The symmetric functions catalog

An overview of symmetric functions and related topics

2020-09-01

Cycle index polynomials

The cycle index polynomial (Zyklenzeiger) of a group $G$ was introduced by G. Pólya in [Eq. (1,5), Pól37]. Let $G \subseteq \symS_n$ be a subgroup. The cycle index polynomial is defined in terms of the powersum symmetric functions as the average

$\cyc_G(\xvec) \coloneqq \frac{1}{|G|} \sum_{\pi \in G} \powerSum_{\type(\pi)}(\xvec).$

Let $\symS_n/G$ denote the set of let cosets of $G,$ and let $\symS_n$ act on $\setC[\symS_n/G].$ This makes $\setC[\symS_n/G]$ into an $\symS_n$-module, and its Frobenius characteristic is given by $\cyc_G(\xvec).$ In particular, $\cyc_G(\xvec)$ is Schur-positive. For more info on the cycle index polynomial, we refer to [LW19b].

Conjecture (Foulkes Conjecture, [Fou50]).

Let $X_{a,b}$ be the set of set-partitions of $\{1,2,\dotsc,ab\}$ into $a$ blocks of size $b.$ The symmetric group $S_{ab}$ act on $X_{a,b}$ in the obvious manner. Let $G_{a,b}$ be the stabilizer subgroup of some fixed element in $X_{a,b}$ under this action. Let

$\cyc_{G_{a,b}}(\xvec) = \sum_\lambda c_{\lambda,a,b}\; \schurS_\lambda(\xvec).$

Foukes conjecture states that whenever $a\leq b,$ then for all $\lambda,$ $c_{\lambda,a,b} \leq c_{\lambda,b,a}.$ This is equivalent with the statement that $\completeH_b[\completeH_a]-\completeH_a[\completeH_b]$ is Schur-positive whenever $a \leq b.$ The notation here indicates plethysm.