The Column Law and Torsion in Chessboard Complexes

1. Introduction

For a covector ϕ ∈ W^*, let d_ϕ act on (W⊕Ke)^⊗n by replacing an ordinary tensor factor w by the odd marker ϕ(w)e, with the Koszul sign. We study the joint kernel ${\mathcal{Z}}_n^p\left(W\right)$ in marker degree p. In characteristic zero, the S^μ-multiplicity space is ${\mathbb{S}}_{{\mu }^{-}}W$ in degree p = 𝓁(μ) and zero in the other degrees, where μ⁻ is obtained by deleting the first column of μ. This Column Law separates the site combinatorics from the invariant theory of any group acting on W.

The characteristic-free form replaces the decomposition by a ballot condition on leading words. It gives the dimension of every content space and every socle layer over an arbitrary field. In multilinear content, these spaces are top cycles of complete chessboard complexes. Their field-independent dimensions imply that ${\tilde{H}}_{m-2}\left(M_{m,n};\mathbb{Z}\right)$ is free for every 1 ≤ m ≤ n. This includes the two boundary families n = 2m − 4,2m − 3 left open in the bottom-homology torsion criterion of Shareshian and Wachs [27, §§7,11], and proves the torsion-free implication of their higher-degree conjecture [27, Conjecture 7.4].

The Schur–Weyl and Pieri rules and the transposition class-sum scalar are classical [9,12,17,20,23]. Berele–Regev [1] and Sergeev [26] provide the super Schur–Weyl setting; see also Cheng–Wang [4]. Related content factors appear in the Pieri inclusions of Olver [21], Sam–Weyman [24], and Eisenbud–Fløystad–Weyman [6]. Marko [18,19] studies even-primitive vectors, meaning highest vectors for the even subgroup [18, §1.5]. Our joint kernel consists instead of invariants of the odd nilradical and contains an entire GL(W)-module in its highest marker degree.

The group-dependent step is Lemma 2.2: on invariant cochains, a frame whose orbit spans W^* detects every contraction. In degree one this identifies invariant closed cochains with symmetric invariants; higher degrees are controlled by invariant multiplicities of Schur functors.

Organisation. Section 2, Section 3 and Section 4 prove the Column Law in characteristic zero, compute the content spectrum, and treat invariant closed cochains. Section 5 establishes the characteristic-free leading-word and socle formulas. Section 6 determines free generators and their hook multiplicities; Section 7 gives the free-range dimensions and Grassmannian evaluation. Section 8 treats smaller orbit spans and several markers. Section 9 applies the formulas to chessboard and row-bounded rook homology. Section 10 and Section 11 record the computations and open questions.

1.1. Conventions

Unless stated otherwise, K is a field of characteristic zero, W is a finite-dimensional K-vector space, and G acts linearly on W. The group may be finite, algebraic, or a specified subgroup of GL(W); no averaging over G is used. Section 5 and Section 9, the freeness and multigraded dimension statements of Section 6, and Proposition 8.4 hold over every field. Schur-module decompositions and Grassmannian evaluation use characteristic zero, except for the good-characteristic extension in Proposition 3.2. Numerical joint-kernel dimension formulas extend to every field by Theorem 5.5. Schur functors vanish when their partitions have more than dimW rows.

We write 𝓁(λ) for the number of nonzero parts of a partition and θ ∈ W^* for the fixed frame.

Write [n] = {1,…,n}, m = n − p, S^λ for a Specht module, and 𝕊_λW for a Schur functor. We use the English convention for Young diagrams and the content

$$c\left(\lambda \right)=\sum _{\left(i,j\right)\in \lambda }\left(j-i\right).$$

(1.1)

For a set of sites T, write W_T for the tensor product with those site labels. Write C^c for the complement of C in [n]. All tensor slots are labelled by sites. An odd marker has degree one; an element of W has degree zero. Permutations of sites use the resulting Koszul signs.

Table 1 records the spaces and actions used throughout the paper.

The fixed-frame operator is not generally a differential on G-invariants. Accordingly, our theorem concerns closed cochains and a joint kernel; it does not identify the cohomology of an invariant subcomplex.

2. The Site Construction and the Column Law

The site construction keeps the orientation of the marked slots separate from the ordinary tensor factors. Define

$$C_n^p\left(W\right)=\underset{\begin{array}{c}C\subseteq \left[n\right]\\ \left|C\right|=p\end{array}}{⨁}\det \left(K^C\right)\otimes W_{C^c}.$$

(2.1)

The orientation of C = {c₁<…<c_p} is $e_{c_1}\wedge \dots \wedge e_{c_p}$. Equivalently, (2.1) is the component with p odd markers of (W⊕Ke)^⊗n. For θ ∈ W^*, the map $d_{\mathsf{\theta }}:C_n^p\left(W\right)\to C_n^{p+1}\left(W\right)$ replaces a slot w_j by e_j, multiplies by θ(w_j), and attaches the sign

(−1)^{|{i∈C:i<j}|}.

(2.2)

Summing this operation over the unmarked sites defines d_θ. The two orders of marking distinct sites have opposite signs. Hence $d_{\mathsf{\theta }}^2=0$, and more generally d_ϕd_ψ + d_ψd_ϕ = 0.

Set

$${\mathcal{Z}}_n^p\left(W\right)=\bigcap _{\varphi \in W^{*}}\ker d_{\varphi }, Z_n^p\left(G,W,\mathsf{\theta }\right)=C_n^p{\left(W\right)}^G\cap \ker d_{\mathsf{\theta }}.$$

(2.3)

Call θ a cyclic frame if span_K(Gθ) = W^*. In particular, a nonzero frame is cyclic when W^* is irreducible.

Theorem 2.1 (Column Law). Let 0 ≤ p ≤ n. There is an isomorphism of GL(W) × S_n-modules

$${\mathcal{Z}}_n^p\left(W\right)\cong \underset{\begin{array}{c}\lambda ⊢n-p\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{⨁}{\mathbb{S}}_{\lambda }W\otimes S^{\lambda +\left(1^p\right)}.$$

(2.4)

If θ is a cyclic frame, then

$$Z_n^p\left(G,W,\mathsf{\theta }\right)\cong \underset{\begin{array}{c}\lambda ⊢n-p\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{⨁}{\left({\mathbb{S}}_{\lambda }W\right)}^G\otimes S^{\lambda +\left(1^p\right)}.$$

(2.5)

Here λ + (1^p) means addition of one box to each of the first p rows, padding λ with zeros. Thus the operation adds a column of height p to the left of the diagram after the usual row alignment. The empty partition and p = 0 are interpreted in the evident way.

Lemma 2.2 (Orbit amplification). Under the cyclic-frame hypothesis,

$$Z_n^p\left(G,W,\mathsf{\theta }\right)={\mathcal{Z}}_n^p{\left(W\right)}^G.$$

(2.6)

Proof. Naturality gives gd_θg⁻¹ = d_gθ. If v is fixed by G and d_θv = 0, then d_gθv = 0 for every g ∈ G. Linearity in the covector and the spanning condition imply d_ϕv = 0 for all ϕ. The reverse inclusion is immediate. □

The lemma identifies exactly where the group is used. Everything needed to determine ${\mathcal{Z}}_n^p\left(W\right)$ is group-free.

2.1. Column Deletion and the Super Tensor Interpretation

For a partition μ, let μ⁻ denote the partition obtained by deleting its first column and discarding zero rows. If 𝓁(λ) ≤ p, then μ = λ + (1^p) has exactly p rows and μ⁻=λ. Consequently, Theorem 2.1 has the equivalent form

$${\mathcal{Z}}_n^p\left(W\right)\cong \underset{\begin{array}{c}\mu ⊢n\\ \mathcal{l}\left(\mu \right)=p\end{array}}{⨁}{\mathbb{S}}_{{\mu }^{-}}W\otimes S^{\mu }.$$

(2.7)

Summing over p,

$$\stackrel{n}{\underset{p=0}{⨁}}{\mathcal{Z}}_n^p\left(W\right)\cong \underset{\mu ⊢n}{⨁}{\mathbb{S}}_{{\mu }^{-}}W\otimes S^{\mu }.$$

(2.8)

Each partition μ indexes one summand in (2.8), in degree p = 𝓁(μ). The nonzero summands satisfy 𝓁(μ⁻)≤d, equivalently μ_d+1 ≤ 1, where d = dimW: this is the (d,1)-hook of super Schur–Weyl duality [1,26]. In the tensor power of W⊕Ke, the Column Law selects the GL(W) multiplicity space obtained by first-column deletion. Table 2 illustrates the decomposition at n = 5.

By cochain degree p = 0,…,5, the same total is 0 + 15 + 80 + 51 + 12 + 1 = 159. Here 𝕊_∅W = K; the p = 0 joint kernel is zero for n > 0, while ${\mathcal{Z}}_0^0\left(W\right)=K$.

Remark 2.3 (Odd nilradical and a second proof). Put V = W⊕Ke and 𝔤 = 𝔤𝔩(W|Ke). Its type-I grading has 𝔤₊₁ = Hom(W,Ke), 𝔤₀ = 𝔤𝔩(W)⊕𝔤𝔩(Ke), and 𝔤₋₁ = Hom(Ke,W) [13]. For φ ∈ W^*, the element X_φ ∈ 𝔤₊₁ sends w to φ(w)e and kills e. Its tensor action is exactly d_φ: the odd coproduct sign is (2.2). Consequently, the direct sum in (2.8) is ${\left(V^{\otimes n}\right)}^{{\mathfrak{g}}_{+1}}$.

In characteristic zero, super Schur–Weyl duality decomposes V^⊗n as ⨁_μL_μ⊗S^μ, where the nonzero L_μ are the irreducible covariant tensor modules indexed by the (d,1)-hook [1,4,26]. Grade each L_μ by the number of odd factors. If a nonzero homogeneous $v\in L_{\mu }^{{\mathfrak{g}}_{+1}}$ has degree a, PBW gives $𝒰$(𝔤)v = $𝒰$(𝔤₋₁)$𝒰$(𝔤₀)v, which has degrees at most a. Irreducibility therefore forces a to be the top degree. Conversely, every vector in that top degree is killed by 𝔤₊₁, since this subalgebra raises degree. The invariant space is graded, so this accounts for every invariant vector.

The hook Schur character in one odd variable is

$$\mathrm{ch}{L}_{\mu }=\sum _{\nu :\hspace{0.17em}\mu /\nu \mathrm{a\; vertical\; strip}}{s}_{\nu }\left(x\right)y^{\left|\mu /\nu \right|}.$$

(2.9)

The largest possible exponent of y is 𝓁(μ), attained uniquely by ν = μ⁻; its coefficient is $s_{{\mu }^{-}}\left(x\right)$. Thus $L_{\mu }^{{\mathfrak{g}}_{+1}}\cong {\mathbb{S}}_{{\mu }^{-}}W$ as a GL(W)-module, in degree 𝓁(μ), proving (2.7). The rational tensor action, Young projectors, multiplicity spaces, and contraction kernels commute with scalar extension, giving the identification over every characteristic-zero field. Section 3 supplies an independent proof and the additional information of every content eigenvalue.

