−1,n−k = Nar(n, k) so that n k=1 dim(DR n ) k−1,n−k = Cat(n).

Equation (6.8) was conjectured by Mike Zabrocki [20] for the Open Problems in Algebraic Combinatorics 2020 Conference 3 . We also have a lattice path basis of the S n -module DR n in Theorem 6.1. For a partition λ ⊢ n, work of Rosas [17] implies that (6.10) grFrob(DR n ; q, t), s λ = 0 unless the partition λ = (λ 1 ≥ λ 2 ≥ λ 3 ≥ · · · ) satisfies λ 3 ≤ 2 (i.e. the Young diagram of λ is a union of two possibly empty hooks). While these multiplicities can be less than aesthetic in general, they are nice when λ is a hook. Recall that the q, t-analog of n is given by (6.11) [n] q,t := q n − t n q − t = q n−1 + q n−2 t + · · · + qt n−2 + t n−1 .

Proposition 6.2. The graded multiplicities of the trivial and sign representations in DR n are given by (6.12) grFrob(DR n ; q, t), s (n) = 1 and grFrob(DR n ; q, t), s (1 n ) = [n] q,t .

If 0 < k < n − 1 we have

Proof. The equation grFrob(DR n ; q, t), s (n) = 1 is immediate since DR n is obtained from ∧{Θ n , Ξ n } by modding out by S n -invariants with vanishing constant term. The multiplicity of the signed representation follows from Theorem 6.1 and the fact that for any partitions λ, µ ⊢ n (6.14)

where λ ′ is the conjugate (transpose) partition of λ. We turn our attention to Equation (6.13). For any statement P , let χ(P ) = 1 if P is true and χ(P ) = 0 if P is false. Rosas proves [17, Proof of Thm. 13 (4) ] that the multiplicity of the Schur function s (n−c,1 c ) in the Kronecker product s (n−a,1 a ) * s (n−b,1 b ) is

whenever 0 < a, b < n and 0 < c < n − 1.

For any 0 ≤ k ≤ n − 1 and all i + j < n, we have (6.16) Frob(DR n ) i,j , s (n−k,1 k ) = s (n−i,1 i ) * s (n−j,1 j ) , s (n−k,1 k ) − s (n−i+1,1 i−1 ) * s (n−j+1,1 j−1 ) , s (n−k,1 k )

A somewhat tedious casework using Equation (6.15) yields

which is equivalent to Equation (6.13).

In order to state a DR n -analog of Theorem 5.2, we need some notation. We define the primed weight wt ′ (s) of a step s to be

The primed weight wt ′ (σ) of a path σ ∈ Π(n) with steps σ = (s 1 , . . . , s n ) is wt ′ (σ) := wt ′ (s 1 ) · · · wt ′ (s n ). Let Π(n) >0 ⊆ Π(n) consist of those paths which only meet the x-axis at their starting point (0, 0) and stay strictly above the x-axis otherwise.

Theorem 6.3. The set {wt ′ (σ) : σ ∈ Π(n) >0 } descends to a basis of DR n . Consequently, we have

Proof. Proposition 4.1 and the discussion prior to Theorem 6.1 imply that the invariant subalgebra ∧{Θ n , Ξ n } Sn is generated by the three elements θ 1 + · · · + θ n , ξ 1 + · · · + ξ n , and θ 1 ξ 1 + · · · + θ n ξ n and consequently (6.21) DR n = ∧{θ 1 , . . . , θ n , ξ 1 , . . . , ξ n }/ θ 1 + · · · + θ n , ξ 1 + · · · + ξ n , θ 1 ξ 1 + · · · + θ n ξ n .

We express DR n as a successive quotient DR n = ∧{θ 1 , . . . , θ n , ξ 1 , . . . , ξ n }/ θ 1 + · · · + θ n , ξ 1 + · · · + ξ n , θ 1 ξ 1 + · · · + θ n ξ n .

Then as graded vector spaces, we identify θ 1 = −θ 2 − · · · − θ n and ξ 1 = −ξ 2 − · · · − ξ n to obtain DR n ∼ = (∧{θ 2 , . . . , θ n } ⊗ ∧{ξ 2 , . . . , ξ n }) / (−θ 2 − · · · − θ n ) ⊗ (−ξ 2 − · · · − ξ n ) + n i=2 θ i ⊗ ξ i (6.24) = (∧{θ 2 , . . . , θ n } ⊗ ∧{ξ 2 , . . . , ξ n }) / n i=2 θ i ⊗ (ξ i + n j=2 ξ j ) (6.25)

The transition matrix from the set {ξ 2 + n j=2 ξ j , . . . , ξ n + n j=2 ξ j } = {ξ ′ 2 , . . . , ξ ′ n } to the standard basis {ξ 2 , . . . , ξ n } of the degree 1 component of ∧{ξ 2 , . . . , ξ n } is

which is easily checked to be invertible. Therefore, the set {ξ ′ 2 , . . . , ξ ′ n } is also a basis of the degree 1 component of ∧{ξ 2 , . . . , ξ n } and we may write (6.26) DR n ∼ = ∧{θ 2 , . . . , θ n , ξ ′ 2 , . . . , ξ ′ n }/ θ 2 ξ ′ 2 + · · · + θ n ξ ′ n .

