A fairly old problem in modular representation theory is to determine the vanishing behavior of the $\mathrm{H}om$ groups and higher $\mathrm{E}xt$ groups of Weyl modules and to compute the dimension of the $\mathbb{Z}/\left(p\right)$-vector space ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ for any partitions $\lambda$, $\mu$ of $r$, which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups ${\mathrm{H}om}_{{\overline{A}}_r}({\overline{K}}_{\lambda },{\overline{K}}_{\mu })$ and provide a new formula for the intertwining number for any...

Soit $G$ un groupe algébrique semi-simple complexe, $U$ un sous-groupe unipotent maximal de $G$, $T$ un tore maximal de $G$ normalisant $U$. Si $V$ est un $G$-module rationnel de dimension finie, alors $G$ opère sur l’algèbre $\mathbf{C}\left[V\right]$ des fonctions polynomiales sur $V$; la structure de $G$-module de $\mathbf{C}\left[V\right]$ est décrite par la $T$-algèbre $\mathbf{C}[V{]}^U$ des $U$-invariants de $\mathbf{C}\left[V\right]$. Cette algèbre est de type fini et multigraduée (par le degré de $\mathbf{C}\left[V\right]$ et le poids par rapport à $T$). On donne une formule intégrale pour la série de Poincaré de cette algèbre graduée....

Let $𝔽$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $𝔽$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,...,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $S{L}_2\left(ℂ\right)$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

We consider problems in invariant theory related to the classification of four vector subspaces of an $n$-dimensional complex vector space. We use castling techniques to quickly recover results of Howe and Huang on invariants. We further obtain information about principal isotropy groups, equidimensionality and the modules of covariants.