3. Proof by a Content Operator

The Column Law is the zero-eigenvalue part of a natural content operator. Its other eigenvalues give an exact spectral bound.

3.1. The Sum of Squares

First work over ℝ and choose an orthonormal basis w₁,…,w_d of W. Give the oriented tensor basis of (2.1) the corresponding positive inner product. Put $D_j=d_{w_j^{*}}$ and

$$H={\displaystyle \sum _{j=1}^d}{D}_j^{\mathrm{t}}{D}_j.$$

(3.1)

Then $\ker H={\bigcap }_j\ker D_j={\bigcap }_{\phi \in W^{*}}\ker d_{\phi }$: the last equality follows because φ ↦ d_φ is linear, so it suffices to test a basis. Although (3.1) uses an inner product, the next expression shows that H is natural in W.

Lemma 3.1. On $C_n^p\left(W\right)$,

$$H=\mathit{mI}-\sum _{\mathrm{one\; marked},\mathrm{one\; unmarked\; site}}\left(\mathit{ij}\right).$$

(3.2)

The transposition in this formula is the signed site transposition, and the set of mixed pairs is evaluated on each summand of (2.1).

Proof. Contracting and then restoring the same site contributes one copy of the identity for each of the m unmarked sites. Contracting one site and restoring a different marked site exchanges those two sites. The marker passes the markers between them with the Koszul sign, while the two differential signs supply the additional minus sign in (3.2). These exhaust the terms in $D_j^{\mathrm{t}}{D}_j$. □

Under the sign identification of Lemma 9.1, H on the content-(1^m) subspace of $C_n^{n-m}\left(K^m\right)$ is the top-degree chessboard Laplacian ∂^t∂ studied by Friedman and Hanlon [8].

3.2. Pieri Decomposition and Content Scalars

Schur–Weyl duality and induction give

$$C_n^p\left(W\right)\cong \underset{\lambda ⊢m}{⨁}{\mathbb{S}}_{\lambda }W\otimes {\mathrm{Ind}}_{S_p\times S_m}^{S_n}\left({\mathrm{sgn}}_p⊠S^{\lambda }\right).$$

(3.3)

By the exterior Pieri rule [9,17,23], each induced factor is the multiplicity-free sum of S^μ for which μ/λ is a vertical p-strip, with at most one new box in each row. Thus the nonzero (λ,μ) blocks are irreducible for the product action.

Let ${\Omega }_n=\sum _{i<j}\left(\mathit{ij}\right)$. On each oriented summand of (2.1), split the pairs into marked–marked, unmarked–unmarked, and mixed pairs. A restriction to one fixed marking set is not by itself a globally S_n-equivariant operator; the marking set is transported with the summand. The marked–marked contribution is the global scalar $-\left({\displaystyle \genfrac{}{}{0pt}{}{p}{2}}\right)$. The unmarked–unmarked contribution is the central class sum of $S_{C^c}$, acting by c(λ) on the unmarked λ-type. Finally Ω_n acts by c(μ) on total site type μ [12,20]. Subtracting these actions in (3.2) is therefore well defined and gives

$$h_{\lambda ,\mu }=m-c\left(\mu \right)+c\left(\lambda \right)-\left({\displaystyle \genfrac{}{}{0pt}{}{p}{2}}\right).$$

(3.4)

3.3. The Zero Eigenvalue

Write the rows receiving new boxes as r₁<…<r_p, padding λ with zeros. The new box in row r has content λ_r + 1 − r. Substitution in (3.4) gives

$$h_{\lambda ,\mu }=\left(m-{\displaystyle \sum _{j=1}^p}{\lambda }_{r_j}\right)+{\displaystyle \sum _{j=1}^p}\left(r_j-j\right).$$

(3.5)

Both summands are nonnegative integers. Equality holds precisely when r_j = j for every j, and every box of λ lies in those first p rows. Thus

h_λ,μ = 0 ⇔ 𝓁(λ) ≤ p and μ = λ + (1^p).

(3.6)

Since H is a sum of squares, its zero eigenspace is exactly the joint kernel. This proves (2.4) over ℝ. All contraction matrices, site permutations, and characteristic-zero Young decompositions used here are defined over ℚ. Equality of their kernels therefore extends to every characteristic-zero field by scalar extension. Taking G-fixed vectors and using Lemma 2.2 proves (2.5). □

The positive inner product determines a universal rational kernel, so no invariant form on W is needed.

3.4. The Exact Spectral Radius and Good Characteristic

Proposition 3.2. Put m = n − p and d = dimW ≥ 1. Every ambient Pieri block satisfies

0 ≤ h_λ,μ ≤ m + p𝓁(λ) ≤ m + pmin(d,m).

(3.7)

The last bound is attained. Thus the spectral radius is m + pmin(d,m); the dimension-independent bound (p + 1)m is attained whenever d ≥ m. If charK > n, the decompositions (2.4), (2.7) and the cyclic-frame formula (2.5) remain valid; this assertion uses Theorem 5.5 below.

Proof. Put r = 𝓁(λ). In (3.5), the jth added box lies in a row r_j ≤ r + j. Hence h_λ,μ ≤ m + pr. Equality holds exactly when the new boxes occupy rows r + 1,…,r + p, equivalently μ = (λ,1^p). A partition of m of length min(d,m) exists when m > 0; use the empty partition when m = 0. Appending p rows realises the bound. The earlier bound follows from min(d,m) ≤ m.

For the characteristic assertion, work with the integral tensor lattice localised at n!. The Young projectors and the Schur–Weyl decomposition are defined there. On reducing modulo q > n, the relevant symmetric-group algebras and polynomial Schur algebras are semisimple [9,23]. The stacked map $D:x\mapsto \sum _a{w}_a\otimes d_{a}x$ is GL(W) × S_n-equivariant, so its kernel splits according to those ambient domain types. Each restricted contraction matrix can only lose rank under reduction. Theorem 5.5 says that the sum of their kernel dimensions does not increase. Therefore no projected kernel dimension increases, and the characteristic-zero block multiplicities persist. Lemma 2.2 then gives the invariant formula without any averaging over G. This assertion concerns the joint contraction kernel; kerH need not equal it in positive characteristic. □

3.5. A Complete Small Block

For p = 2 and λ = (2,1), the induced site representation has dimension $\left({\displaystyle \genfrac{}{}{0pt}{}{5}{2}}\right)\dim S^{\left(2,1\right)}=20$. Table 3 gives the dimensions and eigenvalues of all four Pieri summands.

Figure 1 shows their added strips and the surviving first column.

4. The Invariant Degree-One Poincaré Lemma

The one-marker case identifies invariant closed cochains with symmetric invariants. A comparison with the single-frame kernel shows why cyclicity is needed.

Corollary 4.1. If θ is cyclic, then

$$Z_n^1\left(G,W,\mathsf{\theta }\right)\cong {\left({\mathrm{Sym}}^{n-1}W\right)}^G,$$

(4.1)

and S_n acts trivially on this space.

Proof. The only partition of n − 1 with at most one row is (n − 1). Its column extension is (n), which affords the trivial representation. □

We use the normalised symmetrisation map, with r ≥ 0,

$${\mathrm{Pol}}_r:{\mathrm{Sym}}^{r}W\to {\left(W^{\otimes r}\right)}^{S_r}, {\mathrm{Pol}}_r\left(w_1\dots w_r\right)={\displaystyle \frac{1}{r!}}\sum _{\sigma \in S_r}{w}_{\sigma \left(1\right)}\otimes \dots \otimes w_{\sigma \left(r\right)}.$$

(4.2)

Here Pol₀(1) = 1. If q_r:W^⊗r → Sym^rW is the quotient map, then q_rPol_r = id, while Pol_rq_r is the averaging projector. We abbreviate Pol_r to Pol when the degree is clear.

For P ∈ Symⁿ⁻¹W, put Pol_n−1(P) in the slots other than i, with marker e_i, for every i. The two contractions contributing to a two-marker component cancel. Under cyclicity, Corollary 4.1 makes this a bijection onto invariant closed cochains; its inverse applies q_n−1 to any vertex component. Equivalently, P is a homogeneous invariant polynomial on W^*; polynomials on W require replacing W by its dual.

Example 4.2 (Failure without a cyclic frame). Let G be trivial, let W = Kx⊕Ky, and let θ(x) = 1, θ(y) = 0. At n = 2, p = 1, a cochain is a pair (u,v) ∈ W⊕W, and closure imposes only θ(v − u) = 0. The closed space has dimension three. The diagonal polarisation image has dimension two. For example, (y,0) is closed and is not diagonal. Thus invariance alone does not suffice; the orbit-spanning condition cannot be omitted.

The same proof gives the dimension formula

$$\dim {\mathcal{Z}}_n^p{\left(W\right)}^G=\sum _{\begin{array}{c}\lambda ⊢n-p\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{a}_{\lambda }{f}^{\lambda +\left(1^p\right)}, a_{\lambda }=\dim {\left({\mathbb{S}}_{\lambda }W\right)}^G, f^{\mu }=\dim S^{\mu }.$$

(4.3)

For p = 1, all dimension growth is therefore controlled by the Hilbert series of symmetric invariants. For higher p, the required data are invariant multiplicities of Schur functors, not merely dimensions of symmetric powers.

4.1. The Size of a Single-Frame Kernel

Proposition 4.3 (Single-frame kernel). Let d = dimW ≥ 1, let θ ≠ 0, and put $a_p=\dim \hspace{0.17em}\ker \left(d_{\theta }{|}_{C_n^p\left(W\right)}\right)$. Then

$$a_0={\left(d-1\right)}^n, a_p=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{p-1}}\right)d^{\hspace{0.17em}n-p+1}-a_{p-1} \left(1\le p\le n\right).$$

(4.4)

Proof. Choose u ∈ W with θ(u) = 1. As a two-term complex, V = W⊕Ke with Δ_θ(w) = θ(w)e splits as $\left(\ker \theta ,0\right)\oplus \left(\mathit{Ku}\stackrel{\sim }{\to }\mathit{Ke}\right)$. The oriented site complex is V^⊗n with its total differential. Künneth therefore gives H⁰ = (kerθ)^⊗n and H^p = 0 for p ≥ 1. In positive degree the kernel is the preceding image, whose dimension is $\dim C_n^{p-1}-a_{p-1}$. This proves (4.4), with the usual empty-tensor convention when n = 0. □

For Example 4.2, the single-frame kernel has dimension 4 − 1 = 3, whereas the joint kernel has dimension two. Table 4 gives further comparisons on the full cochain space.

