Partial symmetries of iterated plethysms

TL;DR

This paper investigates partial symmetries in iterated plethystic coefficients of Schur functions, introduces flip-symmetry, and provides explicit formulas demonstrating these symmetries in specific cases.

Contribution

It introduces the concept of flip-symmetry in Schur-positive functions and proves that certain iterated plethysms preserve this symmetry, expanding understanding of plethystic coefficients.

Findings

Partial symmetries exist in large families of iterated plethystic coefficients.

Explicit formulas show flip-symmetry in specific iterated plethysms like $s_2\circ s_b\circ s_a$ and $s_c\circ s_2\circ s_a$.

Open questions on unimodality and asymptotic normality of flip-symmetric sequences are discussed.

Abstract

This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions. The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous $\omega$ involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with $s_\lambda$ preserves flip-symmetry, provided that $\lambda$ is a partition of two. Explicit formulas for the iterated plethysms $s_2\circ s_b\circ s_a$ and $s_c\circ s_2\circ s_a$, with $a,$ $b,$ and $c$ $\ge$ $2$ allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.