Theorem 5.2 applies to complete the proof.

The key result underpinning our analysis of DR W and DR n was the Lefschetz Theorem 3.2. Our proof was combinatorial and ultimately relied on the Boolean poset B(n). Given the importance of Lefschetz elements in geometry, it is natural to ask the following. Modern variants of the HLP and PD were used to great effect in the work of Adiprasito, Huh, and Katz on the Chow rings of matroids [1] . Is there a deeper meaning to the HLP and PD as they apply to exterior algebras? Perhaps the realization of ∧{Θ n , Ξ n } as the holomorphic tangent space to the origin in C n ⊕ C n would be relevant here.

It may also be interesting to consider combining two sets of commuting and anticommuting variables to get a ring (7.1) C[X n , Y n ] ⊗ ∧{Θ n , Ξ n } := C[x 1 , . . . , x n , y 1 , . . . , y n ] ⊗ ∧{θ 1 , . . . , θ n , ξ 1 , . . . , ξ n } which may be identified with the algebra of polynomial-valued holomorphic differential forms on C n ⊕ C n . This ring is quadruply graded, and the diagonal action of S n gives rise to a coinvariant space C[X n , Y n , Θ n , Ξ n ]/ C[X n , Y n , Θ n , Ξ n ] Sn + . Setting the ξ-variables to zero, Zabrocki [19] conjectured that the triply graded Frobenius image of this quotient is given by the Delta Conjecture of Haglund, Remmel, and Wilson [7] . Furthermore, again when the ξ-variables are set to zero, Haglund and Sergel [9] have a conjectural monomial basis of this quotient which would extend a basis of the diagonal coinvariants due to Carlsson and Oblomkov [6] .

Problem 7.2. Find a basis of the quotient C[X n , Y n , Θ n , Ξ n ]/ C[X n , Y n , Θ n , Ξ n ] Sn + which generalizes the basis of C[X n , Y n ]/ C[X n , Y n ] Sn + due to Carlsson-Oblomkov [6] and the conjectural basis of C[X n , Y n , Θ n ]/ C[X n , Y n , Θ n ] Sn + due to Haglund-Sergel [9] .

A solution to Problem 7.2 might be obtained by interpolating between the parking function 'schedules' present in [6, 9] and our Motzkin-like paths Π(n) >0 .

Let X k×n = (x i,j ) 1≤i≤k,1≤j≤n be a k × n matrix of commuting variables and let C[X k×n ] be the polynomial ring in these variables. The ring C[X k×n ] carries a S n -module structure inherited from column permutation and the quotient C[X k×n ]/ C[X k×n ] Sn + is a (Z ≥0 ) k -graded S n -module. When k = 2, we recover the classical diagonal coinvariants. F. Bergeron has many fascinating conjectures about this object obtained by letting the parameter k grow [3] .

We can carry out the contstruction of the previous paragraph with a matrix Θ k×n = (θ i,j ) 1≤i≤k,1≤j≤n of anticommuting variables. We still have an action of S n on columns and can still consider the quotient (7.2) R(k × n) := ∧{Θ k×n }/ ∧{Θ k×n } Sn + . In the case k = 2 we recover DR n . For stability results involving such quotients, and corresponding quotients using both commuting and anticommuting variables, see [15] . Question 7.3. Find the multigraded isomorphism type of R(k × n).

It is unclear how to use Lefschetz Theory to solve Question 7.3 for k > 2. For any set S ⊆ {1, 2, . . . , k} of rows, we have a S n -invariant (7.3) δ S := i∈S θ i,1 + i∈S θ i,2 + · · · + i∈S θ i,n ∈ ∧{Θ k×n } where the products are taken in increasing order of i ∈ S. When |S| is even, this has the potential to be a Lefschetz element, but δ 2 S = 0 when |S| is odd. For |S| = 1, the row sum δ S may be easy to handle, but the situation becomes more complicated as an odd-sized set S grows. Furthermore, one would have to understand how the various images of multiplication by the δ S between bidegrees intersect as S varies.

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