The Weyl groupoid in Type A, Young diagrams and Borel subalgebras

Abstract

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero. Let ${\stackrel{{\rm o}}{{\mathfrak{g}}}}$ be the Lie superalgebra ${\mathfrak{sl}}(n|m)$ and let $\mathfrak{W}$ be the Weyl groupoid introduced by Sergeev and Veselov using the root system of ${\stackrel{{\rm o}}{{\mathfrak{g}}}}$. An important subgroupoid $\mathfrak T_{iso}$ of ${\mathfrak{W}}$ has base $\Delta_{iso}$, the set of all the isotropic roots. Motivated by deformed quantum Calogero-Moser problems, the same authors considered an action of $\mathfrak{W}$ on $\mathtt{k}^{n|m}$ depending on a parameter $\kappa$. %When $\kappa$ is negative special, they showed this action has infinite orbits. In the case $m>n$, with $m,n $ relatively prime and $\kappa=-n/m$ we study a particular infinite orbit of $\mathfrak T_{iso}$ with some special properties. This orbit, thought of as a directed graph is isomorphic to the graph of an orbit for the action of $\mathfrak T_{iso}$ on certain Borel subalgebras of the affinization ${\widehat{L}(\stackrel{{\rm _o}}{{\mathfrak{g}}})}$ of ${\stackrel{{\rm o}}{{\mathfrak{g}}}}$. %The root groupoid has a base consisting of Borel subalgebras with fixed even part, and morphisms are given by odd reflections. The underlying reason for this graph isomorphism is that both have combinatorics which can be described using Young diagrams and tableaux drawn on the surface of a rotating cylinder with circumference $n$ and length $m$. Allowing the cylinder to rotate produces an infinite orbit. This leads to a third graph which is isomorphic to the other two.