Shirshov’s theorem and representations of semigroups

Abstract

In [AS], Amitsur and Small observed that if G is a finitely generated group (on a set X), then they were only finitely many possible irreducible representations of dimension n over a field F if the characteristic polynomials of all words (in X) of length at most 2n were fixed. This followed immediately from Shirshov’s theorem (cf. [Pr] or [Ro]). However, Shirshov’s Theorem does not give a condition which determines the isomorphism class of the irreducible module. In this note, we investigate the connection between the character of a finite dimensional representation of a semigroup and its composition factors. It is folklore that, over a field of characteristic 0, the character of a representation determines its composition factors. In positive characteristic p, this can fail for several reasons – a composition factor may have multiplicity a multiple of p, there can be inseparable extensions and there can be division algebras whose degree is a multiple of p. In Section 2, we prove that these are the only reasons for this failure. We prove (Theorem 2.6) that if F is a perfect field, S is a semigroup and V is a finite dimensional F [S]-module such that every composition factor has multiplicity less than p, then the composition factors are determined by its character. Let X be a generating set for a subalgebra of Mn(F ) with F a field. Let g(X,n) denote the smallest positive integer g such that the subalgebra is spanned by all words (in X) of length at most g. Let g(n) be the maximum of of g(X,n) for all possible generating sets X. We in fact prove that the composition factors are determined by the traces of elements of length at most 2g(n)+1. Clearly, g(n) ≤ n2−1. Paz [Pz] showed that g(n) is bounded by n/3 (approximately). This has recently been improved by Pappacena [Pa] to show that g(n) ≤ √2n3/2 + 3n (see [Pa] for a slightly more precise estimate).

Locations

• Pacific Journal of Mathematics - View - PDF