Restriction of irreducible modules for $\mbox{Spin}_{2n+1}(K)$ to $\mbox{Spin}_{2n}(K)$

TL;DR

This paper classifies finite-dimensional irreducible modules of the group Spin(2n+1) over an algebraically closed field, focusing on those modules where a subgroup of type D_n acts with exactly two composition factors.

Contribution

It provides a complete classification of irreducible modules with a specific restriction property for Spin(2n+1) to Spin(2n).

Findings

Classified modules with exactly two composition factors under subgroup action.

Established isomorphism classes for these modules.

Extended understanding of module restrictions in algebraic groups.

Abstract

Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $Y=\mbox{Spin}_{2n+1}(K)$ $(n\geqslant 3)$ be a simply connected simple algebraic group of type $B_n$ over $K.$ Also let $X$ be the subgroup of type $D_n,$ embedded in $Y$ in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for $Y.$ In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational $KY$-modules on which $X$ acts with exactly two composition factors.