Fluctuations of Young diagrams for symplectic groups
时间
2025年01月24日 13:00 至 14:30
地点
A14-201
线上
Zoom 242 742 6089
(BIMSA)
摘要
Consider an $n\times k$ matrix of i.i.d. Bernoulli random numbers with some value of $p$. Dual RSK algorithm gives a bijection of this matrix to a pair of Young tableaux of conjugate shape, which is manifestation of skew Howe $GL_{n}\times GL_{k}$-duality. Thus the probability measure on zero-ones matrix leads to the probability measure on Young diagrams proportional to the ratio of the dimension of $GL_{n}\times GL_{k}$-representation and the dimension of the exterior algebra $\bigwedge\left(C^{n}\otimes C^{k}\right)$. Similarly, by applying Proctor's algorithm based on Berele's modification of the Schensted insertion, we get skew Howe duality for the pairs of groups $Sp_{2n}\times Sp_{2k}$. In the limit when $n,k\to\infty$, $GL$-case is relatively easily studied by use of free-fermionic representation for the correlation kernel. But for the symplectic groups there is no convenient free-fermionic representation. We use Christoffel transformation to obtain the semiclassical orthogonal polynomials for $Sp_{2n}\times Sp_{2k}$.