Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials

TL;DR

This paper studies the asymptotic behavior of Young diagrams associated with symplectic groups, using orthogonal polynomials derived from Krawtchouk polynomials, to understand their limit shapes and fluctuations.

Contribution

It introduces a novel approach to analyze symplectic group Young diagrams via semiclassical orthogonal polynomials obtained through Christoffel transformations.

Findings

Derived integral representation for semiclassical orthogonal polynomials.

Described limit shapes of Young diagrams for symplectic groups.

Analyzed fluctuations of Young diagrams using asymptotic analysis.

Abstract

Consider an $n\times k$ matrix of i.i.d. Bernoulli random numbers with $p=1/2$. 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 $\bigwedge\left(\mathbb{C}^{n}\otimes\mathbb{C}^{k}\right)$. 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 $Sp_{2n}\times Sp_{2k}$ from Krawtchouk polynomials that describe $GL_{2n}\times GL_{2k}$ case. We derive an integral representation for semiclassical polynomials. The study of the asymptotic of this integral representation gives us the description of the limit shapes and fluctuations of the random Young diagrams for symplectic groups.