For d = 3 and p = 2, the recurrence and the hook formulas simplify to

$$\dim C_n^2=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)3^{n-2}, a_2=\left(n-3\right)3^{n-1}+2^n, \dim {\mathcal{Z}}_n^2=n\left(n-1\right)2^{n-3} \left(n\ge 2\right).$$

(4.5)

At n = 2, the factor 2ⁿ⁻³ = 1/2 still gives the joint-kernel dimension 1. For the last identity, write λ = (n − 2 − b,b). Its contribution is ${\displaystyle \frac{n}{2}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{b}}\right){\left(n-1-2b\right)}^2$. Pairing b with n − 1 − b and using the binomial second moment gives (4.5). The ambient and single-frame dimensions have exponential base three, while the joint kernel has base two; the joint-to-ambient ratio is exactly (2/3)ⁿ⁻².

5. Characteristic-Free Leading Words and Socle Layers

We now describe the joint kernel and every layer of its socle filtration by leading words. Let K be any field, W = K^d, and order the alphabet by e < 1<…<d, with e odd and the letters even. We take the lexicographically largest supported word as the leading word LW(x).

For a letter a, let

$$d_a\left(u\right)=\sum _{j:u_j=a}{\left(-1\right)}^{\#\{i<j:u_i=e\}}u\left[j\mapsto e\right].$$

These operators satisfy $d_a^2=0$ and d_ad_b = −d_bd_a, including in characteristic two. For a subset S = {a₁<…<a_r}, put $d_S=d_{a_r}\dots d_{a_1}$, so that $d_{a_1}$ acts first. Define

$${\mathrm{soc}}^k\left(n\right)=\bigcap _{\left|S\right|=k}\ker d_S.$$

The intersection is the whole tensor power when k > d. In particular, ${\mathrm{soc}}^1\left(n\right)={⨁}_p{\mathcal{Z}}_n^p\left(W\right)$.

For a word u, let E_u(j)=#{i > j:u_i = e}. Write $ℬ$^(k) for the words satisfying

t ≤ k − 1 + E_u(j₁)

(5.1)

for every strictly increasing subsequence $u_{j_1}<\dots <u_{j_t}$ of nonmarker letters, with j₁<…<j_t. Set $ℬ$ = $ℬ$⁽¹⁾; its elements are called ballot words (the term refers to (5.1), not the classical Yamanouchi condition). For p ≥ 0 and α ∈ ℕ^d with p + |α| = n, let $ℬ$_k;p,α be the words in $ℬ$^(k) having p markers and nonmarker content α, and put $ℬ$_p,α = $ℬ$_1;p,α. The condition concerns all increasing subsequences, not only adjacent runs.

Lemma 5.1 (the marker-free tail). Let 0 ≠ x ∈ soc^k(n) and u = LW(x). If an increasing subsequence starts at j₁ and no marker occurs after j₁, its length is at most k − 1.

Proof. Suppose its first k letters are a₁<…<a_k, in positions j₁<…<j_k. Replace these letters by markers to obtain ω. A word contributing to the coefficient of ω in d_Sx, where S = {a₁,…,a_k}, is obtained by restoring these k letters in marker positions of ω. Every original marker of u lies before j₁. Restoring a letter in such a position produces a word larger than u. If only the positions j₁,…,j_k are used, the increasing assignment is the lexicographically smallest assignment of the k distinct letters. Thus every assignment except the original one again gives a larger word. Only u contributes, with coefficient ±x_u ≠ 0. This contradicts d_Sx = 0. □

Lemma 5.2 (deleting a marker). For a position f, define

$${\partial }_f^{e}x=\sum _{v:v_f=e}{\left(-1\right)}^{E_v\left(f\right)}{x}_v\left(v\setminus f\right).$$

Then ${\partial }_f^e\left({\mathrm{soc}}^k\left(n\right)\right)\subseteq {\mathrm{soc}}^{k+1}\left(n-1\right)$.

Proof. The socle conditions are homogeneous in marker degree, so it suffices to take x homogeneous of marker degree p. Move slot f to the last position by the signed permutation of tensor factors. The signed adjacent interchange is τ(v⊗w) = (−1)^|v||w|w⊗v. Substitution in the Koszul rule d(v⊗w) = dv⊗w + (−1)^|v|v⊗dw gives τd = dτ term by term; composing adjacent interchanges proves the same for the slot move. Write the resulting tensor as

$$\mathit{\tau x}=y\otimes e+\sum _a{y}_a\otimes a.$$

Moving an e past the markers on its right gives exactly the sign in the definition of ${\partial }_f^e$, so $y={\partial }_f^{e}x$. The tensors y_a have marker degree p. For S = {a₁<…<a_k}, the component with final letter a in d_Sτx is d_Sy_a⊗a. Its component with final marker is

$$\left(d_{S}y+{\displaystyle \sum _{i=1}^k}{\left(-1\right)}^{p+i-1}{d}_{S\setminus \left\{a_i\right\}}{y}_{a_i}\right)\otimes e.$$

(5.2)

Indeed, at most one contraction can act in the final slot. In the term arising from $y_{a_i}\otimes a_i$, the first i − 1 contractions act in the left factor, changing its parity from p to p + i − 1. Applying the i-th contraction in the final slot therefore contributes (−1)^p+i−1 by the Koszul rule. Every later contraction again acts in the left factor, with no further tensor-product sign. Since d_Sτx = 0, every y_a belongs to soc^k(n − 1). For a set T of size k + 1, choose t ∈ T and apply d_t to (5.2) with S = T∖{t}. Every term on its right contains k distinct contractions applied to some y_a, and therefore vanishes. Hence d_Ty = 0. □

Theorem 5.3 (leading-word bound). Over every field,

LW(soc^k(n)∖{0}) ⊆ $ℬ$^(k).

Proof. Induct on n, the empty word being immediate. Let u = LW(x). If u contains no marker, Lemma 5.1 applies to every increasing subsequence. Otherwise let f be its last marker. A subsequence starting after f again satisfies the required bound by Lemma 5.1. For a subsequence starting before f, use Lemma 5.2. Among words having a marker in slot f, deletion of that slot is injective and preserves lexicographic order. The coefficient of u∖f in ${\partial }_f^{e}x$ is exactly ${\left(-1\right)}^{E_u\left(f\right)}{x}_u\ne 0$. Consequently

$$\mathrm{LW}\left({\partial }_f^{e}x\right)=u\setminus f.$$

The induction hypothesis gives the bound k + E_u∖f(j₁). Since E_u∖f(j₁) = E_u(j₁) − 1, this is (5.1). □

The Robinson–Schensted–Knuth (RSK) correspondence, using ordinary row insertion [23,25], gives a count uniform in the socle index. Write K_να for the number of semistandard tableaux of shape ν and content α, and f^μ for the number of standard tableaux of shape μ.

Proposition 5.4 (standardised RSK). Fix a word u with p markers. Replace its i-th marker from the right by ε_i, where

ε_p<…<ε₁ < 1<…<d.

Apply ordinary row insertion to the resulting word read from right to left, obtaining an insertion tableau P(u), a recording tableau R(u) and a shape μ(u). The marker entries of P(u) occupy exactly the first p boxes of its first column. Moreover,

$$\mathcal{l}\left(\mu \left(u\right)\right)=p+\Delta \left(u\right), \Delta \left(u\right)=\max \left(0,\underset{\left(j_1,\dots ,j_t\right)}{\max }\{t-E_u\left(j_1\right)\}\right).$$

The inner maximum ranges over the strictly increasing subsequences of nonmarker letters. Consequently u ∈ $ℬ$^(k) if and only if 𝓁(μ(u)) ≤ p + k − 1. Removing the marker entries induces a bijection

$${\mathcal{B}}_{k;p,\alpha }\leftrightarrow \coprod _{\begin{array}{c}\mu ⊢n,\hspace{0.25em}\left(1^p\right)\subseteq \mu \\ \mathcal{l}\left(\mu \right)\le p+k-1\end{array}}\mathrm{SSYT}\left(\mu /\left(1^p\right),\alpha \right)\times \mathrm{SYT}\left(\mu \right).$$

(5.3)

For k = 1 this gives SSYT(λ,α) × SYT(λ + (1^p)), summed over λ⊢n − p with 𝓁(λ) ≤ p.

Proof. Restricting an insertion tableau to its smallest symbols gives the insertion tableau of their subword. Here that subword is ε₁,…,ε_p, which is strictly decreasing, so its tableau is the first-column shape (1^p).

By Schensted’s theorem [25], 𝓁(μ(u)) is the longest strictly decreasing subsequence in the reversed standardised word. Such a subsequence consists of letters followed by markers. Reading its letters in the original order gives a strictly increasing subsequence starting at j₁, of some length t. Every marker to the left of j₁ can then be included: they occur in decreasing order and are smaller than every letter. The longest length is therefore

$$\max \left(p,\underset{\left(j_1,\dots ,j_t\right)}{\max }\{t+p-E_u\left(j_1\right)\}\right)=p+\Delta \left(u\right).$$

The term p accounts for subsequences consisting only of markers; for the empty word its value is zero.

To verify surjectivity in (5.3), start with any displayed pair (T,R). Fill the removed column, from top to bottom, by ε_p,…,ε₁. The result is an ordinary semistandard tableau P, because every marker is smaller than every letter. Inverse RSK applied to (P,R) recovers a word whose marker subword has insertion tableau a single column. Since the marker symbols are distinct, that subword is necessarily ε₁,…,ε_p in this order. Reverse the recovered word and replace all ε_i by e. This is precisely the standardisation convention above and gives the unique inverse. The row bound identifies the words in $ℬ$_k;p,α.

When k = 1, the marker column forces 𝓁(μ) = p. Subtracting one box from each of these rows gives λ, and shifting the remaining tableau one column to the left identifies its skew shape with λ. □

The Pieri rule gives

$$s_{\mu /\left(1^p\right)}=\sum _{\begin{array}{c}\nu \subseteq \mu \\ \mu /\nu \mathrm{a\; vertical}p-\mathrm{strip}\end{array}}{s}_{\nu }.$$

Thus the number of skew tableaux in (5.3) is the sum of K_να over the indicated partitions.

Theorem 5.5 (characteristic-free Column Law). For every field K, every n,p ≥ 0, and every nonmarker content α ∈ ℕ^d with p + |α| = n,

$$\mathrm{LW}\left({\mathcal{Z}}_n^p{\left(W\right)}_{\alpha }\setminus \left\{0\right\}\right)={\mathcal{B}}_{p,\alpha }, {\dim }_K{\mathcal{Z}}_n^p{\left(W\right)}_{\alpha }=\sum _{\begin{array}{c}\lambda ⊢n-p\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{K}_{\mathit{\lambda \alpha }}\hspace{0.17em}{f}^{\lambda +\left(1^p\right)}.$$

(5.4)

Proof. Gaussian elimination identifies the dimension of a subspace with the number of its possible leading words. Theorem 5.3 and the case k = 1 of Proposition 5.4 therefore bound the left side of (5.4) above by its stated right side. Over ℚ, equality is the weight-space form of the characteristic-zero Column Law. On each content space the stacked contraction matrix has integer entries. Reduction modulo a prime can only decrease its rank, so its kernel over that prime field has dimension at least its rational kernel dimension. Extension of the coefficient field preserves rank. The upper and lower bounds coincide in every characteristic, forcing both the dimension formula and equality of the leading-word sets. □

Theorem 5.6 (characteristic-free socle series). For every field K, every k ≥ 1 and every n,p,α as in Theorem 5.5,

LW(soc^k(n)_p,α∖{0}) = $ℬ$_k;p,α,

$${\dim }_K{\mathrm{soc}}^k{\left(n\right)}_{p,\alpha }=\sum _{\begin{array}{c}\mu ⊢n\\ \mathcal{l}\left(\mu \right)\le p+k-1\end{array}}{f}^{\mu }\sum _{\begin{array}{c}\nu \subseteq \mu \\ \mu /\nu \mathrm{a\; vertical}p-\mathrm{strip}\end{array}}{K}_{\mathit{\nu \alpha }}.$$

(5.5)

In particular, the dimensions of every socle term and every successive socle quotient are independent of the field.

