Irreducibility criterion for the set of two matrices

2008/07/29 by Alexandre Kosyak, Kosyak, Alexandre
Mathematics · #16Gxx) #20G05 sep (20Cxx #22E46 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20G05 #msc:22E46

paper · pdf · doi:10.48550/arxiv.0807.4696

13 pages

arxiv created 2008/11/02 · arxiv updated 2009/12/01

Abstract

We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two n× n matrices Λ_n and A_n in terms of the subalgebra associated with the "support" of the matrix A_n, where Λ_n is a diagonal matrix with different non zeros eigenvalues and A_n is an arbitrary one. The list of all maximal subalgebras of the algebra \rm Mat(n,\mathbb C) and the list of the corresponding invariant subspaces connected with these two matrices is also given. The properties of the corresponding subalgebras are expressed in terms of the graphs associated with the support of the second matrix. For arbitrary n we describe all minimal subsets of the elementary matrices E_km that generate the algebra \rm Mat(n,\mathbb C).

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