The original Foulkes‘ conjecture, which dates back to 1949, was stated as a “theorem” (at page 206 of paper) but with no claim that there actually was a proof. As it is,  more than 75 years later, the conjecture is still open (with several interesting advances). The conjecture asserts that the difference of plethysms of complete homogeneous symmetric functions

$\large h_b\circ h_a – h_a\circ h_b$

expands as a positive coefficient linear composition of Schur symmetric functions, for any pair of integers $a<b$.  One says that this is a Schur-positive expression. Equivalently, in terms of character theory, the conjecture asserts that there is an inclusion of $GL(V)$-modules

$\large S^a(S^b(V)) \rightarrow S^b(S^a(V)),$

where $S^a(V)$ stands for the $a^{\rm th}$-symmetric power of the vector space $V$. Thus, the above Schur positivity statement corresponds to inequalities between the number of copies of irreducible modules occurring in each of these two (polynomial) $GL(V)$-representations. A nice generalization of this conjecture (see Vessenes) states that for any $a\leq c,d\leq b$, sucht that $ab=cd$, the analogous difference of plethysms

$\large h_c\circ h_d – h_a\circ h_b$

is also Schur positive.

New conjecture

Several interesting statements appear as specializations of the following conjecture that I have recently proposed. For  any $a\leq c,d\leq b$, such that $ab=cd$, and any $k$, then the divided difference of plethysms

$\displaystyle \frac{\large H_c\circ H_{[d\times k]}(\mathbf{x};q,t) – H_a\circ H_{[b\times k]}(\mathbf{x};q,t)}{\large 1-q},$

expands in the Schur function basis with coefficients in $\mathbb{N}[q,t]$. Here $[d\times k]$ (resp. $[b\times k]$) stands for the integer partition that has $k$ parts of size $d$ (resp. $k$ parts of size $b$); and $H_{\mu}(\mathbf{x};q,t)$ denotes the $\mu$-indexed Macdonald polynomial. In view of well-known properties of these, one gets back the previous (generalized) conjecture of Foulkes by setting $q=0$ and $k=1$. There are several other interesting specializations.

References