Reidemeister spectra of free nilpotent groups and plethysms of Schur functions

TL;DR

This paper explores the connection between Reidemeister spectra of free nilpotent groups and plethysm coefficients of Schur functions, revealing that spectral computations involve sums of specific plethysms of Schur functions.

Contribution

It establishes a novel link between two open problems in algebraic combinatorics and group theory, connecting Reidemeister spectra with plethysm coefficients of Schur functions.

Findings

Reidemeister spectrum expressions are sums of plethysms of the form s_{1^{i}}[g]

The computations involve Schur functions or Schur positive functions

Provides a new perspective linking group theory and symmetric functions

Abstract

We establish a strong link between two open problems: determining the Reidemeister spectrum of free nilpotent groups and determining the coefficients in the Schur expansion of plethysms of Schur functions. Specifically, we show that the expressions occurring in the computations for the Reidemeister spectrum are sums of plethysms of the form $s_{1^{i}}[g]$, where $g$ is a Schur function or a Schur positive function.