Dynamical Spin Limit Shape of Young Diagram and Spin Jucys-Murphy Elements for Symmetric Groups

TL;DR

This paper studies the evolution of limit shapes of strict partitions associated with spin representations of symmetric groups, using probabilistic and free probability tools to describe their dynamics over time.

Contribution

It introduces a framework for analyzing the time evolution of limit shapes of strict partitions in spin symmetric groups, connecting representation theory with free probability.

Findings

Derived the limit shape $_t$ for the evolving strict partitions.

Established a spin version of Biane's formula for Jucys--Murphy elements.

Described the propagation of initial concentration to all times $t>0$.

Abstract

The branching rule for spin irreducible representations of symmetric groups gives rise to a Markov chain on the spin dual $(\widetilde{\mathfrak{S}}_n)^\wedge_{\mathrm{spin}}$ of symmetric group $\mathfrak{S}_n$ through restriction and induction of spin irreducible representations. This further produces a continuous time random walk $(X_s^{(n)})_{s\geqq 0}$ on $(\widetilde{\mathfrak{S}}_n)^\wedge_{\mathrm{spin}}$ by introducing an appropriate pausing time. Taking diffusive scaling limit for these random walks under $s=tn$ and $1/\sqrt{n}$ reduction as $n\to\infty$, we consider a concentration phenomenon at each macroscopic time $t$. Since an element of $(\widetilde{\mathfrak{S}}_n)^\wedge_{\mathrm{spin}}$ is regarded as a strict partition of $n$ with $\pm 1$ indices, the limit shapes of profiles of strict partitions appear. In this paper, we give a framework in which initial concentration at $t=0$ is propagated to concentration at any $t>0$. We thus obtain the limit shape $\omega_t$ depending on macroscopic time $t$, and describe the time evolution by using devices in free probability theory. Included is the case where Kerov's transition measure has non-compact support but determinate moment problem. A spin version of Biane's formula for spin Jucys--Murphy elements is shown, which plays an important role in our analysis.