Proof. First work in characteristic zero. Under super Schur–Weyl duality, write V^⊗n = ⨁_μL_μ⊗S^μ, with μ_d+1 ≤ 1. Grade L_μ by the number of markers. The hook-Schur branching formula gives

$$\mathrm{ch}{L}_{\mu }\left[p\right]=\sum _{\begin{array}{c}\nu \subseteq \mu \\ \mu /\nu \mathrm{a\; vertical}p-\mathrm{strip}\end{array}}{s}_{\nu }.$$

Its highest marker degree is 𝓁(μ): a vertical strip removes at most one box per row, and removal of the whole first column leaves a partition with at most d rows.

Let 𝔤 = 𝔤𝔩(W∣Ke) and write 𝔤 = 𝔤₋₁⊕𝔤₀⊕𝔤₊₁, where 𝔤₊₁ consists of the contractions. It is odd abelian, with enveloping algebra Λ(𝔤₊₁), and raises marker degree by one. Suppose 0 ≠ v ∈ L_μ[p] is annihilated by all products of k contractions. Longer products annihilate it as well. The PBW theorem, ordered with the raising part acting first, gives

$𝒰$(𝔤)v = $𝒰$(𝔤₋₁)$𝒰$(𝔤₀)Λ^<k(𝔤₊₁)v.

Every vector on the right has marker degree at most p + k − 1. Irreducibility implies $𝒰$(𝔤)v = L_μ, so p + k − 1 ≥ 𝓁(μ). Conversely, every vector of degree p ≥ 𝓁(μ) − k + 1 is killed by k contractions, since they would raise its degree above the largest one present. It follows that

$${\mathrm{soc}}^k\left(L_{\mu }\right)=\underset{p\ge \mathcal{l}\left(\mu \right)-k+1}{⨁}{L}_{\mu }\left[p\right].$$

(5.6)

Combining (5.6), the branching formula and the multiplicity f^μ proves (5.5) over ℚ. Partitions outside the (d,1) hook contribute zero to its right side because every associated Kostka number is zero.

For an arbitrary field, Theorem 5.3 and Proposition 5.4 now give

dim_Ksoc^k(n)_p,α ≤ |$ℬ$_k;p,α| = dim_ℚsoc^k(n)_p,α ≤ dim_Ksoc^k(n)_p,α.

The last inequality follows from the integer matrix obtained by stacking all d_S with |S| = k, exactly as in Theorem 5.5. Equality throughout proves both assertions. Subtracting the dimensions of successive terms proves the claim about the quotients. □

The empty-partition conventions cover n = p = 0. If d = 0, the only word is eⁿ; if k > d, the intersection defining soc^k is the whole tensor power and every word satisfies the ballot inequality. Formula (5.5) includes both cases.

6. The Algebra of Closed Cochains

Let $\mathcal{Z}={⨁}_{n,p}{\mathcal{Z}}_n^p\left(W\right)$, including ${\mathcal{Z}}_0^0=K$, and let 𝒵₊ be its augmentation ideal. Since each contraction is a parity-skew derivation, concatenation makes 𝒵 a graded subalgebra of T(W⊕Ke). Write

$$Q={\mathcal{Z}}_{+}/{\mathcal{Z}}_{+}^2$$

for its space of indecomposables.

6.1. Prime Ballot Words

A nonempty ballot word is prime if it is not the concatenation of two nonempty ballot words. Denote the set of prime ballot words by Π.

Theorem 6.1 (freeness). For each π ∈ Π, choose a closed cochain g_π with leading word π, homogeneous in its length, marker number and letter content. Then the elements g_π freely generate 𝒵.

Proof. Every suffix of a ballot word is ballot: the number of markers to the right of the first letter of any increasing subsequence is unchanged on passing to a suffix. Ballot words are also closed under concatenation. Indeed, split an increasing subsequence in uv into its portions in u and v; their lengths are bounded by the relevant markers in u and by the total number of markers in v, respectively.

Consequently, a ballot word is prime precisely when it has no proper nonempty ballot prefix. Factoring at the shortest nonempty ballot prefix, and repeating on the remaining suffix, gives a unique factorisation into primes. Equivalently, the prime words form a prefix code, and the ballot monoid is the free monoid on Π.

Leading words multiply under concatenation. Thus distinct products of the g_π have distinct leading words, namely the distinct ballot words, and are linearly independent. In each multidegree, the characteristic-free leading-word theorem says that their number is dim𝒵 in that multidegree. They therefore form a basis. □

Let $\mathscr{H}$(t,u;w) be the character series of 𝒵, where t records length, u records marker number and w records letter content. Then

$$\mathrm{ch}Q=\sum _{\pi \in \Pi }{t}^{\left|\pi \right|}{u}^{p\left(\pi \right)}{\mathbf{w}}^{\mathrm{content}\left(\pi \right)}=1-{\mathcal{H}}^{-1}.$$

In characteristic zero, the Column Law gives the equivalent symmetric-function expression

$$\mathcal{H}=1+\sum _{n\ge 1}{\displaystyle \sum _{p=1}^n}\sum _{\begin{array}{c}\lambda ⊢n-p\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{f}^{\lambda +1^p}{s}_{\lambda }\hspace{0.17em}{t}^n{u}^p.$$

Here Schur functions with more than dimW rows vanish on specialisation. In particular, 1 − $\mathscr{H}$⁻¹ is Schur-positive. A degreewise GL(W)-stable lift of Q to 𝒵₊ identifies 𝒵 equivariantly with T(Q).

Proposition 6.2 (support). If p ≥ 2, then $Q_n^p=0$ for n < 2p.

Proof. For a prefix u, let

Δ(u) = max(0, t−#{markers to the right of j₁ in u}),

where the maximum ranges over all strictly increasing subsequences $u_{j_1}<\dots <u_{j_t}$ of letters. Proposition 5.4 identifies this deficit with 𝓁(μ(u)) − p(u), where p(u) is the marker count. Thus u is ballot exactly when Δ(u) = 0. Appending a letter increases Δ by at most one, whereas appending a marker replaces it by max(0,Δ−1).

A prime word other than e has positive deficit at every proper nonempty prefix. Each of its p markers therefore lowers the deficit by exactly one, and its letters must supply at least p increases. It has at least p letters, so its length is at least 2p. Theorem 6.1 now gives the assertion. □

6.2. Generator Representations

Every positive-degree closed cochain has at least one marker, so the one-marker part is indecomposable. The Column Law gives

$$Q_n^1\cong {\mathrm{Sym}}^{n-1}W \left(n\ge 1\right).$$

Proposition 6.3 (two markers). For n ≥ 2,

$$Q_n^2\cong \stackrel{⌊\left(n-2\right)/2⌋}{\underset{j=0}{⨁}}{\mathbb{S}}_{\left(n-2-j,j\right)}{W}^{\oplus m\left(n,j\right)}, m\left(n,j\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j+1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)-\left(n-1-2j\right).$$

Zero parts of a partition are omitted; in particular m(n,0) = 0.

Proof. The multiplicity of 𝕊_{(n−2−j,j)}W in ${\mathcal{Z}}_n^2$ is

$$f^{\left(n-1-j,j+1\right)}=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j+1}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right).$$

By Theorem 6.1, the decomposable part with two markers is the direct sum, over a = 1,…,n − 1, of Sym^a−1W⊗Sym^n−a−1W. Pieri’s rule gives one copy of the indicated Schur module exactly when j + 1 ≤ a ≤ n − j − 1, hence n − 1 − 2j copies in total. Subtracting proves the formula. □

Proposition 6.4 (Catalan diagonal). The multiplicity of Λ^pW in $Q_{2p}^p$ is

$${\mathrm{Cat}}_{p-1}={\displaystyle \frac{1}{p}}\left({\displaystyle \genfrac{}{}{0pt}{}{2p-2}{p-1}}\right) \left(p\ge 1\right),$$

with the usual convention that Λ^pW = 0 for p > dimW. Apart from the scalar generator e, these are the only column-shaped generator representations.

Proof. A column Λ^rW can occur in ${\mathcal{Z}}_n^p$ only when n = p + r and r ≤ p. Proposition 6.2 forces r = p for generators with p ≥ 2, while the one-marker formula gives only e and W among column-shaped generators.

A Littlewood–Richardson product can contain a column only if each factor is a column, and Λ^aW⊗Λ^bW contains Λ^a+bW with multiplicity one whenever the latter is nonzero. In a product of such generators, each non-scalar factor has length minus twice its marker number equal to zero, whereas e has this difference equal to −1. Hence e cannot occur in a product contributing to Λ^pW at (n,p) = (2p,p). The Column Law gives multiplicity $f^{\left(2^p\right)}={\mathrm{Cat}}_p$ in ${\mathcal{Z}}_{2p}^p$. If B(z) records the corresponding generator multiplicities and $C\left(z\right)=\sum _{p\ge 0}{\mathrm{Cat}}_p{z}^p$, freeness gives

$$C\left(z\right)={\displaystyle \frac{1}{1-B\left(z\right)}}, B\left(z\right)=1-C{\left(z\right)}^{-1}=\mathit{zC}\left(z\right).$$

The coefficient of z^p is Cat_p−1. □

Theorem 6.5 (hook generators). Assume that K has characteristic zero. For p ≥ 1 and n ≥ 2p, the hook-shaped part of $Q_n^p$ consists of

$${\mathbb{S}}_{\left(n-2p+1,1^{p-1}\right)}W \mathrm{with\; multiplicity} {\mathrm{Cat}}_{p-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n-2}{2p-2}}\right).$$

Together with the scalar generator $Q_1^1=K$, these are all hook-shaped and trivial generator representations. As usual, a Schur module with more than dimW rows is zero.

Proof. Work first with the stable symmetric-function character series $\mathscr{H}$(t,u) of 𝒵, where t records length and u records marker number. Let ρ:Λ_ℚ → ℚ[x,y] be the ring homomorphism defined on power sums by

ρ(p_j) = x^j + (−1)^j−1y^j.

