Young diagrams, Borel subalgebras and Cayley graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912159112364032 |
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| author | Musson, Ian M. |
| author_facet | Musson, Ian M. |
| contents | Let $\mathtt{k}$ be an algebraically closed field of characteristic zero and $n, m$ coprime positive integers. Let ${\stackrel{\rm o}{\mathfrak{g}}}$ be the Lie superalgebra ${\mathfrak{sl}}(n|m)$ and let $\mathfrak T_{iso}$ be the groupoid introduced by
Sergeev and Veselov \cite{SV2} with base the set of odd roots of ${\stackrel{\rm o}{\mathfrak{g}}}$. We show the Cayley graphs for three actions of $\mathfrak T_{iso}$ are isomorphic, These actions originate in quite different ways. Consider the set $X$ of Young diagrams contained in a rectangle with $n$ rows and $m$ columns. By adding or deleting rows and columns from certain diagrams and keeping track of the total number of boxes added or deleted, we obtain an equivalence relation on $X\times {\mathbb Z}$ such that $\mathfrak T_{iso}$ acts on the set of equivalence classes $[X\times {\mathbb Z}]$. We compare the action on $[X\times {\mathbb Z}]$ to an action on Borel subalgebras of the affinization ${\widehat{L}(\stackrel{\rm _o}{\mathfrak{g}})}$ of ${\stackrel{\rm o}{\mathfrak{g}}}$ which are related by odd reflections. The third action comes from an action of $\mathfrak T_{iso}$ on $\mathtt{k}^{n|m}$ defined by
Sergeev and Veselov, motivated by deformed quantum Calogero-Moser problems \cite{SV1}. This action will be considered in \cite{M24}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12141 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Young diagrams, Borel subalgebras and Cayley graphs Musson, Ian M. Representation Theory 17B65 Let $\mathtt{k}$ be an algebraically closed field of characteristic zero and $n, m$ coprime positive integers. Let ${\stackrel{\rm o}{\mathfrak{g}}}$ be the Lie superalgebra ${\mathfrak{sl}}(n|m)$ and let $\mathfrak T_{iso}$ be the groupoid introduced by Sergeev and Veselov \cite{SV2} with base the set of odd roots of ${\stackrel{\rm o}{\mathfrak{g}}}$. We show the Cayley graphs for three actions of $\mathfrak T_{iso}$ are isomorphic, These actions originate in quite different ways. Consider the set $X$ of Young diagrams contained in a rectangle with $n$ rows and $m$ columns. By adding or deleting rows and columns from certain diagrams and keeping track of the total number of boxes added or deleted, we obtain an equivalence relation on $X\times {\mathbb Z}$ such that $\mathfrak T_{iso}$ acts on the set of equivalence classes $[X\times {\mathbb Z}]$. We compare the action on $[X\times {\mathbb Z}]$ to an action on Borel subalgebras of the affinization ${\widehat{L}(\stackrel{\rm _o}{\mathfrak{g}})}$ of ${\stackrel{\rm o}{\mathfrak{g}}}$ which are related by odd reflections. The third action comes from an action of $\mathfrak T_{iso}$ on $\mathtt{k}^{n|m}$ defined by Sergeev and Veselov, motivated by deformed quantum Calogero-Moser problems \cite{SV1}. This action will be considered in \cite{M24}. |
| title | Young diagrams, Borel subalgebras and Cayley graphs |
| topic | Representation Theory 17B65 |
| url | https://arxiv.org/abs/2412.12141 |