Speaker
Description
Let ${\mathtt{k}}$ be an algebraically closed field of characteristic zero.
Let ${\stackrel{{\mathrm o}}{{\mathfrak{g}}}}$ be the Lie superalgebra ${\mathfrak{sl}}(n|m)$ and let $\mathfrak T_{iso}$ be the groupoid introduced by Sergeev and Veselov with base the set of odd roots of ${\stackrel{{\mathrm o}}{{\mathfrak{g}}}}$.
We show the Cayley graphs for three actions of $\mathfrak T_{iso}$ are isomorphic.
These actions originate in quite different ways.
The first arises from Young diagrams contained in a rectangle with $n$ rows and $m$ columns, the second from Borel subalgebras of the affinization $\widehat{L}(\stackrel{{\mathrm o}}{{\mathfrak{g}}})$ of ${\stackrel{{\mathrm o}}{{\mathfrak{g}}}}$ 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.