We provide a formal framework for the theory of representations of finite groups, as modules over the group ring. Along the way, we develop the general theory of groups (relying on the group add class for the basics), modules, and vector spaces, to the extent required for theory of group representations. We then provide formal proofs of several important introductory theorems in the subject, in...

Let F be a field, V a 6-dimensional F-vector space and f a nondegenerate alternating bilinear form on V . We consider a 14-dimensional module for the symplectic group Sp(V, f) ∼= Sp(6,F) associated with (V, f), and classify the orbits on vectors. For characteristic distinct from 2, this module is irreducible and isomorphic to the Weyl module of Sp(V, f) for the fundamental weight λ3. If the cha...

In this article, we study the restriction of Zuckerman’s derived functor (g,K)-modules Aq(λ) to g′ for symmetric pairs of reductive Lie algebras (g, g′). When the restriction decomposes into irreducible (g′,K′)-modules, we give an upper bound for the branching law. In particular, we prove that each (g′,K′)-module occurring in the restriction is isomorphic to a submodule of Aq′ (λ ′) for a parab...

Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote...

Let l be a prime, and let Γ be a finite subgroup of GLn(Fl) = GL(V ). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad V there exists an element g ∈ Γ with an eigenvalue α such that tr eg,αW 6= 0. Here, eg,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the definition of adequacy in section 2. Let Γ denote...

An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an ir...

(1) For every sup-semilattice L and for all elements x, y of L holds d−eL(↑x∩↑y) = xt y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = xu y. (3) Let L be a non empty relational structure and x, y be elements of L. If x is maximal in (the carrier of L)\↑y, then ↑x\{x}= ↑x∩↑y. (4) Let L be a non empty relational structure and x, y be elements of L. If x is minimal ...