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Reidemeister spectra of free nilpotent groups and plethysms of Schur functions

Pieter Senden

TL;DR

The article links the open problem of determining Reidemeister spectra for free nilpotent groups with the combinatorial problem of plethysms of Schur functions, proposing a route through Lie algebras and symmetric polynomials. It shows that Reidemeister numbers are governed by symmetric polynomials evaluated at the eigenvalues of the induced abelianization map, and that these polynomials can be interpreted within the framework of symmetric functions and Schur expansions. A key achievement is proving Schur positivity of certain Hall-basis-derived functions $F_n$ and connecting them to Frobenius characters, which ties Reidemeister calculations to plethysm coefficients, including concrete cases like $s_{1^i}[s_{k-1,1}]$. The paper further extends the discussion to the free nilpotent metabelian groups, where analogous symmetric-function expressions govern the spectrum, highlighting both the method and the remaining open combinatorial problems surrounding plethysms. Overall, the work provides a structured bridge between group-theoretic Reidemeister theory and the rich algebraic-combinatorial landscape of symmetric functions and plethysms, with practical implications for computation of spectra via coefficient data of characteristic polynomials.

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.

Reidemeister spectra of free nilpotent groups and plethysms of Schur functions

TL;DR

The article links the open problem of determining Reidemeister spectra for free nilpotent groups with the combinatorial problem of plethysms of Schur functions, proposing a route through Lie algebras and symmetric polynomials. It shows that Reidemeister numbers are governed by symmetric polynomials evaluated at the eigenvalues of the induced abelianization map, and that these polynomials can be interpreted within the framework of symmetric functions and Schur expansions. A key achievement is proving Schur positivity of certain Hall-basis-derived functions and connecting them to Frobenius characters, which ties Reidemeister calculations to plethysm coefficients, including concrete cases like . The paper further extends the discussion to the free nilpotent metabelian groups, where analogous symmetric-function expressions govern the spectrum, highlighting both the method and the remaining open combinatorial problems surrounding plethysms. Overall, the work provides a structured bridge between group-theoretic Reidemeister theory and the rich algebraic-combinatorial landscape of symmetric functions and plethysms, with practical implications for computation of spectra via coefficient data of characteristic polynomials.

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 , where is a Schur function or a Schur positive function.
Paper Structure (12 sections, 28 theorems, 166 equations, 1 figure, 1 table)

This paper contains 12 sections, 28 theorems, 166 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Starting point: Reidemeister spectra of free nilpotent groups
  3. Destination: plethysm of Schur functions
  4. From Reidemeister spectra to symmetric polynomials
  5. From symmetric polynomials to plethysms of Schur functions
  6. Symmetric functions
  7. From symmetric polynomials to symmetric functions
  8. Schur functions
  9. Schur positivity of $F_{n}$
  10. From plethysms of Schur functions back to Reidemeister spectra
  11. Main objectives
  12. Reidemeister spectrum of free nilpotent metabelian groups

Key Result

Theorem 1.1

Let $r \geq 2$ and $c \geq 1$ be integers. Then $N_{r, c}$ has the $R_{\infty}$-property if and only if $c \geq 2r$.

Figures (1)

  • Figure 3: Standard Young tableau of shape $(4, 2, 1)$

Theorems & Definitions (55)

  • Theorem 1.1: DekimpeGoncalves14
  • Conjecture : DekimpeTertooyVargas20
  • Lemma 2.1: Romankov11
  • Theorem 2.2: DekimpeGoncalvesOcampo21
  • Remark
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Witt37
  • Lemma 2.6
  • proof
  • ...and 45 more