Classifying partitions of full rank: injectivity of the standard (Foulkes–Howe) map between Young tabloid modules

For a partition $\lambda$ of $n$, let $H^\lambda$ be the set of horizontal tableaux (tabloids) of shape $\lambda$ — set partitions of $\{1,\dots,n\}$ whose block sizes are the parts of $\lambda$ — and let $V^\lambda=H^{\lambda'}$ be the vertical tableaux of the conjugate shape. The orthogonality matrix $K_\lambda$ records which horizontal and vertical tableaux intersect in at most one entry per row–column pair, and the standard map $\psi_\lambda:\mathbb{C}[H^\lambda]\to\mathbb{C}[V^\lambda]$ is the $S_n$-equivariant map it defines; for rectangular $\lambda$ this is the classical Foulkes–Howe map between Foulkes modules. Stanley, and independently Siemons and Wagner, conjectured that $\psi_\lambda$ is injective whenever $\lambda$ dominates its conjugate; Pylyavskyy disproved this and proposed instead that $K_\lambda$ has full rank for every $\lambda$, a reformulation that also fails in general. The problem left open by Pylyavskyy's reformulation is to determine, with proof, exactly which partitions $\lambda$ have full rank $K_\lambda$ — equivalently, to characterize the partitions for which full rank fails. Its rectangular subcase asks exactly for the pairs $(a,b)$ at which the Foulkes–Howe map is injective — a refinement of Foulkes' 1950 conjecture, verified only for $a\leqslant 5$.

Background

A partition $\lambda=(\lambda_1,\lambda_2,\dots)$ is a weakly decreasing sequence of positive integers summing to $n$; its Young diagram is the array of left-justified rows of boxes with $\lambda_i$ boxes in row $i$, and the conjugate partition $\lambda'$ is obtained by exchanging the roles of rows and columns. The dominance order $\lambda\geq\lambda'$ means $\lambda_1+\cdots+\lambda_i\geq\lambda'_1+\cdots+\lambda'_i$ for all $i$.

A tableau of shape $\lambda$ is a filling of its Young diagram with the numbers $1,\dots,n$, each used once. Two tableaux are $h$-equivalent if one is obtained from the other by permuting entries within rows and permuting rows of equal length; a horizontal tableau (tabloid) $\mu\in H^\lambda$ is such an equivalence class, i.e. a set partition of $\{1,\dots,n\}$ into blocks whose sizes are the parts of $\lambda$. Vertical tableaux $\nu\in V^\lambda$ are defined analogously using columns; equivalently $V^\lambda=H^{\lambda'}$. The complex vector spaces $\mathbb{C}[H^\lambda]$ and $\mathbb{C}[V^\lambda]$ are permutation modules for the symmetric group $S_n$; concretely $\mathbb{C}[H^\lambda]\cong 1\!\uparrow_{N_{S_n}(S_\lambda)}^{S_n}$, the induction of the trivial representation from the normalizer of the Young subgroup $S_\lambda=S_{\lambda_1}\times S_{\lambda_2}\times\cdots$. One counts $|H^\lambda|=n!/(\lambda_1!\lambda_2!\cdots\,m_1!m_2!\cdots)$, where $m_j$ is the number of parts of $\lambda$ equal to $j$.

