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A Frobenius-Type Formula for Compact Lie Groups

Shripad Garge, Uday Bhaskar Sharma

Abstract

Let $G$ be a group and $α: G \times G \to G$ denote the commutator map. In the case of finite groups, Frobenius gave the formula to compute the cardinalities of the fibres $α^{-1}(g)$ in terms of the character values $χ(g)$ for irreducible characters $χ$ of $G$. We generalise this formula to compact Lie groups. Further, we connect this generalised formula to the commutator probability of the concerned groups.

A Frobenius-Type Formula for Compact Lie Groups

Abstract

Let be a group and denote the commutator map. In the case of finite groups, Frobenius gave the formula to compute the cardinalities of the fibres in terms of the character values for irreducible characters of . We generalise this formula to compact Lie groups. Further, we connect this generalised formula to the commutator probability of the concerned groups.

Paper Structure

This paper contains 6 sections, 15 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Finite groups
  3. Connected compact Lie groups
  4. FC Compact Groups
  5. Compact Lie Groups with Open FC-Centre
  6. The commutator probability

Key Result

Theorem 1.1

Let $G$ be a connected, compact, Lie group with normalised Haar measure $\mu$. Let $d$ denote the dimension of the group $G$ and let $r$ denote the rank of $G$. Let $Irr(G)_n$ denote the set of irreducible characters of $G$, whose weight-sum is less than or equal to $n$. Then for each $g \in G$,

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 20 more