Triangular Products of Group Representations and Their Applications

1. Front Matter

   Pages i-ix

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2. Triangular Products

   • Samuel M. Vovsi

   Pages 1-56

3. Applications

   • Samuel M. Vovsi

   Pages 57-114

4. Back Matter

   Pages 115-131

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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 .

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

• Riga Polytechnic Institute, Riga, USSR

  Samuel M. Vovsi

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