2023/09/12 by Hora, Akihito · 1 citation
#20C30 #46L54 #60J27 #82C41 #FOS: Mathematics #Probability (math.PR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2309.06059
The branching rule for spin irreducible representations of symmetric groups gives rise to a Markov chain on the spin dual (\widetilde\mathfrakSn)^\wedgespin of symmetric group \mathfrakSn through restriction and induction of spin irreducible representations. This further produces a continuous time random walk (Xs(n))s\geqq 0 on (\widetilde\mathfrakSn)^\wedgespin by introducing an appropriate pausing time. Taking diffusive scaling limit for these random walks under s=tn and 1/√(n) reduction as n→∞, we consider a concentration phenomenon at each macroscopic time t. Since an element of (\widetilde\mathfrakSn)^\wedgespin is regarded as a strict partition of n with ± 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 ωt 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.