Big composition factors in restrictions of~representations of the special linear group to~subsystem subgroups with two simple components

I. D. Suprunenko

Trudy Instituta matematiki, Tome 23 (2015) no. 2, pp. 123-136

Résumé

The article is devoted to constructing composition factors with certain special properties in the restrictions of modular irreducible representations of the special linear group to subsystem subgroups with two simple components. The goal is to find factors big in some sense for both components. For an irreducible representation $\varphi$ of the group $A_l(K)$ with highest weight $\sum_{i=1}^la_i\omega_i$ set $s(\varphi)=\sum_{i=1}^la_i$ and if $l>2$, put $t(\varphi)=\sum_{i=2}^{l-1}a_i$. We show that the restriction of $\varphi$ to a maximal subsystem subgroup with two simple components $H_1$ and $H_2$ has a composition factor of the form $\varphi_1\otimes\varphi_2$ where $\varphi_i$ is an irreducible representation of $H_i$, $s(\varphi_1)=s(\varphi)$, and $s(\varphi_2)=t(\varphi)$, and prove that for all such factors $\tau_1\otimes\tau_2$ the sum $s(\tau_1)+s(\tau_2)\leqslant s(\varphi)+t(\varphi)$ and $s(\tau_i)\leqslant s(\varphi)$. If the ground field characteristic is a prime $p$, the ranks of the components are $>2$, the representation $\varphi$ is $p$-restricted and its highest weight is large with respect to $p$, we almost always can construct a factor where the highest weight of $\varphi_1$ is large with respect to $p$ and $s(\varphi_i)$ are not very far from the maximal possible values. The existence of such factors yield effective tools for solving a number of questions, in particular, for finding or estimating various parameters of the images of individual elements in representations of such groups.