Call $\mu\in H^\lambda$ and $\nu\in V^\lambda$ orthogonal, written $\mu\perp\nu$, if every row of $\mu$ meets every column of $\nu$ in at most one entry — equivalently, if some tableau is simultaneously consistent with $\mu$ and $\nu$. The orthogonality matrix $K_\lambda$ is the 0/1 matrix with rows indexed by $H^\lambda$, columns by $V^\lambda$, and $K_\lambda(\mu,\nu)=1$ exactly when $\mu\perp\nu$. The standard map $\psi_\lambda:\mathbb{C}[H^\lambda]\to\mathbb{C}[V^\lambda]$ sends $\mu$ to $\sum_{\nu\perp\mu}\nu$; its matrix with respect to the natural bases is $K_\lambda$. These objects were introduced in this form by Black and List (1989), and $\psi_\lambda$ is the $S_n$-avatar of the map $\Psi$ on symmetric powers studied classically: for a complex vector space $U$, $\operatorname{Sym}^a(\operatorname{Sym}^b U)$ embeds into tensor space, the tensor factors are reordered, and one projects to $\operatorname{Sym}^b(\operatorname{Sym}^a U)$. For the rectangle $\lambda=(b,b,\dots,b)$ with $a$ rows, $\mathbb{C}[H^\lambda]$ is the Foulkes module and $\psi_\lambda=\Psi^{a\times b}$ is the Foulkes–Howe map; by Schur–Weyl duality (the fundamental dictionary between polynomial representations of the general linear group and actions of the symmetric group on tensor powers) its injectivity for $a\leqslant b$ implies the containment of plethysm representations $\operatorname{Sym}^a(\operatorname{Sym}^b U)\hookrightarrow\operatorname{Sym}^b(\operatorname{Sym}^a U)$.

The surrounding conjectures have a long history. Foulkes (1950) conjectured that for $a\leqslant b$ every plethysm coefficient of $h_b\circ h_a$ dominates the corresponding coefficient of $h_a\circ h_b$ (coefficients are taken in the Schur basis, the standard basis of the ring of symmetric functions indexed by partitions), where $h_d$ is the complete homogeneous symmetric function and $\circ$ denotes plethysm, i.e. composition of symmetric functions; equivalently, $h_b\circ h_a-h_a\circ h_b$ is Schur-positive, meaning that all of its Schur coefficients are nonnegative. This is equivalent to the existence of an equivariant inclusion $\operatorname{Sym}^a(\operatorname{Sym}^b U)\hookrightarrow\operatorname{Sym}^b(\operatorname{Sym}^a U)$ and to the corresponding containment of Foulkes modules. It is known for $a\leqslant 2$ (Thrall 1942), $a=3$ (Dent and Siemons 2000), $a=4$ (Müller and Neunhöffer 2005, combined with McKay's propagation theorem), $a=5$ (Cheung, Ikenmeyer and Mkrtchyan 2015/2017, by computer), and for $b$ sufficiently large relative to fixed $a$ (Brion 1993); it remains open for $a\geqslant 6$. Hadamard (1897), and later Howe (1987) in modern language, conjectured the stronger statement that $\Psi^{a\times b}$ itself is injective for all $a\leqslant b$ (the Foulkes–Howe conjecture); Müller and Neunhöffer (2005) refuted it by computing that $\Psi^{5\times 5}$ has a nontrivial kernel (while $\Psi^{4\times 4}$ is injective), and Cheung–Ikenmeyer–Mkrtchyan determined $\ker\Psi^{5\times 5}$ completely as a $GL$-representation and showed $\Psi^{6\times 6}$ is also non-injective. Injectivity of the standard map is thus strictly stronger than the Foulkes coefficient inequality, which may hold even where the canonical map has a kernel.

