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Global Representation Ring and Knutson Index

Dylan Johnston, Diego Martín Duro, Dmitriy Rumynin

TL;DR

The paper generalizes Knutson's index from characters to finitely generated commutative rings with a dimension map and a regular element, and develops a finite-computation approach for reduced rings. It then specializes to the Burnside ring to relate the Knutson subindex to splitting of Sylow-subgroup sequences in group extensions, establishing a p-part factorization and a precise equivalence between 𝒦_r^S(G/N)=1 and universal Sylow splitting. Building on this, the authors define the reduced global representation ring R(G,N) and its global table T(G,N), which simultaneously encode the character table and the table of marks, yielding substantial but not complete knowledge of the underlying group. They show that 𝒦_r(Burnside) divides 𝒦_r(T(G,N)) and provide extensive computational examples (including Brauer-pairs) demonstrating both recoverable and non-unique information from the global table, along with open questions about when the global Knutson Index equals 1 and how the global table compares to the other classical invariants.

Abstract

Global representation rings were discovered by Sarah Witherspoon in 1995 and the Knutson Index was introduced by the second author in 2022. In the present paper we introduce the Knutson Index for general commutative rings and study it for Burnside rings and global representation rings. We also introduce the global table of a finite group, that encompasses both the character table and the Burnside table of marks. We discuss what properties of a group can be recovered from its global table.

Global Representation Ring and Knutson Index

TL;DR

The paper generalizes Knutson's index from characters to finitely generated commutative rings with a dimension map and a regular element, and develops a finite-computation approach for reduced rings. It then specializes to the Burnside ring to relate the Knutson subindex to splitting of Sylow-subgroup sequences in group extensions, establishing a p-part factorization and a precise equivalence between 𝒦_r^S(G/N)=1 and universal Sylow splitting. Building on this, the authors define the reduced global representation ring R(G,N) and its global table T(G,N), which simultaneously encode the character table and the table of marks, yielding substantial but not complete knowledge of the underlying group. They show that 𝒦_r(Burnside) divides 𝒦_r(T(G,N)) and provide extensive computational examples (including Brauer-pairs) demonstrating both recoverable and non-unique information from the global table, along with open questions about when the global Knutson Index equals 1 and how the global table compares to the other classical invariants.

Abstract

Global representation rings were discovered by Sarah Witherspoon in 1995 and the Knutson Index was introduced by the second author in 2022. In the present paper we introduce the Knutson Index for general commutative rings and study it for Burnside rings and global representation rings. We also introduce the global table of a finite group, that encompasses both the character table and the Burnside table of marks. We discuss what properties of a group can be recovered from its global table.
Paper Structure (6 sections, 15 theorems, 12 equations, 1 table)

This paper contains 6 sections, 15 theorems, 12 equations, 1 table.

Table of Contents

  1. Introduction
  2. Knutson Index for Rings
  3. Knutson Index for Burnside Rings
  4. The Global Representation Ring and its Knutson Index
  5. Further counterexamples to the converse of Lemmas \ref{['lemma p-split']}( ii) & \ref{['lemma index']}
  6. Groups with same character table and same table of marks

Key Result

Theorem 1

Let $1 \to N \to G \to \Gamma \to 1$ be an exact sequence. Then $\mathcal{K}_r^S(G/N) = 1$ if and only if the sequence of Sylow $p$-subgroups $1 \to N_p \to G_p \to \Gamma_p \to 1$ splits for all primes $p$, where $G_p$ is a Sylow $p$-subgroup of $G$, $N_p = N \cap G_p$ and $\Gamma_p = (G_pN)/N$.

Theorems & Definitions (31)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.6
  • proof
  • Lemma 3.1
  • ...and 21 more