On types of matrices and centralizers of matrices and permutations

2013/10/19 by John R. Britnell, Mark Wildon, Britnell, John R. +1
Mathematics · #15A21 #20B35 #FOS: Mathematics #Group Theory (math.GR) #Primary 15A27 #Secondary 12F15 #math.GR #msc:12F15 #msc:15A21 #msc:15A27 #msc:20B35

paper · pdf · doi:10.48550/arxiv.1310.5261

12 pages

arxiv created 2013/10/19 · arxiv updated 2013/10/22

Abstract

It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.

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