Essential Semigroups and Branching Rules
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arXiv
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| Format: | Preprint |
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2024
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| author | Gornitskii, Andrei |
| author_facet | Gornitskii, Andrei |
| contents | Let $\mathfrak{g}$ be a semisimple complex Lie algebra of finite dimension and $\mathfrak{h}$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $\mathfrak{g}\supset\mathfrak{h}$. According to an idea of Zhelobenko the information on restriction to $\mathfrak{h}$ of all irreducible representations of $\mathfrak{g}$ is contained in one associative algebra, which we call the \emph{branching algebra}. We use an \emph{essential semigroup} $Σ$, which parametrizes some bases in every finite-dimensional irreducible representations of $\mathfrak{g}$, and describe the branching rules for $\mathfrak{g}\supset\mathfrak{h}$ in terms of a certain subsemigroup $Σ'$ of $Σ$. If $Σ'$ is finitely generated, then the semigroup algebra corresponding to $Σ'$ is a toric degeneration of the branching algebra. We propose the algorithm to find a description of $Σ'$ in this case. We give examples by deriving the branching rules for $A_n\supset A_{n-1}$, $B_n\supset D_n$, $G_2\supset A_2$, $B_3\supset G_2$, and $F_4\supset B_4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Essential Semigroups and Branching Rules Gornitskii, Andrei Representation Theory Let $\mathfrak{g}$ be a semisimple complex Lie algebra of finite dimension and $\mathfrak{h}$ be a semisimple subalgebra. We present an approach to find the branching rules for the pair $\mathfrak{g}\supset\mathfrak{h}$. According to an idea of Zhelobenko the information on restriction to $\mathfrak{h}$ of all irreducible representations of $\mathfrak{g}$ is contained in one associative algebra, which we call the \emph{branching algebra}. We use an \emph{essential semigroup} $Σ$, which parametrizes some bases in every finite-dimensional irreducible representations of $\mathfrak{g}$, and describe the branching rules for $\mathfrak{g}\supset\mathfrak{h}$ in terms of a certain subsemigroup $Σ'$ of $Σ$. If $Σ'$ is finitely generated, then the semigroup algebra corresponding to $Σ'$ is a toric degeneration of the branching algebra. We propose the algorithm to find a description of $Σ'$ in this case. We give examples by deriving the branching rules for $A_n\supset A_{n-1}$, $B_n\supset D_n$, $G_2\supset A_2$, $B_3\supset G_2$, and $F_4\supset B_4$. |
| title | Essential Semigroups and Branching Rules |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2407.07756 |