Multiplicity-free representations of algebraic groups II

TL;DR

This paper classifies certain multiplicity-free representations of algebraic groups, focusing on cases where a type A subgroup is irreducibly embedded in groups of types B, C, or D, revealing few triples for high ranks but interesting exceptions at small ranks.

Contribution

It extends previous work by classifying multiplicity-free triples involving type A subgroups embedded in types B, C, or D, highlighting new exceptional cases at small ranks.

Findings

Few triples for high ranks of X

Several interesting exceptional triples at small ranks

Complete classification for the specified cases

Abstract

We continue our work (started in ``Multiplicity-free representations of algebraic groups", arXiv:2101.04476), on the program of classifying triples $(X,Y,V)$, where $X,Y$ are simple algebraic groups over an algebraically closed field of characteristic zero with $X<Y$, and $V$ is an irreducible module for $Y$ such that the restriction $V\downarrow X$ is multiplicity-free. In this paper we handle the case where $X$ is of type $A$, and is irreducibly embedded in $Y$ of type $B,C$ or $D$. It turns out that there are relatively few triples for $X$ of arbitrary rank, but a number of interesting exceptional examples arise for small ranks.