Linear preservers of Schur functions

Abstract

Let M n be the vector space of the n × n matrices over the field F . Let H be a subgroup of the symmetric group of degree n and x an F -valued character of H. If A = [ a ij ] ϵ M n , the Schur function of A is A is d H x(A) = ∑ σϵH X(σ) Π i a iσ(i) . A linear mapping T: M n → M n is called a Schur function preserver if d H x(T(X)) = d H x(X) for every X in M n . We survey some very recent results on Schur function preservers with special emphasis on the case in which H is the full symmetric group and F is the complex field. In this case a complete description of the preservers is possible.