A generalization of the Burnside theorem and of Schur's lemma for reducible representations

Abstract

We show that any reducible group generated by a finite number of matrices from GL(n,ℂ) can be conjugated to a block uppertriangular form such that

• (1) if the matrices from the centralizer of the group are block-decomposed as the group itself, then their blocks are either scalar or 0 (generalization of Schur's lemma),

• (2) the linear spaceS spanned in gl(n,ℂ) by the matrices of the group is described as follows: all entries of a part of the blocks of the matrices ofS are parameters and the rest of the blocks are linearly expressed by them by means of equations of the kind α₁ Q ₁+ ... +α_s Q _s=0, where α _j ∈ℂ andQ _j denote blocks of the matrices ofS (generalization of the Burnside theorem).