Radicals of $S_n$-invariant positive semidefinite hermitian forms

Clara Franchi,Mario Mainardis,Alexander A Ivanov

doi:10.5802/alco.24

Abstract

Let G be a finite group, V a complex permutation module for G over a finite G-set 𝒳, and f:V×V→ℂ a G-invariant positive semidefinite hermitian form on V. In this paper we show how to compute the radical V ⊥ of f, by extending to nontransitive actions the classical combinatorial methods from the theory of association schemes. We apply this machinery to obtain a result for standard Majorana representations of the symmetric groups.

Highlights

A major difficulty in studying linear representations of certain finite groups, such as the large sporadic simple groups, arises when the degrees of these representations become so large that applying the general methods from linear algebra gets hard, if not practically impossible, even by machine computation
In this paper we cope with a frequent problem when dealing with the usual representation of the Monster on the Norton–Conway–Griess algebra, or, more generally, with Majorana representations of finite groups, and can be stated as follows: given a finite group G, a complex permutation module V on a finite G-set X, and a G-invariant positive semidefinite hermitian form f, determine the radical V ⊥ of f from the Gram matrix Γ associated to f with respect to X
By [5, p. 11] or [15, §2.6 and §2.7], Γ is equivalent to a block diagonal matrix Γ, whose blocks have sizes corresponding to the multiplicities of the irreducible C[G]-submodules of V, so that the decomposition of V ⊥ into irreducible C[G]-submodules can be recovered from the ranks of the diagonal blocks of Γ

Summary

Introduction

A major difficulty in studying linear representations of certain finite groups, such as the large sporadic simple groups, arises when the degrees of these representations become so large that applying the general methods from linear algebra gets hard, if not practically impossible, even by machine computation. In this paper we cope with a frequent problem when dealing with the usual representation of the Monster (and many if its simple subgroups) on the Norton–Conway–Griess algebra, or, more generally, with Majorana representations of finite groups (see [7]), and can be stated as follows: given a finite group G, a complex permutation module V on a finite G-set X , and a G-invariant positive semidefinite hermitian form f , determine the radical V ⊥ of f from the Gram matrix Γ associated to f with respect to X.

Strategy

Transitive non multiplicity-free actions