arXiv subjects

Akihito Hora

Publications and source records attributed to Akihito Hora.

7 recordsLinked to original sources

arXiv2026

Jucys--Murphy Elements for Wreath Products and Their Application to Dynamical Random Multi-Diagrams

The equivalence classes of irreducible representations of wreath product $\mathfrak{S}_n(T) = T^n \rtimes \mathfrak{S}_n$ of finite group $T$ with respect to symmetric group $\mathfrak{S}_n$ are parametrized by $\mathbb{Y}_n(\widehat{T})$, the $\lvert \widehat{T}\rvert$-tuple Young diagrams with total size $n$. We show a formula connecting the Kerov transition measures of these Young diagrams with the Jucys--Murphy elements of $\mathfrak{S}_n(T)$. This formula is due to Biane in the case of symmetric groups. The formula enables us to investigate asymptotic property of the shapes of multi-diagrams through combinatorial analysis for the Jucys--Murphy elements. On the other hand, a Markov chain is introduced on $\mathbb{Y}_n(\widehat{T})$, canonically reflecting the branching rule for the tower of wreath product groups. We have a continuous time stochastic process on $\mathbb{Y}_n(\widehat{T})$ from this chain by replacing the discrete time by a counting process. Our project is to specify the deterministic limit shape of multi-diagrams at each macroscopic time through appropriate space-time scaling limit, and to describe evolution of related quantities characterizing the shape. Especially, we derive dynamical concentrated limit shapes in the case of abelian $T$ by using free probability tools under the assumption of approximate factorization property for initial ensembles with an additional property of a pausing time distribution.

math.PR↗

arXiv2025

Time evolution of averaged limit shapes of random multiple Young diagrams

The branching rule for the tower of wreath products of a finite group by the symmetric groups induces a stochastic process on the set of multiple Young diagrams through random transitions of boxes of the diagrams between one another. We observe dynamical multiple averaged limit shapes resulting from appropriate scaling limits, either diffusive or non-diffusive. We describe time evolution of the macroscopic multiple averaged limit shapes in terms of Voiculescu's $R$-transforms and free Lévy measures of corresponding Kerov transition measures. Our microscopic dynamics admits non-exponential pausing time distributions.

math.PR↗

arXiv2024

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

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 $ω_t$ 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.

math.PR↗

arXiv2024

$q$-Racah probability distribution

We introduce a certain discrete probability distribution $P_{n,m,k,l;q}$ having non-negative integer parameters $n,m,k,l$ and quantum parameter $q$ which arises from a zonal spherical function of the Grassmannian over the finite field $\mathbb{F}_q$ with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single $q$-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating ${}_4 ϕ_3$-hypergeometric series.

math.RT↗

arXiv2023

Stochastic behavior of outcome of Schur-Weyl duality measurement

We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. As the first setting, we discuss the asymptotic behavior of the measurement outcome when the state is given as the permutation mixture $ρ_{mix,n,l}$ of the state $| 1^{l} \, 0^{n-l} \rangle := | 1 \rangle^{\otimes l} \otimes |0\rangle^{\otimes (n-l)}$. In contrast, when the state is given as the Dicke state $|Ξ_{n,l}\rangle$, the measurement outcome takes one deterministic value. These two cases have completely different behaviors. As the second setting, we study the case when the state is given as the tensor product of the permutation mixture $ρ_{mix,k,l}$ and the Dicke state $| Ξ_{n-k,m-l} \rangle$. We derive various types of asymptotic distribution including a kind of central limit theorem when $n$ goes to infinity.

math-ph↗

arXiv2019

Effect of microscopic pausing time distributions on the dynamical limit shapes for random Young diagrams

The irreducible decomposition of successive restriction and induction of irreducible representations of a symmetric group gives rise to a Markov chain on Young diagrams keeping the Plancherel measure invariant. Starting from this Res-Ind chain, we introduce a not necessarily Markovian continuous time random walk on Young diagrams by considering a general pausing time distribution between jumps according to the transition probability of the Res-Ind chain. We show that, under appropriate assumptions for the pausing time distribution, a diffusive scaling limit brings us concentration at a certain limit shape depending on macroscopic time which leads to a similar consequence to the exponentially distributed case studied in our earlier work. The time evolution of the limit shape is well described by using free probability theory. On the other hand, we illustrate an anomalous phenomenon observed with a pausing time obeying a one-sided stable distribution, heavy-tailed without the mean, in which a nontrivial behavior appears under a non-diffusive regime of the scaling limit.

math.PR↗

arXiv2018

Projective representations and spin characters of complex reflection groups $G(m, p, n)$ and $G(m, p, \infty)$, III

This paper is a continuation of two previous papers in MSJ Memoirs, Vol.\,29 (Math. Soc. Japan, 2013) with the same title and numbered as I and II. Based on the hereditary property given there, from mother groups $G(m,1,n)$, the generalized symmetric groups, to child groups $G(m,p,n)$, the complex reflection groups, we study in detail classification and construction of irreducible projective representations (= spin representations) and their characters of $G(m,1,n)$ for $n$ finite. Then, taking limits as $n$ tends to infinity, we obtain spin characters of the inductive limit groups $G(m,1,\infty)$. By the heredity studied further, this gives the main kernel of the results for $G(m,p,\infty)$ with $p|m, p>1$.

math.RT↗