On uniqueness of branching to fixed point Lie subalgebras

Abstract: Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\theta$ be a finite order automorphism of $\mathfrak{g}$. We assume that any ${\rm A}_{2n}$-type $\theta$-stable indecomposable ideal of $\mathfrak{g}$ is simple and any ${\rm D}_k$, ${\rm A}_{2k+1}$ and ${\rm E}_6$-type $\theta$-stable indecomposable ideal of $\mathfrak{g}$ has length at most $2$. Let $\mathfrak{g}_0$ be the fixed point subalgebra of $\mathfrak{g}$. In this talk, for any irreducible finite dimensional representations $V_1$ and $V_2$ of $\mathfrak{g}$, we show that ${\rm res}_{\mathfrak{g}_0}V_1\simeq {\rm res}_{\mathfrak{g}_0}V_2$ if and only if $V_2$ is isomorphic to $V_1^\sigma$, for some outer automorphism $\sigma$ of $\mathfrak{g}$.

combinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

( video )

ARCSIN - Algebra, Representations, Combinatorics and Symmetric functions in INdia

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