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

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.