Young diagrams, deformed Calogero-Moser systems and Cayley graphs
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2024
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| _version_ | 1866929643655790592 |
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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 ${\mathfrak{gl}}(n|m)$ with root system $Δ$. Using $Δ$, Sergeev and Veselov, \cite{SV2} introduced an action of the Weyl groupoid ${\mathcal{W}}$, in connection with their study of the the Grothendieck group of finite dimensinonal graded $\mathfrak{g}$-modules. We denote the subgroupoid of ${\mathcal{W}}$ with morphisms corresponding to isotropic roots by $\mathfrak T_{iso}$.
Later, \cite{SV101} the same authors defined an action of ${\mathcal{W}}$ on $X={\mathtt{k}}^{n|m}$ such that the invariant algebra ${\mathcal{O}}(X)^{\mathcal{W}}$ is isomorphic to the algebra of quantum integrals for the deformed Calogero-Moser system introduced in \cite{SV1}. This completely integrable system depends on a non-zero parameter $κ$. When $κ=-m/n$ we study a certain infinite $\mathfrak T_{iso}$-orbit {\bf O} for this action. %which appears in \cite{SV101} Equation (14). The Cayley graph for this orbit is isomorphic to the Cayley graphs for two other actions of $\mathfrak T_{iso}$ which were studied in \cite{M23}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Young diagrams, deformed Calogero-Moser systems and Cayley graphs Musson, Ian M. Representation Theory Combinatorics Dynamical Systems 17B35 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 ${\mathfrak{gl}}(n|m)$ with root system $Δ$. Using $Δ$, Sergeev and Veselov, \cite{SV2} introduced an action of the Weyl groupoid ${\mathcal{W}}$, in connection with their study of the the Grothendieck group of finite dimensinonal graded $\mathfrak{g}$-modules. We denote the subgroupoid of ${\mathcal{W}}$ with morphisms corresponding to isotropic roots by $\mathfrak T_{iso}$. Later, \cite{SV101} the same authors defined an action of ${\mathcal{W}}$ on $X={\mathtt{k}}^{n|m}$ such that the invariant algebra ${\mathcal{O}}(X)^{\mathcal{W}}$ is isomorphic to the algebra of quantum integrals for the deformed Calogero-Moser system introduced in \cite{SV1}. This completely integrable system depends on a non-zero parameter $κ$. When $κ=-m/n$ we study a certain infinite $\mathfrak T_{iso}$-orbit {\bf O} for this action. %which appears in \cite{SV101} Equation (14). The Cayley graph for this orbit is isomorphic to the Cayley graphs for two other actions of $\mathfrak T_{iso}$ which were studied in \cite{M23}. |
| title | Young diagrams, deformed Calogero-Moser systems and Cayley graphs |
| topic | Representation Theory Combinatorics Dynamical Systems 17B35 |
| url | https://arxiv.org/abs/2412.16259 |