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

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.

See also https://arxiv.org/pdf/2008.13070.pdf.

## References

- [Fou50] H. O. Foulkes. Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. Journal of the London Mathematical Society, s1-25(3):205–209, July 1950.
- [LW19b] Nicholas A. Loehr and Gregory S. Warrington. Quasisymmetric and Schur expansions of cycle index polynomials. Discrete Mathematics, 342(1):113–127, January 2019.
- [Pól37] G. Pólya. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica, 68(0):145–254, 1937.