Young diagrams, Borel subalgebras and Cayley graphs

Ian M. Musson

Published 2024-12-09Version 1

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}.