Growth of root multiplicities along imaginary root strings in Kac--Moody algebras
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909978271416320 |
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| author | Carbone, Lisa Coelho, Terence Murray, Scott H. Thurman, Forrest Zhu, Songhao |
| author_facet | Carbone, Lisa Coelho, Terence Murray, Scott H. Thurman, Forrest Zhu, Songhao |
| contents | Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root $α$ and a real root $β$ of $\mathfrak{g}$, it is known that the $β$-string through $α$, denoted $R_α(β)$, is finite. Given an imaginary root $β$, we show that $R_α(β)=\{β\}$ or $R_α(β)$ is infinite. If $(β,β)<0$, we also show that the multiplicity of the root ${α+nβ}$ grows at least exponentially as $n\to\infty$. If $(β,β)=(α, β) = 0$, we show that $R_α(β)$ is bi-infinite and the multiplicities of $α+nβ$ are bounded. If $(β,β)=0$ and $(α, β) \neq 0$, we show that $R_α(β)$ is semi-infinite and the muliplicity of $α+nβ$ or $α-nβ$ grows faster than every polynomial as $n\to\infty$. We also prove that $\dim \mathfrak{g}_{α+β} \geq \dim \mathfrak{g}_α+ \dim \mathfrak{g}_β-1$ whenever $α\neq β$ with $(α, β)<0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01687 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Growth of root multiplicities along imaginary root strings in Kac--Moody algebras Carbone, Lisa Coelho, Terence Murray, Scott H. Thurman, Forrest Zhu, Songhao Representation Theory 17B67, 17B22 Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root $α$ and a real root $β$ of $\mathfrak{g}$, it is known that the $β$-string through $α$, denoted $R_α(β)$, is finite. Given an imaginary root $β$, we show that $R_α(β)=\{β\}$ or $R_α(β)$ is infinite. If $(β,β)<0$, we also show that the multiplicity of the root ${α+nβ}$ grows at least exponentially as $n\to\infty$. If $(β,β)=(α, β) = 0$, we show that $R_α(β)$ is bi-infinite and the multiplicities of $α+nβ$ are bounded. If $(β,β)=0$ and $(α, β) \neq 0$, we show that $R_α(β)$ is semi-infinite and the muliplicity of $α+nβ$ or $α-nβ$ grows faster than every polynomial as $n\to\infty$. We also prove that $\dim \mathfrak{g}_{α+β} \geq \dim \mathfrak{g}_α+ \dim \mathfrak{g}_β-1$ whenever $α\neq β$ with $(α, β)<0$. |
| title | Growth of root multiplicities along imaginary root strings in Kac--Moody algebras |
| topic | Representation Theory 17B67, 17B22 |
| url | https://arxiv.org/abs/2403.01687 |