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Triangular Products of Group Representations and Their Applications

  • Samuel M. Vovsi

Part of the Progress in Mathematics book series (PM, volume 17)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Samuel M. Vovsi
    Pages 1-56
  3. Samuel M. Vovsi
    Pages 57-114
  4. Back Matter
    Pages 115-131

About this book

Introduction

The construction considered in these notes is based on a very simple idea. Let (A, G ) and (B, G ) be two group representations, for definiteness faithful and finite­ 1 2 dimensional, over an arbitrary field. We shall say that a faithful representation (V, G) is an extension of (A, G ) by (B, G ) if there is a G-submodule W of V such that 1 2 the naturally arising representations (W, G) and (V/W, G) are isomorphic, modulo their kernels, to (A, G ) and (B, G ) respectively. 1 2 Question. Among all the extensions of (A, G ) by (B, G ), does there exist 1 2 such a "universal" extension which contains an isomorphic copy of any other one? The answer is in the affirmative. Really, let dim A = m and dim B = n, then the groups G and G may be considered as matrix groups of degrees m and n 1 2 respectively. If (V, G) is an extension of (A, G ) by (B, G ) then, under certain 1 2 choice of a basis in V, all elements of G are represented by (m + n) x (m + n) mat­ rices of the form (*) ~1-~ ~-J lh I g2 I .

Keywords

Finite Group representation Matrix Natural automorphism construction eXist field form group kernel presentation semigroup

Authors and affiliations

  • Samuel M. Vovsi
    • 1
  1. 1.Riga Polytechnic InstituteRigaUSSR

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4684-6721-5
  • Copyright Information Birkhäuser Boston 1981
  • Publisher Name Birkhäuser Boston
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4684-6723-9
  • Online ISBN 978-1-4684-6721-5
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
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