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Induced representations and harmonic analysis on finite groups

Fabio Scarabotti, Filippo Tolli

Abstract

Given a finite group $G$ and a subgroup $K$, we study the commutant of $\text{Ind}_K^Gθ$, where $θ$ is an irreducible $K$-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in such commutant and an associated Fourier transform. Then we translate our results in the corresponding Hecke algebra, an isomorphic algebra in the group algebra of $G$. Again a complete Fourier analysis is developed and, as particular cases, we obtain some results of Curtis and Fossum on the irreducible characters of Hecke algebras. Finally, we develop a theory of Gelfand-Tsetlin bases for Hecke algebras.

Induced representations and harmonic analysis on finite groups

Abstract

Given a finite group and a subgroup , we study the commutant of , where is an irreducible -representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in such commutant and an associated Fourier transform. Then we translate our results in the corresponding Hecke algebra, an isomorphic algebra in the group algebra of . Again a complete Fourier analysis is developed and, as particular cases, we obtain some results of Curtis and Fossum on the irreducible characters of Hecke algebras. Finally, we develop a theory of Gelfand-Tsetlin bases for Hecke algebras.

Paper Structure

This paper contains 6 sections, 20 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orthogonality relations for Frobenius reciprocity
  4. Harmonic analysis in $\hbox{\rm Hom}_G(\hbox{\rm Ind}_K^GV, \hbox{\rm Ind}_K^GV)$
  5. Harmonic analysis in the Hecke algebra
  6. Gelfand-Tsetlin bases

Key Result

Lemma 2.1

Suppose that $(\sigma,W)$ is irreducible. Then the operators $T_1, T_2, \ldots, T_m$ give rise to an isometric orthogonal decomposition of the $W$ component of $U$ if and only if $T_1, T_2, \ldots, T_m$ form an orthonormal basis for $\hbox{\rm Hom}_G(W,U)$. Moreover, if this is the case, then we hav

Theorems & Definitions (46)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 3.1
  • Theorem 3.2: Frobenius reciprocity revisited
  • proof
  • Corollary 3.3: The other side of Frobenius reciprocity
  • ...and 36 more