The one-even-variable, one-odd-variable specialisation of the Schur functions gives

$$\rho \left(s_{\mathrm{⌀}}\right)=1, \rho \left(s_{\left(a,1^b\right)}\right)=x^{a-1}{y}^b\left(x+y\right), \rho \left(s_{\lambda }\right)=0 \mathrm{if}{\lambda }_2\ge 2.$$

These identities also follow by expanding $\rho \left(s_{\lambda }\right)=\sum _{\nu \subseteq \lambda }{s}_{\nu }\left(x\right)s_{\left(\lambda /\nu \right)\mathrm{\text{'}}}\left(y\right)$: the surviving ν are rows and λ/ν must be a vertical strip. The displayed nonzero images, including 1, are linearly independent: the nonconstant ones are (x + y) times distinct monomials, while every such polynomial vanishes at x = y = 0. Thus ρ retains exactly the scalar and hook coefficients. By Theorem 6.1,

ρ(chQ) = 1 − ρ($\mathscr{H}$)⁻¹.

To compute ρ($\mathscr{H}$), put $f\left(a,b,c\right)=f^{\left(a+1,2^b,1^c\right)}$ for a ≥ 1, b,c ≥ 0, and put f(0,0,c) = 1. Set f = 0 on all other triples, and define

$$\Psi \left(X,v,w\right)=\sum _{a,b,c\ge 0}f\left(a,b,c\right)X^a{v}^b{w}^c.$$

Here f(0,0,0) = 1 represents the one-box shape. Removing the box containing the largest entry of a standard tableau gives, at every other valid triple,

f(a,b,c) = f(a − 1,b,c) + f(a,b − 1,c + 1) + f(a,b,c − 1).

The zero convention includes the missing first-row corner when a = 1 and b > 0. Summing this recursion gives

$$\Psi -1=\mathit{X\Psi }+\mathit{w\Psi }+{\displaystyle \frac{v}{w}}\left(\Psi -{\displaystyle \frac{1}{1-w}}-\Psi \left(X,v,0\right)+1\right).$$

The expression in parentheses is divisible by w, so this is an identity of formal power series. Equivalently,

$$\Psi \left(w-\mathit{Xw}-w^2-v\right)=w-{\displaystyle \frac{\mathit{vw}}{1-w}}-\mathit{v\Psi }\left(X,v,0\right).$$

Let ω ∈ vℚ[[X,v]] be the root of

ω² − (1 − X)ω + v = 0,  B = 1 − X − ω,  v = ωB.

Substitution w = ω is valid formally and determines vΨ(X,v,0) = ω(1 − v/(1 − ω)). The two sides of the preceding functional equation then contain the factor w − ω; cancelling it and using v = ωB yields the convenient form

$$\Psi \left(X,v,w\right)-{\displaystyle \frac{1}{1-w}}={\displaystyle \frac{X}{\left(1-w\right)\left(1-\omega \right)\left(B-w\right)}}.$$

All three denominator factors have constant term 1.

For a hook λ = (a,1^b) in the Column Law, write c = p − 1 − b. Then λ + 1^p = (a + 1,2^b,1^c). Consequently, with

X = xt,  v = yut²,  w = tu,

the specialised Column Law is

$$\rho \left(\mathcal{H}\right)={\displaystyle \frac{1}{1-w}}+\left(x+y\right){\displaystyle \frac{w}{x}}\left(\Psi \left(X,v,w\right)-{\displaystyle \frac{1}{1-w}}\right)={\displaystyle \frac{B}{\left(1-\omega \right)\left(B-w\right)}}.$$

There is no division by a nonunit in this power series: the parentheses are divisible by X = xt. For the last equality, the displayed expression for Ψ − 1/(1 − w) reduces the numerator to (1 − ω)(B − w) + (x + y)wt = B(1 − w), using v = ywt = ωB. It follows directly that

$$\rho \left(\mathrm{ch}Q\right)=1-{\displaystyle \frac{\left(1-\omega \right)\left(B-w\right)}{B}}=\omega +{\displaystyle \frac{w\left(1-\omega \right)}{B}}.$$

Equivalently,

$$\rho \left(\mathrm{ch}Q\right)=w+{\displaystyle \frac{\left(x+y\right)\omega }{y}}=\mathit{tu}+{\displaystyle \frac{x+y}{2y}}\left(1-\mathit{xt}-\mathcal{D}\right),$$

where

$$\mathcal{D}=\sqrt{{\left(1-\mathit{xt}\right)}^2-4\mathit{yu}{t}^2}.$$

The square root is the formal one with constant term 1; ω is divisible by y, so the quotient by y is again a power series. Writing $C\left(z\right)=\sum _{q\ge 0}{\mathrm{Cat}}_q{z}^q$, we have

$${\displaystyle \frac{\omega }{y}}={\displaystyle \frac{u{t}^2}{1-\mathit{xt}}}\hspace{0.17em}C\left({\displaystyle \frac{\mathit{yu}{t}^2}{{\left(1-\mathit{xt}\right)}^2}}\right)=\sum _{p\ge 1}\sum _{n\ge 2p}{\mathrm{Cat}}_{p-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n-2}{2p-2}}\right)x^{n-2p}{y}^{p-1}{t}^n{u}^p.$$

Comparing the linearly independent hook images proves the formula and excludes every other scalar or hook constituent. Specialising the symmetric functions to dimW variables gives the assertion for W. □

Theorem 6.5 recovers the hook part of Proposition 6.3 and the Catalan diagonal of Proposition 6.4.

For example, in characteristic zero the first generator beyond the one-marker polarisations has (n,p) = (4,2) and type Λ²W. For a,b ∈ W, a representative is

abee − aeeb − baee + beea − eabe + ebae + eeab − eeba.

In characteristic two this integral representative becomes decomposable; homogeneous generators with prime leading words are supplied by Theorem 6.1.

6.3. Hilbert Series and Invariant Algebras

Put $T_n=\sum _p\dim {\mathcal{Z}}_n^p$, ${\mathcal{H}}_d\left(x\right)=1+\sum _{n\ge 1}{T}_n{x}^n$, and $q_d\left(x\right)=\sum _{n\ge 1}\left(\dim Q_n\right)x^n$. Freeness gives $q_d=1-{\mathcal{H}}_d^{-1}$. For n ≥ 1 and d = 1,

$$T_n=2^{n-1}, q_1\left(x\right)={\displaystyle \frac{x}{1-x}}.$$

For d = 2,

$$T_n={\displaystyle \frac{3^n+2n-1}{4}}, q_2\left(x\right)={\displaystyle \frac{x\left(1-2x\right)}{1-4x+5x^2-3x^3}}.$$

These identities follow by summing the Column Law. More explicitly, for d = 1, $\dim {\mathcal{Z}}_n^p=\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{p-1}}\right)$. For d = 2, the one-marker part has dimension n, while the free-range formula gives

$${\displaystyle \sum _{p=2}^n}\dim {\mathcal{Z}}_n^p={\displaystyle \sum _{k=2}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)2^{k-2}={\displaystyle \frac{3^n-1-2n}{4}}.$$

The generator series follow by inversion. For every d ≥ 1, the algebra 𝒵 is infinitely generated, since $Q_n^1\ne 0$ in every length.

The invariant algebra 𝒵(W)^G is free for every subgroup G ≤ GL(W). Choose a degreewise GL(W)-stable lift of Q to identify 𝒵(W) equivariantly with T(Q). The Lane–Kharchenko theorem [15,16], in the arbitrary-rank form of Ferreira–Murakami–Paques [7, Corollary 3.3], then gives freeness. For a cyclic frame, Lemma 2.2 identifies this algebra with the invariant closed cochains.

Figure 2 shows the support of the generator dimensions and the two-marker multiplicities, including the boundary values of Proposition 6.3.

7. The Free Range and Grassmannian Sections

The dimension formula simplifies when p ≥ d; for p < d it is realised by sections of a vector bundle on a Grassmannian.

7.1. The Free Range

Put d = dimW ≥ 1. The marker operators make (W⊕Ke)^⊗n a graded module over the exterior algebra E = Λ(W^*).

Proposition 7.1 (Free-range dimensions). For d ≤ p ≤ n,

$$\dim {\mathcal{Z}}_n^p\left(W\right)={\displaystyle \sum _{k=p}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(d-1\right)}^{n-k}\left({\displaystyle \genfrac{}{}{0pt}{}{k-d}{p-d}}\right).$$

(7.1)

Equivalently, this dimension is the coefficient of t^p−d in the polynomial quotient obtained by dividing (d + t)ⁿ by (1 + t)^d. Consequently,

$$\sum _{n\ge 0}{\displaystyle \sum _{p=d}^n}\dim {\mathcal{Z}}_n^p\left(W\right)x^n{y}^p={\displaystyle \frac{{\left(\mathit{xy}\right)}^d}{{\left(1-\left(d-1\right)x\right)}^d\left(1-\mathit{dx}-\mathit{xy}\right)}}.$$

(7.2)

Proof. In the super Schur–Weyl decomposition, a covariant tensor module L_μ is typical precisely when 𝓁(μ) ≥ d. For the even-first Borel, its Kac presentation is Λ(Hom(W,Ke))⊗L⁰, so it is free over E; its generators have marker degree 𝓁(μ) − d, and its socle has degree 𝓁(μ). These are the standard typicality and Kac-module facts for 𝔤𝔩(d|1), with the raising odd nilradical used in Remark 2.3; see Cheng–Wang [5, (2.2), Theorem 2.7 and Proposition 3.11]. The remaining summands have top marker degree 𝓁(μ) < d, by (2.9). Thus

(d + t)ⁿ = (1 + t)^dG_n(t) + A_n(t),  degA_n < d,

where G_n counts the generators of the typical summands, including their Specht multiplicities. In degree p ≥ d, the joint kernel is their socle, so $\dim {\mathcal{Z}}_n^p=\left[t^{p-d}\right]G_n\left(t\right)$.

Expand

$${\left(d+t\right)}^n={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(d-1\right)}^{n-k}{\left(1+t\right)}^k.$$

The terms with k < d form the remainder and the others give the quotient. Taking its coefficient proves (7.1). Summing the quotient over n, using

$$\sum _{n\ge k}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\left(d-1\right)}^{n-k}{x}^n={\displaystyle \frac{x^k}{{\left(1-\left(d-1\right)x\right)}^{k+1}}},$$

and then summing the geometric series over k ≥ d, gives (7.2). □

Here freeness refers to the exterior-algebra action of the contractions; it is separate from the associative-algebra freeness of Section 6.

Corollary 7.2 (A primitive in the free range). Choose a basis $w_1^{*},\dots ,w_d^{*}$ of W^* and put $\Delta =d_{w_1^{*}}\dots d_{w_d^{*}}$. For p ≥ d,

$${\mathcal{Z}}_n^p\left(W\right)=\Delta \left(C_n^{p-d}\left(W\right)\right).$$

(7.3)

Proof. On a free E-module the joint kernel of multiplication by its generators is the image of its top exterior power. Apply this to every typical summand. Atypical summands have no component in degree p ≥ d. □

7.2. Sections on a Grassmannian

Let 1 ≤ p < d, let X = Gr_p(W) denote the Grassmannian of rank-p quotients of W, and let $𝒬$ be its tautological quotient bundle. The symmetric group acts on $𝒬$^⊗n by ordinary place permutations.

