Weak Foulkes conjecture: multiset-partition counts and monomial positivity of the Foulkes plethysm difference

For integers $1 \le a < b$, Harman (2018) formulated a weak form of Foulkes' 1950 conjecture: for every multiset $M$ of total size $ab$, the number of multiset partitions of $M$ into $b$ blocks each of size $a$ is at least the number of multiset partitions of $M$ into $a$ blocks each of size $b$. Equivalently, the plethysm difference $h_b[h_a] - h_a[h_b]$ of complete homogeneous symmetric functions has nonnegative coefficients in the monomial basis, whereas the full Foulkes conjecture asserts the same in the Schur basis. Since Schur positivity implies monomial positivity, every known case of full Foulkes (smallest parameter $a \le 5$, plus stability and propagation regimes) transfers to the weak form, but no proof or counterexample is known for the general statement.

Background

A symmetric function is a bounded-degree formal power series in variables $x_1, x_2, \ldots$ that is invariant under all permutations of the variables; the symmetric functions of degree $n$ form a vector space with several standard bases. The monomial symmetric function $m_\mu$ is the sum of all distinct monomials obtained by permuting the multiset of exponents given by a partition $\mu$ of $n$. The complete homogeneous symmetric function $h_n$ is the sum of all monomials of total degree $n$. The Schur functions $s_\lambda$, indexed by partitions $\lambda$, form the basis underlying the representation theory of symmetric groups, since the Frobenius characteristic map sends the irreducible Specht module of shape $\lambda$ to $s_\lambda$. Plethysm is a special substitution operation on symmetric functions, denoted with square brackets $f[g]$, which builds the Frobenius characteristic of a wreath-product induction: for positive integers $a$ and $b$, the Foulkes module $H^{(a^b)}$ is the induced representation $\mathrm{Ind}_{S_a \wr S_b}^{S_{ab}} \mathbf{1}$, where $S_a \wr S_b$ (a wreath product) is the imprimitive subgroup of $S_{ab}$ consisting of permutations that permute $b$ blocks of size $a$ among themselves; its Frobenius characteristic is the plethysm $h_b[h_a]$. In 1950 Foulkes conjectured, equivalently, that for $a < b$ the difference $h_b[h_a] - h_a[h_b]$ has nonnegative coefficients in the Schur basis; this is the Foulkes conjecture, one of the central open problems on plethysm coefficients. It is classical for the smaller parameter $a = 2$ (Thrall, 1942), was settled for $a \le 4$ by the early 2000s (with Dent and Siemons handling $a = 3$; surveyed in Pylyavskyy's 2004 paper), and was verified for $a = 5$ by Cheung, Ikenmeyer, and Mkrtchyan (2017) using a computer calculation together with a propagation theorem of McKay (2008); Brion (1993) initiated the geometric and stable study of the conjecture, and Müller and Neunhöffer (2005) verified the coefficientwise comparison computationally in a range of degrees. A multiset partition of a multiset $M$ is an unordered collection of sub-multisets, called blocks, whose multiset union is $M$. The coefficient of $m_\mu$ in $h_b[h_a]$ counts exactly the multiset partitions of a multiset of content $\mu$ into $b$ blocks each of size $a$, so the monomial coefficients of the two plethysms carry a purely combinatorial comparison. Because each Schur function expands into the monomial basis with nonnegative Kostka coefficients, Schur positivity of the Foulkes difference implies its monomial positivity, so the statement below is genuinely weaker than the full Foulkes conjecture. Harman (2018) derived this weak form from his study of polynomial representations of the general linear group $GL_n$ restricted to the subgroup $T \rtimes S_n$ of monomial matrices (the torus $T$ of diagonal matrices semidirect product with permutation matrices): his framework identifies the relevant multiplicities with dimensions of spaces of permutation invariants in symmetrized weight spaces, and his combinatorial interpretation of the restriction of tensor products of symmetric powers reduces the monomial-coefficient comparison to counting multiset partitions. He records the resulting inequality as his Conjecture 4.4, the weak Foulkes conjecture, introduced as a purely combinatorial formulation of a weak version of Foulkes' conjecture for tensor products of symmetric powers; his remark that the formulation seems to be missing from much of the Foulkes literature concerns Conjecture 4.3, the mixed $GL_n$/$S_n$ form of the full conjecture. A nearby but distinct line of work (Black and List 1989; a rank conjecture of Stanley studied by Pylyavskyy 2004 and Sivek 2006) attacks the full conjecture through ranks of 0-1 incidence matrices between set-partition classes; both Stanley's original formulation and Pylyavskyy's reformulation were shown false in general, and that rank-based line leaves the monomial-coefficient inequality untouched.

Problem Statement

Prove or disprove the weak Foulkes conjecture of Harman (arXiv:1804.04702, Conjecture 4.4), a strictly weaker consequence of the 1950 Foulkes conjecture: for all integers $1 \le a < b$ and every multiset $M$ of total cardinality $ab$, the number of multiset partitions of $M$ into $b$ blocks each of size $a$ is at least the number of multiset partitions of $M$ into $a$ blocks each of size $b$ (blocks are sub-multisets and the collection of blocks is unordered). Equivalently: for all $a < b$, the plethysm difference $h_b[h_a] - h_a[h_b]$ of complete homogeneous symmetric functions has nonnegative coefficients when expanded in the monomial basis $\{m_\mu\}$. Currently the inequality is established only in the cases inherited from verified cases of the full (Schur-basis) Foulkes conjecture, namely smallest parameter $a \le 5$ and the stability/propagation regimes; the general statement is open.

References

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