For general shapes, Stanley (2000) — and independently Siemons and Wagner in 1986, in a private communication recounted in McKay's paper — conjectured (the SWS conjecture) that $\psi_\lambda$ is injective whenever $\lambda\geq\lambda'$ in dominance order; for rectangles this recovers the Foulkes–Howe statement, and it was known for hook shapes (partitions of the form $(k,1^{n-k})$, whose diagram is a single row together with a single column). Pylyavskyy (2004) proved the $2\times n$ case and disproved the conjecture as stated: for $\lambda=(6,2,2,1,1)$ one has $\lambda\geq\lambda'$ yet $|H^\lambda|>|V^\lambda|$ (a counterexample suggested by Stanley as the smallest), so injectivity fails for sheer dimension reasons. Pylyavskyy proposed the alternative that $K_\lambda$ has full rank for every $\lambda$ (i.e. rank $\min(|H^\lambda|,|V^\lambda|)$; note $K_{\lambda'}=K_\lambda^{\mathsf T}$, so full rank is a conjugation-symmetric property), while reporting evidence from Neunhöffer's unpublished calculations that even this fails for some $\lambda$. McKay (2008) developed the principal positive technique: adding a column of admissible ('good') type to the left of the diagram of $\lambda$ preserves injectivity of $\psi_\lambda$. In the rectangular case this propagation theorem reads: if $\Psi^{a\times(b-1)}$ is injective then $\Psi^{a\times c}$ is injective for all $c\geqslant b$ (retranslated elegantly by Ikenmeyer 2015); hence verifying Foulkes' conjecture for fixed $a$ reduces to the single base case $b=a+1$. McKay's column-addition criterion is sufficient but is not known to be necessary, and beyond such sufficient conditions and the known failure cases no characterization of the shapes with injective (or full-rank) standard map is known: this is the open problem recorded here, and it is also the framework in which later partial work sits — Tétreault's generalized Foulkes–Howe maps between plethystic tabloid modules for arbitrary partition pairs and his conjectured partial order $\trianglelefteq$ on partitions defined by Schur-positivity of $h_\mu[h_\nu]-h_\nu[h_\mu]$, Vessenes' generalized Foulkes modules for arbitrary partitions, and the partition-algebra approach of Bowman and Paget, which proves the stable (saturated) version of Foulkes' conjecture but leaves every finite-$n$ injectivity question untouched.

Problem Statement

Determine, with proof, exactly which partitions $\lambda$ have full rank $K_\lambda$ and which do not — the classification left open by Pylyavskyy's reformulation of the question in his 2004 Electronic Journal of Combinatorics paper. Conjecture 1.3 there asserts full rank for every $\lambda$, but that universal assertion is already false: the self-conjugate rectangles $\lambda=(5^5)$ and $\lambda=(6^6)$ have $|H^\lambda|=|V^\lambda|$ with $\Psi^{5\times5}$ and $\Psi^{6\times6}$ non-injective, so $K_\lambda$ is not of full rank there, and the Black–List approach does not work in general; the question asks for a classification of the full-rank (and rank-deficient) partitions rather than a single witness. Formally: for a partition $\lambda$ of $n$ with conjugate $\lambda'$, let $K_\lambda$ be the matrix with rows indexed by horizontal tableaux $\mu\in H^\lambda$ and columns by vertical tableaux $\nu\in V^\lambda$, with $K_\lambda(\mu,\nu)=1$ if and only if $\mu\perp\nu$ (every row of $\mu$ meets every column of $\nu$ in at most one entry), and let $\psi_\lambda:\mathbb{C}[H^\lambda]\to\mathbb{C}[V^\lambda]$ be the $S_n$-equivariant standard map with matrix $K_\lambda$ (equivalently, the homomorphism $1\!\uparrow_{N_{S_n}(S_\lambda)}^{S_n}\to 1\!\uparrow_{N_{S_n}(S_{\lambda'})}^{S_n}$). Say that $\lambda$ is full-rank if $\operatorname{rank}K_\lambda=\min(|H^\lambda|,|V^\lambda|)$, i.e. $\psi_\lambda$ is injective whenever $|H^\lambda|\leqslant|V^\lambda|$ (the property is conjugation-symmetric since $K_{\lambda'}=K_\lambda^{\mathsf T}$). The rectangular subcase $\lambda=(b,b,\dots,b)$ with $a$ rows is the same question for the Foulkes–Howe map $\Psi^{a\times b}$: injectivity is currently known for $a=b\leqslant 4$ and for $a\leqslant 5<b$ (via McKay's propagation theorem from $\Psi^{4\times 4}$ and $\Psi^{5\times 6}$), non-injectivity at $(5,5)$ and $(6,6)$, and the status is open for all remaining pairs $(a,b)$; this subcase refines Foulkes' 1950 conjecture, itself open for $a\geqslant 6$. Since the conjecture ranges over all partitions, finite verification data (rank computations for bounded $n$) may support but cannot substitute for a general proof or refutation.

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