Theorem 7.3 (Grassmannian evaluation). There is a natural GL(W) × S_n-equivariant isomorphism

$${\mathcal{Z}}_n^p\left(W\right)\hspace{0.33em}\stackrel{\hspace{0.25em}\mathrm{ev}\hspace{0.25em}}{\to }\hspace{0.33em}{H}^0\left(X,{\mathcal{Q}}^{\otimes n}\otimes {\left(\det \mathcal{Q}\right)}^{-1}\right).$$

At a quotient A:W↠U, choose a basis u₁,…,u_p of U. Replace the p markers of a word by the alternating sum of their assignments to u₁,…,u_p, apply A to its letters in W, and tensor with ${\left(u_1\wedge \dots \wedge u_p\right)}^{*}$. This prescription is independent of the basis and defines ev.

Proof. The prescription is polynomial in the quotient map and is unchanged by a change of quotient frame, since the determinant factor cancels the change in the alternating tensor. Interchanging two markers changes sign on both sides, so evaluation intertwines the signed place action on the source with the ordinary place action on the target.

Schur–Weyl duality gives

$${\mathcal{Q}}^{\otimes n}\otimes {\left(\det \mathcal{Q}\right)}^{-1}\cong \underset{\begin{array}{c}\mu ⊢n\\ \mathcal{l}\left(\mu \right)\le p\end{array}}{⨁}{\mathbb{S}}_{\mu -\left(1^p\right)}\mathcal{Q}\otimes S^{\mu },$$

where the notation on the right permits a final part equal to −1. If 𝓁(μ) = p, Borel–Weil identifies its space of sections with ${\mathbb{S}}_{\mu -\left(1^p\right)}W$. If 𝓁(μ) < p, the corresponding weight is singular: after adding the usual ρ, its p-th and (p + 1)-st entries coincide. Borel–Weil–Bott [2,29] therefore gives zero cohomology in every degree for that summand. Consequently the target is

$$\underset{\begin{array}{c}\mu ⊢n\\ \mathcal{l}\left(\mu \right)=p\end{array}}{⨁}{\mathbb{S}}_{\mu -\left(1^p\right)}W\otimes S^{\mu },$$

which is the multiplicity-free decomposition of the source supplied by the Column Law.

It remains to show that the specified map is nonzero on each summand. First take a real vector space U of dimension p, with an orthonormal basis, and let $\Delta =d_{u_p^{*}}\dots d_{u_1^{*}}:C_n^0\left(U\right)\to C_n^p\left(U\right)$. The free-range identity gives ${\mathcal{Z}}_n^p\left(U\right)=\mathrm{im}\Delta$. Up to a fixed overall sign, evaluation at the identity quotient is the transpose Δ^t: the transpose contractions replace markers with the alternating basis frame. Since imΔ∩kerΔ^t = 0, this evaluation is injective on ${\mathcal{Z}}_n^p\left(U\right)$.

Now choose a split quotient W↠U. Its splitting induces ${\mathcal{Z}}_n^p\left(U\right)\hookrightarrow {\mathcal{Z}}_n^p\left(W\right)$. Every partition μ with 𝓁(μ) = p occurs in ${\mathcal{Z}}_n^p\left(U\right)$, so evaluation on W is nonzero on every summand. Schur’s lemma proves the isomorphism over ℝ. The maps and decompositions are defined over ℚ, hence the result holds over every characteristic-zero field by scalar extension. □

Corollary 7.4 (exact polynomial growth). Put D = p(d − p). There is a polynomial P_p,d(n) of degree D such that

$$\dim {\mathcal{Z}}_n^p\left(W\right)=p^n{P}_{p,d}\left(n\right), \left[n^D\right]P_{p,d}\left(n\right)={\displaystyle \frac{\mathrm{deg}{\mathrm{Gr}}_p\left(W\right)}{D!p^D}}=p^{-D}{\displaystyle \prod _{i=0}^{p-1}}{\displaystyle \frac{i!}{\left(d-p+i\right)!}}.$$

The equality holds for every nonnegative integer n, with both sides zero when n < p.

Proof. The preceding Borel–Weil–Bott computation also proves that all higher cohomology vanishes. Hirzebruch–Riemann–Roch [10] therefore gives

$$\dim {\mathcal{Z}}_n^p\left(W\right)={\int }_X\mathrm{ch}{\left(\mathcal{Q}\right)}^n{e}^{-c_1\left(\mathcal{Q}\right)}\mathrm{td}\left(X\right).$$

Write ch($𝒬$) = p + N, where N has positive cohomological degree. Since N^D+1 = 0,

$$\dim {\mathcal{Z}}_n^p\left(W\right)=p^n{\displaystyle \sum _{j=0}^D}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)p^{-j}{\int }_X{N}^j{e}^{-c_1\left(\mathcal{Q}\right)}\mathrm{td}\left(X\right).$$

Only the term j = D contributes to the leading coefficient, and its integral is ${\int }_X{c}_1{\left(\mathcal{Q}\right)}^D=\mathrm{deg}X>0$. The usual hook-length expression for the degree of the Plücker embedding gives the displayed product. □

For p = 1 this recovers $\dim {\mathcal{Z}}_n^1=\left({\displaystyle \genfrac{}{}{0pt}{}{n+d-2}{d-1}}\right)$, with n ≥ 1. At the other boundary,

$$\dim {\mathcal{Z}}_n^{d-1}=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d-1}}\right){\left(d-1\right)}^{n-d+1}.$$

For the latter identity, identify the Grassmannian with projective space and use the tautological sequence 0 → $𝒪$(−1) → W⊗$𝒪$ → $𝒬$ → 0. In its Grothendieck group, [$𝒬$] = d − [$𝒪$(−1)] and (det$𝒬$)⁻¹ differs from $𝒪$(−1) only by a constant one-dimensional factor. Thus its Euler characteristic is

$${\displaystyle \sum _{j=0}^n}{\left(-1\right)}^{j+d-1}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)d^{n-j}\left({\displaystyle \genfrac{}{}{0pt}{}{j}{d-1}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{d-1}}\right){\left(d-1\right)}^{n-d+1}.$$

Corollary 7.5 (rationality). For fixed d, put

$${\mathcal{H}}_d\left(x,y\right)=\sum _{n,p\ge 0}\dim {\mathcal{Z}}_n^p\left(W\right)x^n{y}^p.$$

This series is rational. More precisely,

$${\mathcal{H}}_d\left(x,y\right)=1+{\displaystyle \sum _{p=1}^{d-1}}{y}^p\sum _{n\ge 0}{p}^n{P}_{p,d}\left(n\right)x^n+{\displaystyle \frac{{\left(\mathit{xy}\right)}^d}{{\left(1-\left(d-1\right)x\right)}^d\left(1-\mathit{dx}-\mathit{xy}\right)}}.$$

For 1 ≤ p < d, the series $\sum _n\dim {\mathcal{Z}}_n^p{x}^n$ has a pole of order exactly p(d − p) + 1 at x = 1/p.

Proof. A polynomial of degree D multiplied by pⁿ has a rational generating function with denominator (1 − px)^D+1; its nonzero leading coefficient prevents cancellation of that pole. The final term is the free-range series. □

The evaluation map depends on the characteristic: when p = 1, the closed cochain $\mathit{we}+\mathit{ew}\in {\mathcal{Z}}_2^1$ evaluates to 2A(w) and hence vanishes in characteristic two.

8. Variations

The frame reduction also applies when its orbit spans only part of W^*. Allowing several marker types replaces first-column deletion by deletion of the first k columns.

8.1. Frames with a Smaller Orbit Span

Let G ⊆ GL(W) and θ ∈ W^*. Set

A = span_K(Gθ),  W₀ = A^⊥ ⊆ W.

Thus A and W₀ are G-stable. The orbit-amplification argument does not require A = W^*.

Proposition 8.1 (Orbit-span decomposition). One has

$$Z_n^p\left(G,W,\theta \right)={\left(\bigcap _{\phi \in A}\ker d_{\phi }\right)}^G.$$

(8.1)

Suppose that W = W₀⊕W' with W' G-stable. Then, as G × S_n-modules,

$$\bigcap _{\phi \in A}\ker d_{\phi }\cong \stackrel{n}{\underset{k=0}{⨁}}{\mathrm{Ind}}_{S_{n-k}\times S_k}^{S_n}\left(W_0^{\otimes \left(n-k\right)}⊠{\mathcal{Z}}_k^p\left(W\mathrm{\text{'}}\right)\right).$$

(8.2)

Here ${\mathcal{Z}}_k^p=0$ for p > k, ${\mathcal{Z}}_0^0=K$, and the action of G on each tensor product is diagonal. If $c_{\alpha ,\beta }^{\mu }$ denotes a Littlewood–Richardson coefficient, then

$$Z_n^p\left(G,W,\theta \right)\cong \underset{\mu ⊢n}{⨁}{\left(S^{\mu }\right)}^{\oplus b_{\mu }},$$

(8.3)

where

$$b_{\mu }={\displaystyle \sum _{k=p}^n}\sum _{\alpha ⊢n-k}\sum _{\begin{array}{c}\lambda ⊢k-p\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{c}_{\alpha ,\lambda +1^p}^{\mu }\hspace{0.17em}\dim {\left({\mathbb{S}}_{\alpha }{W}_0\otimes {\mathbb{S}}_{\lambda }W\mathrm{\text{'}}\right)}^G.$$

(8.4)

Proof. If v is G-invariant and d_θv = 0, then equivariance gives d_gθv = 0 for every g, and linearity gives (8.1). Restriction identifies A with W'^*. Expand

$${\left(W\oplus \mathit{Ke}\right)}^{\otimes n}=\underset{S\subseteq \left[n\right]}{⨁}{W}_0^{\otimes S^c}\otimes {\left(W\mathrm{\text{'}}\oplus \mathit{Ke}\right)}^{\otimes S}.$$

The operators d_φ, φ ∈ A, preserve each summand, kill its W₀-slots, and act on the remaining slots as the full family of marker operators for W'. Taking their joint kernel and collecting the subsets S by cardinality proves (8.2). All W₀-slots are even, so their permutation with the other slots introduces no additional sign. Schur–Weyl duality for $W_0^{\otimes \left(n-k\right)}$, the Column Law for ${\mathcal{Z}}_k^p\left(W\mathrm{\text{'}}\right)$, and the Littlewood–Richardson induction rule give (8.3)–(8.4). Taking G-invariants here requires no averaging: the decomposition is already G-equivariant, and G acts only on the displayed Schur multiplicity spaces. □

The complement hypothesis holds for finite G in characteristic zero.

