Young diagrams, Borel subalgebras and Cayley graphs

TL;DR

This paper explores the isomorphisms between Cayley graphs arising from three distinct actions of a groupoid on Young diagrams, Borel subalgebras, and a vector space, revealing deep structural connections in Lie superalgebra theory.

Contribution

It establishes the isomorphism of Cayley graphs for three different actions of the groupoid on combinatorial and algebraic structures related to Lie superalgebras.

Findings

Cayley graphs for three actions are isomorphic

Action on Young diagrams relates to Borel subalgebras via odd reflections

Third action connects to deformed quantum Calogero-Moser problems

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 ${\mathfrak{sl}}(n|m)$ 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}.