Combinatorial dimensions: Indecomposability on certain local finite-dimensional trivial extension algebras

We study problems related to indecomposability of modules over certain local finite-dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.

Invariant valuations on quaternionic vector spaces

The spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.