Example 8.2. Let G = C₂ act on W = Kx⊕Ky by gx = x and gy = −y, and take θ₀ = x^*. Then W₀ = Ky, W' = Kx, and, for p ≥ 1,

$$\dim Z_n^p\left(C_2,W,{\theta }_0\right)=\sum _{\begin{array}{c}p\le k\le n\\ k\equiv n\hspace{0.25em}(\mathrm{mod}\hspace{0.25em}2)\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{p-1}}\right).$$

(8.5)

Indeed, $\dim {\mathcal{Z}}_k^p\left(\mathit{Kx}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{k-1}{p-1}}\right)$, while the n − k copies of y contribute the character (−1)^n−k. In particular, for n ≥ 1,

$$\dim Z_n^1\left(C_2,W,{\theta }_0\right)=2^{n-1}-1_{2\mid n}.$$

(8.6)

For the cyclic frame θ₊=x^* + y^*, Corollary 4.1 instead gives

$$\dim Z_n^1\left(C_2,W,{\theta }_{+}\right)=⌊\left(n+1\right)/2⌋.$$

(8.7)

Here 𝕊_(a,b)W≅(detW)^⊗b⊗Sym^a−bW, so the multiplicity a_(a,b) in (4.3) counts the integers 0 ≤ j ≤ a − b with j + b even. Table 5 compares the two frames at p = 1,2.

Thus failure of cyclicity can leave an exponentially larger invariant closed space.

8.2. Several Odd Markers

Let U be an odd vector space of dimension k ≥ 1, let W be even of dimension d ≥ 1, and put V = W⊕U. For u⊗ϕ ∈ U⊗W^*, let d_u,ϕ be the odd derivation that sends w ∈ W to ϕ(w)u and kills U. Write

$$\mathcal{Z}{\left(W,U\right)}_n=\bigcap _{u,\varphi }\ker \left(d_{u,\varphi }:V^{\otimes n}\to V^{\otimes n}\right).$$

Theorem 8.3 (the k-column law). In characteristic zero,

$$\mathcal{Z}{\left(W,U\right)}_n\cong \underset{\begin{array}{c}\mu ⊢n\\ {\mu }_{d+1}\le k\end{array}}{⨁}{\mathbb{S}}_{\left({\left({\mu }_1-k\right)}_{+},{\left({\mu }_2-k\right)}_{+},\dots \right)}W\otimes {\mathbb{S}}_{\left(\mu {\mathrm{\text{'}}}_1,\dots ,\mu {\mathrm{\text{'}}}_k\right)}U\otimes S^{\mu }.$$

Here 𝕊_νU means the ordinary Schur module of the underlying k-dimensional space; its parity is the parity of |ν|.

The odd-first highest-weight rule is due to Cheng–Lam–Wang [3]; we use the formulation in Cheng–Wang [5, Theorem 3.14 and Example 3.16(2)].

Proof. Super Schur–Weyl duality decomposes V^⊗n as ${⨁}_{{\mu }_{d+1}\le k}{L}_{\mu }\otimes S^{\mu }$. The contractions form the odd nilradical Hom(W,U) of the parabolic whose ordered blocks are U,W. The invariant subspace of a simple polynomial module L_μ is a simple module for the even Levi subgroup GL(W) × GL(U): an additional Levi highest-weight vector would be an additional highest-weight vector for the full Borel in the simple module. For the odd-first Borel, the highest-weight rule reads the first k columns of μ as the U-weight and then reads the remaining rows as the W-weight. This gives exactly the two Schur modules displayed above. □

Proposition 8.4 (freeness with several markers). The algebra 𝒵(W,U) = ⨁_n𝒵(W,U)_n is free associative over every field.

Proof. Suppose first that charK ≠ 2. The contractions and the parity operator σ give a homogeneous action on T(W⊕U) of the Hopf algebra

$ℰ$ = Λ(U⊗W^*)⋊K[C₂],  Δ_H(d) = d⊗1 + σ⊗d.

The subalgebra A generated by the σd is a right coideal: its generators satisfy

Δ_H(σd) = σd⊗σ + 1⊗σd ∈ A⊗$ℰ$.

Its invariant algebra is exactly the joint contraction kernel, since σ is invertible. Ferreira–Murakami–Paques [7, Corollary 3.2] prove that the invariant algebra of a right coideal under a homogeneous Hopf action on a free algebra is free. In characteristic two the contractions are ordinary commuting square-zero derivations. Use instead the exterior Hopf algebra on U⊗W^*, with primitive generators, and take the whole Hopf algebra as the coideal. The same corollary applies. □

Conjecture 8.5 (characteristic-free several-marker leading words). Order the alphabet by e₁<…<e_k < 1<…<d. Standardise each marker type separately: the j-th occurrence of e_i from the right receives ε_i,j, with ε_i,j+1 < ε_i,j; all symbols of type i precede those of type i + 1, and all marker symbols precede the letters. Apply ordinary RSK to the reversed standardised word. Let μ(u) be its shape and $\tilde{\kappa }\left(u\right)$ the shape occupied by marker entries in its insertion tableau.

For every field K, the leading words in 𝒵(W,U)_n are exactly those satisfying

$$\tilde{\kappa }{\left(u\right)}_i=\min \left(k,\mu {\left(u\right)}_i\right) \mathrm{for\; every\; row}i.$$

Equivalently, the markers fill precisely the first k columns of the insertion tableau, including every box of rows shorter than k. For fixed letter content α and marker content β, this would give

$${\dim }_K\mathcal{Z}{\left(W,U\right)}_n^{\left(\alpha ;\beta \right)}=\sum _{\mu ⊢n}{f}^{\mu }{K}_{{\lambda }^{\left(k\right)}\left(\mu \right),\alpha }\hspace{0.17em}{K}_{{\kappa }^{\left(k\right)}\left(\mu \right),\beta },$$

where ${\lambda }^{\left(k\right)}\left(\mu \right)={\left({\left({\mu }_i-k\right)}_{+}\right)}_i$, with zero parts removed, and κ^(k)(μ) = (μ'₁,…,μ'_k), again omitting zeros. Here |α| + |β| = n.

The proposed set has the displayed cardinality: standardised RSK gives a semistandard letter tableau of shape λ^(k)(μ), a semistandard tableau of shape κ^(k)(μ) obtained by transposing the marker part, and a standard recording tableau of shape μ. The inverse-descent property forces the prescribed order of the standardised occurrences of each marker type, giving the inverse bijection.

Theorem 8.3 gives this dimension in characteristic zero. A leading-word inclusion over every field would prove the conjecture by the integer-rank argument of Theorem 5.5.

9. Chessboard Complexes

The multilinear content spaces encode placements of nonattacking rooks. Let M_m,n be the chessboard complex of the complete rectangular m × n board, with 1 ≤ m ≤ n; it has dimension m − 1. All homology groups below are reduced.

Lemma 9.1 (the top-degree dictionary). Over any field K, the space ${\tilde{H}}_{m-1}\left(M_{m,n};K\right)$ is isomorphic to the subspace of ${\mathcal{Z}}_n^{n-m}\left(K^m\right)$ with each nonmarker letter occurring exactly once.

Proof. Encode a placement of m rooks by the word having letter a in column j when its row-a rook lies in column j, and marker e in each empty column. Order the vertices of a simplex by increasing column. The simplicial boundary sign for removing a rook in column j is

(−1)^{#{rooks in columns <j}},

whereas the corresponding contraction sign is

(−1)^{#{empty columns <j}}.

Their ratio is (−1)^j−1. For any placement R, set

$$\epsilon \left(R\right)=\prod _{\left(a,j\right)\in R}{\left(-1\right)}^{j-1}.$$

Let ϕ(R) = ε(R)word(R). Since the number of empty columns before j plus the number of occupied columns before j is j − 1, the signs give $\varphi \circ \partial =\left(\sum _a{d}_a\right)\circ \varphi$. In top degree, the outputs obtained by deleting different row labels lie in distinct content spaces. Consequently the boundary vanishes exactly when every individual contraction vanishes. There are no simplices of dimension m, so the top homology equals the top cycle space. □

Theorem 9.2 (field-independent top homology). For every field K,

$${\dim }_K{\tilde{H}}_{m-1}\left(M_{m,n};K\right)=\sum _{\begin{array}{c}\lambda ⊢m\\ \mathcal{l}\left(\lambda \right)\le n-m\end{array}}{f}^{\lambda }{f}^{\lambda +\left(1^{n-m}\right)}.$$

(9.1)

In particular, this dimension is independent of the characteristic.

Proof. Apply Theorem 5.5 with p = n − m and content (1,…,1). A semistandard tableau with this content is a standard tableau, so $K_{\lambda ,\left(1^m\right)}=f^{\lambda }$. Lemma 9.1 identifies the resulting joint kernel with the required homology. □

Over ℚ, (9.1) is Garst’s top-homology formula [27, Corollary 6.2], with conjugate partitions. The Friedman–Hanlon decomposition was known over fields of characteristic greater than m and n [28, Remark 3.4].

Corollary 9.3. The group ${\tilde{H}}_{m-2}\left(M_{m,n};\mathbb{Z}\right)$ is free abelian for every 1 ≤ m ≤ n.

Proof. For m ≥ 2, the homological universal coefficient theorem gives, for every prime q, a short exact sequence

$$0\to {\tilde{H}}_{m-1}\left(M_{m,n};\mathbb{Z}\right)\otimes {\mathbb{F}}_q\to {\tilde{H}}_{m-1}\left(M_{m,n};{\mathbb{F}}_q\right)\to {\mathrm{Tor}}_1^{\mathbb{Z}}\left({\tilde{H}}_{m-2}\left(M_{m,n};\mathbb{Z}\right),{\mathbb{F}}_q\right)\to 0.$$

(9.2)

The integral top homology is a subgroup of the free abelian top chain group and is therefore free. Its rank is its rational dimension. By Theorem 9.2, the first two terms of (9.2) have the same dimension, so the Tor term vanishes for every prime. The finitely generated group ${\tilde{H}}_{m-2}\left(M_{m,n};\mathbb{Z}\right)$ consequently has no torsion and is free abelian. When m = 1, the complex is nonempty and ${\tilde{H}}_{-1}=0$, which gives the remaining case. □

Since top homology is free and homology vanishes above dimension m − 1, Corollary 9.3 gives ${\tilde{H}}_i\left(M_{m,n};\mathbb{Z}\right)$ free for every i ≥ m − 2. This proves the torsion-free implication of Shareshian–Wachs Conjecture 7.4 [27].

Put

ν_m,n = min{m,n,⌊(m + n + 1)/3⌋} − 1.

Shareshian and Wachs [27, Conjecture 7.1] conjectured that ${\tilde{H}}_{{\nu }_{m,n}}\left(M_{m,n};\mathbb{Z}\right)$ is torsion-free exactly when n ≥ 2m − 4. The families n = 2m − 4 and n = 2m − 3 were left open in [27, §§7,11].

An integral top-homology basis alone leaves the Tor term in (9.2) undetermined. The leading-word bound controls the top-cycle dimension over every field and makes that term vanish.

