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}.
Categories: math.RT