Triangular Products of Group Representations and Their Applications

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 .

Inhalt

1. Triangular products.- 1. Preliminaries.- 2. The definition and basic properties of triangular products.- 3. The Embedding Theorem. Connections with closure operations.- 4. Generalized triangular products.- 5. Isomorphisms and automorphisms of triangular products.- 6. Identities of triangular products.- 2. Applications.- 7. Identities of triangular matrix groups and their canonical representations.- 8. Augmentation powers and dimension subgroups.- 9. The semigroup of varieties of group representations.- 10. The semigroups of radical classes and prevarieties of group representations.- 11. Infinite products of radical classes and prevarieties.- 12. Invariant subspaces in representations.- References.