Corollary 9.4. For 2 ≤ m ≤ n,

$${\tilde{H}}_{{\nu }_{m,n}}\left(M_{m,n};\mathbb{Z}\right)\mathrm{is\; torsion}-\mathrm{free} \iff n\ge 2m-4.$$

Proof. If n ≥ 2m − 1, then ν_m,n = m − 1 and the group is top homology, hence free. The case n = 2m − 2 is Shareshian–Wachs [27, Theorem 7.2]. For n = 2m − 4,2m − 3, one has ν_m,n = m − 2, so Corollary 9.3 applies. When n ≤ 2m − 5, Friedman–Hanlon [8; 27, Theorem 1.5] and Shareshian–Wachs [27, Theorem 3.1] give finite nonzero bottom homology except at (6,6),(7,7),(8,9). The first two have 3-torsion [27, Table 1.2]; the last has 3-torsion by Jonsson [11, Theorem 2]. Thus every board in this last range has torsion. □

This settles the i = 3,4 families in Wachs’s problem [28, Problem 6.10], where n = 2m − i; their ranks follow from Friedman–Hanlon [8; 27, Theorem 6.1]. The other listed families, i = 6,7,10, have bottom-degree 3-torsion by Jonsson [11, Theorem 2; 27, Table 1.2].

Example. On the 5 × 6 board, which lies in n = 2m − 4, the relevant words have five distinct letters and one marker. There is exactly one ballot word, 54321e: the marker must be last and no increasing letter pair is allowed. Equivalently, (9.1) has only λ = (5) and gives f⁽⁵⁾f⁽⁶⁾ = 1. Hence the top homology has dimension one over every field, and ${\tilde{H}}_3\left(M_{5,6};\mathbb{Z}\right)$ is free. The face counts in degrees 0,1,2,3,4 are 30,300,1200,1800,720. Their reduced Euler characteristic is −151; the known connectivity below degree 3 [11, Theorem 1.1] then gives ${\tilde{H}}_3\left(M_{5,6};\mathbb{Z}\right)\cong {\mathbb{Z}}^{152}$.

In the proof of their top-homology basis theorem, Shareshian and Wachs [27, Theorem 6.3, (6.5)–(6.6)] pair integral cycles with single-simplex cocycles v(S,T) to obtain a unitriangular matrix, giving the same lower bound over every field. Theorem 5.3 supplies the corresponding upper bound on top cycles.

9.1. The Cartan Complex and Row Bounds

Let W = K^d, M = (W⊕Ke)^⊗n, and E = Λ(W^*), acting on M by the contractions d_a. In this subsection Ext_E means Ext for ordinary left modules over the exterior algebra, with the content grading retained and the super grading forgotten. Give d_a content degree −e_a and its dual variable w_a degree e_a. The Cartan cochain complex is

$${\mathcal{C}}^i\left(M\right)=M\otimes {\mathrm{Sym}}^{i}W, \Delta \left(m\otimes f\right)={\displaystyle \sum _{a=1}^d}{d}_{a}m\otimes w_{a}f.$$

The total content is preserved. The relations $d_a^2=0$ and d_ad_b + d_bd_a = 0 give Δ² = 0, including in characteristic two.

Proposition 9.5 (the dictionary in every degree). For α ∈ ℕ^d, let r = |α|, and let Δ_α,n be the complex of sets of squares on the complete d × n board containing at most α_a squares in row a and at most one in each column. For every field K, every n ≥ 0, and every i ≥ 0,

$${\mathrm{Ext}}_E^i{\left(K,M\right)}_{\alpha }\cong {\tilde{H}}_{r-1-i}\left({\Delta }_{\alpha ,n};K\right).$$

There is no restriction r ≤ n in this assertion. In particular, with d = m and α = (1^m),

$${\mathrm{Ext}}_{\Lambda \left({\left(K^m\right)}^{*}\right)}^i(K,{\left(K^m\oplus \mathit{Ke}\right)}^{\otimes n}{)}_{\left(1^m\right)}\cong {\tilde{H}}_{m-1-i}\left(M_{m,n};K\right).$$

Proof. Use the Cartan resolution P_i = E⊗Γⁱ(W^*) of the trivial module, where Γ denotes divided powers. For its divided-power basis γ_b, the differential is

$$\partial \left(a\otimes {\gamma }_{\mathbf{b}}\right)=\sum _{j:b_j>0}a{d}_j\otimes {\gamma }_{\mathbf{b}-{\mathbf{e}}_j}.$$

This is left E-linear. Its exactness can be checked in each multidegree: in a nonzero multidegree the basis is indexed by subsets of the nonzero coordinates, and the differential is the signed operation of adjoining a coordinate, an acyclic augmented simplex complex. Degree zero gives the augmentation onto K. The divided powers ensure that the formula has no coefficients b_j, so it works over every field. Applying Hom_E(−,M) gives precisely ($𝒞$^•(M),Δ).

The content-α component has basis

u⊗w^α−β,  content(u) = β ≤ α,  |β| = r − i.

Here u encodes a placement R of r − i rooks, with a letter in each occupied column and e in each empty column. Order the vertices by increasing column and put

$$\epsilon \left(R\right)=\prod _{\left(a,j\right)\in R}{\left(-1\right)}^{j-1}.$$

The map u⊗w^α−β ↦ ε(R)[R] intertwines Δ with the simplicial boundary. Indeed, deleting the rook in column j has contraction sign (−1)^{#{empty columns before j}}, boundary sign (−1)^{#{occupied columns before j}}, and ε(R∖{(a,j)}) = (−1)^j−1ε(R). This calculation is independent of the row and does not require squarefree content. The empty face corresponds to eⁿ⊗w^α in cochain degree r, so the chain complex is augmented. Taking homology proves the assertion. □

Under Proposition 9.5, for 2 ≤ m ≤ n the bottom degree ν_m,n corresponds to Ext degree i = m − 1 − ν_m,n = max(0,m − ⌊(m + n + 1)/3⌋), and n ≥ 2m − 4 is equivalent to i ≤ 1. Thus Corollary 9.4 says that bottom homology is torsion-free exactly when it lies in Ext degree 0 or 1 (Figure 3).

Theorem 9.6 (row-bounded complexes). Suppose r = |α| ≤ n, and put p = n − r. Then for every field K,

$${\dim }_K{\tilde{H}}_{r-1}\left({\Delta }_{\alpha ,n};K\right)=\sum _{\begin{array}{c}\lambda ⊢r\\ \mathcal{l}\left(\lambda \right)\le p\end{array}}{K}_{\mathit{\lambda \alpha }}\hspace{0.17em}{f}^{\lambda +1^p}.$$

In particular, this dimension is independent of K, and

$${\tilde{H}}_{r-2}\left({\Delta }_{\alpha ,n};\mathbb{Z}\right) \mathrm{is\; free\; abelian}.$$

Proof. For r > 0, the complex has dimension r − 1: all row capacities can be filled in distinct columns. Its top chains have word content exactly α, and there are no chains one degree higher. By Proposition 9.5 its top cycles form ${\mathcal{Z}}_n^{n-r}{\left(W\right)}_{\alpha }$. Equivalently, the terms d_au⊗w_a lie in distinct summands, so their sum vanishes exactly when all contractions vanish. Theorem 5.5 gives the displayed dimension. The universal coefficient argument of Corollary 9.3 now proves the assertion about integral homology. If r = 0, the complex consists only of the empty face; by the augmented convention ${\tilde{H}}_{-1}=K$ and ${\tilde{H}}_{-2}=0$, in agreement with the formula. If p = 0 < r, the sum is empty and the top homology vanishes. □

These complexes are the chessboard complexes with multiplicities ${\Delta }_{\alpha ,1^n}$ of Reiner and Roberts [22, Proposition 3.2], for every α. Karaguezian, Reiner and Wachs [14, §5] realise these complexes as weight spaces of a GL(W) × GL_n-complex with terms Λ^q(W⊗Kⁿ)⊗Sym^*W⊗Sym^*(Kⁿ).

The convention in [22] uses A = K[z_aj] and the Segre module

$$\mathrm{Segre}\left(d,n,r-n\right)=\underset{a,b\ge 0,\hspace{0.25em}a-b=r-n}{⨁}{\mathrm{Sym}}^a{K}^d\otimes {\mathrm{Sym}}^b{K}^n.$$

The corresponding relation is

$${\mathrm{Tor}}_q^A(\mathrm{Segre}\left(d,n,r-n\right),K{)}_{\left(\alpha ,1^n\right)}\cong {\tilde{H}}_{q-1}\left({\Delta }_{\alpha ,n};K\right).$$

When r = n, the module is A/I₂, where I₂ is the ideal of 2 × 2 minors. Thus the homological index is q − 1 and the module is the quotient by the ideal. Proposition 9.5 gives an exterior-algebra Ext description of the same complexes.

10. Computations

The proofs are independent of the computations. The supplied verification files are run by verification/run_all.py. Rational row reduction gives the fifteen cases in Table 6, the seven single-frame comparisons in Table 4, and the six C₂ frame kernels in Table 5; the formula (4.5) is also checked for 2 ≤ n ≤ 30. Spectral checks cover nine full matrices and 539 content triples, and a size-limited sweep covers 153 pieces at primes 2,3,5,7. Ballot leading words are tested at primes 2,3 for (d,n_max) = (2,7),(3,6),(4,5),(5,5); exact character arithmetic checks hook multiplicities through n = 36. Socle checks cover 647 rational content-and-layer cases for d = 0,1,2,3 through lengths 4,5,5,4, and leading words at primes 2,3,1000003 for d = 2,n ≤ 6,k ≤ 2 and d = 3,n ≤ 5,k ≤ 3. Several-marker leading words are checked at the same three primes for (d,k,n_max) = (1,2,7),(2,2,6),(1,3,6),(2,3,5),(3,2,5). Chessboard top boundaries are checked on fifteen boards at primes 2,3,5,7; the Cartan dictionary is checked in every degree for eight contents at primes 3,1000003, and ten row-bounded complexes at primes 2,3,5,1000003. Grassmannian evaluation has the predicted rank at sampled quotients over 𝔽_1000003 in five cases.

11. Open Questions

Theorem 6.1 indexes the free generators by prime ballot words, while Theorem 6.5 determines their hook-shaped constituents. Can the character identity in §6.1 be turned into a direct combinatorial rule for every Schur multiplicity? This asks, in particular, how the nonhook constituents are reflected in the prime-word basis.

Conjecture 8.5 proposes a tableau description of leading words for several marker types. The proposed set has the required cardinality in characteristic zero; the remaining step is to establish the leading-word bound over every field. An analogue of the argument in §5 would then extend the characteristic-free Column Law to this setting.

The chessboard complex is the matching complex of a complete bipartite graph. Can a comparable leading-word description be found for matching complexes of complete graphs? The question is whether the same approach can identify characteristic-independent top-cycle dimensions and, through the universal coefficient theorem, yield torsion-freeness one degree below.

Use of AI tools. Claude (Anthropic), ChatGPT/Codex (OpenAI) were used for mathematical exploration, proof development, literature searches, writing the code for the computations in Section 10, and drafting and editing the text. The author takes full responsibility for the arguments, the computations and the final text.