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Minimal Polynomials in Spin Representations of Symmetric and Alternating Groups

Amritanshu Prasad, Velmurugan S, Alexey Staroletov

TL;DR

This work classifies minimal polynomials of elements in irreducible spin representations of the double covers of the symmetric and alternating groups. Using the interplay of Schur covers, shifted Young tableaux, and Clifford algebra frameworks, it proves that for most elements $g$ and irreducible spin representations, the minimal polynomial of $\rho(g)$ is of the form $x^{k}\pm1$ with $k$ equal to the least common multiple of the cycle lengths of $g$ modulo the center, while four infinite families and finitely many sporadic exceptions arise. The results are organized around generic families (n-cycles, other specific element families) and then extended to general elements via dominant partitions and shifted Littlewood–Richardson calculus, with wreath-product and parabolic-subgroup methods driving the spectral analysis. The paper further extends the symmetric-group results to the double cover of the alternating group, detailing how conjugacy class splitting and restriction behavior affect minimal polynomials, and provides explicit exceptional cases for small $n$ through tables. Overall, the work deepens our understanding of eigenvalue structures in spin representations and has implications for arithmetic, geometry, and combinatorics where these representations play a role.

Abstract

We determine the minimal polynomial of each element of the double cover $G$ of the symmetric or alternating group in every irreducible spin representation of $G$.

Minimal Polynomials in Spin Representations of Symmetric and Alternating Groups

TL;DR

This work classifies minimal polynomials of elements in irreducible spin representations of the double covers of the symmetric and alternating groups. Using the interplay of Schur covers, shifted Young tableaux, and Clifford algebra frameworks, it proves that for most elements and irreducible spin representations, the minimal polynomial of is of the form with equal to the least common multiple of the cycle lengths of modulo the center, while four infinite families and finitely many sporadic exceptions arise. The results are organized around generic families (n-cycles, other specific element families) and then extended to general elements via dominant partitions and shifted Littlewood–Richardson calculus, with wreath-product and parabolic-subgroup methods driving the spectral analysis. The paper further extends the symmetric-group results to the double cover of the alternating group, detailing how conjugacy class splitting and restriction behavior affect minimal polynomials, and provides explicit exceptional cases for small through tables. Overall, the work deepens our understanding of eigenvalue structures in spin representations and has implications for arithmetic, geometry, and combinatorics where these representations play a role.

Abstract

We determine the minimal polynomial of each element of the double cover of the symmetric or alternating group in every irreducible spin representation of .
Paper Structure (13 sections, 46 theorems, 110 equations, 1 figure, 4 tables)

This paper contains 13 sections, 46 theorems, 110 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries for the double cover of the symmetric group
  3. Double covers of Parabolic subgroups and their irreducible representations
  4. Some known results
  5. Combinatorics on shifted Young tableaux
  6. Minimal polynomial of $n$-cycles in $\tilde{S_n}$.
  7. Minimal polynomial of some family of elements of Sn in some irreducible spin representations
  8. The case $\lambda=(n)$
  9. The case $\lambda=(n-1,1)$
  10. Minimal polynomials of an element with all cycles of odd length
  11. Minimal polynomials of an element with all cycles of even length
  12. Minimal polynomials of a general element
  13. Proof of results on double cover of alternating groups

Key Result

Theorem 1.1

Let $\lambda\in DP_n$ and $(\rho_\lambda,V_\lambda)$ be the corresponding irreducible spin representation of $\tilde{S}_n$. Let $g\in \tilde{S}_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and let $k=\mathop{\mathrm{lcm}}\nolimits(\mu)$. Then the minimal polynomial $p_\lambda(g)

Figures (1)

  • Figure 1: Shifted tableaux for $\lambda=(n)$

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2: Canonical representatives
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Example 2.1
  • Proposition 2.5
  • ...and 66 more