Triangular Products of Group Representations and Their Applications

Vovsi, S.M.

ISBN 10: 1468467239 / ISBN 13: 9781468467239

Published by Birkhauser 2012-02, 2012

About this Item

Seller Inventory # 6666-IUK-9781468467239

Bibliographic Details

• Title: Triangular Products of Group Representations and Their Applications
• Publisher: Birkhauser 2012-02
• Publication Date: 2012
• Language: English
• ISBN 10: 1468467239
• ISBN 13: 9781468467239
• Binding: PF
• Condition: New

About this title

Synopsis

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 .