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...

Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$. Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$-theory, Chow groups, Grothendieck’s $K_0$, etc.) with formal group law $F$. We construct a ring from $F$ and the characters of $T$, that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁(G/B)$ where $G/B$ is the variety of Borel subgroups of $G$. Our main result says that when the torsion index...