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Zeros of $S$-characters

Thomas Breuer, Michael Joswig, Gunter Malle

TL;DR

The paper investigates whether $S$-characters of finite groups must vanish on an element of prime power order, extending known results for transitive permutation characters and non-linear irreducible characters. It reframes the problem in terms of lattice points in the $S$-character simplex $S(G)$ and develops an explicit computational approach using polyhedral geometry, including Barvinok-type and project-and-lift methods, implemented via OSCAR. A main theoretical result shows that every non-trivial ordinary $S$-character of a finite solvable group vanishes on a prime power order element, while a concrete counterexample is found in ${\mathfrak{A}}_8$: there exists a non-trivial ordinary $S$-character that does not vanish on any element of prime power order, making ${\mathfrak{A}}_8$ the smallest such group by the number of conjugacy classes. The work also demonstrates the complexity of the problem, reports extensive computational findings (mostly rational examples) and leaves open questions about non-rational $S$-characters and solvable cases.

Abstract

The concept of $S$-characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial $S$-character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. We present examples that this does not generalise to $S$-characters, thereby answering a question posed by J-P. Serre.

Zeros of $S$-characters

TL;DR

The paper investigates whether -characters of finite groups must vanish on an element of prime power order, extending known results for transitive permutation characters and non-linear irreducible characters. It reframes the problem in terms of lattice points in the -character simplex and develops an explicit computational approach using polyhedral geometry, including Barvinok-type and project-and-lift methods, implemented via OSCAR. A main theoretical result shows that every non-trivial ordinary -character of a finite solvable group vanishes on a prime power order element, while a concrete counterexample is found in : there exists a non-trivial ordinary -character that does not vanish on any element of prime power order, making the smallest such group by the number of conjugacy classes. The work also demonstrates the complexity of the problem, reports extensive computational findings (mostly rational examples) and leaves open questions about non-rational -characters and solvable cases.

Abstract

The concept of -characters of finite groups was introduced by Zhmud' as a generalisation of transitive permutation characters. Any non-trivial -character takes a zero value on some group element. By a deep result depending on the classification of finite simple groups a non-trivial transitive permutation character even vanishes on some element of prime power order. We present examples that this does not generalise to -characters, thereby answering a question posed by J-P. Serre.
Paper Structure (7 sections, 5 theorems, 11 equations, 1 table)

This paper contains 7 sections, 5 theorems, 11 equations, 1 table.

Table of Contents

  1. Introduction
  2. Characters as lattice points in polytopes
  3. $S$-character simplices
  4. Enumerating lattice points
  5. Characters of solvable groups
  6. Results
  7. Concluding remarks

Key Result

Theorem 1

The alternating group ${\mathfrak{A}}_8$ possesses a non-trivial ordinary $S$-character which does not vanish on any element of prime power order. This is the smallest example, in terms of number of conjugacy classes. In particular, no other group with at most 14 conjugacy classes shares that proper

Theorems & Definitions (14)

  • Theorem 1
  • Proposition 2
  • proof
  • Remark 3
  • Example 4
  • Proposition 5
  • proof
  • Example 6
  • Remark 7
  • Lemma 8
  • ...and 4 more