Young diagrams, deformed Calogero-Moser systems and Cayley graphs

Abstract

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 $\Delta$. Using $\Delta$, 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 $\kappa$. When $\kappa=-m/n$ 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}.