2022/09/02 by Zhihong Jin, Naihuan Jing, Jin, Zhihong +5
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2209.00767
In this paper, we first introduce a family of universal symplectic functions spλ(x±;z) that include symplectic Schur functions spλ(x±), odd symplectic characters spλ(x±;z), universal symplectic characters spλ(z) and intermediate symplectic characters as subfamilies. We then realize the universal symplectic functions by vertex operators, which naturally lead to their skew versions, and show that spλ(x±;z) obey the general branching rules. This also gives the Gelfand-Tsetlin representations of odd symplectic characters and a transition formula between odd symplectic characters and symplectic Schur functions. Secondly we introduce a family of universal orthogonal functions oλ(x±;z) and their skew versions in a similar manner, and we provide their vertex operator realizations and obtain transition formulas and the branching rule. The universal orthogonal functions oλ(x±;z) generalize orthogonal Schur functions oλ(x±), odd orthogonal Schur functions soλ(x±), universal orthogonal characters oλ(z) as well as intermediate orthogonal characters. Thirdly, we give vertex operator realizations for the CB-interpolating Schur functions sCBλ(x;β) introduced by Bisi and Zygouras (Adv. Math., 2022) and the DB-interpolating Schur functions sDBλ(x;β) interpolating between characters of type D and B. As an application, we show sCBλ(x;β) are equal to the orthosymplectic Schur polynomials spoλ(x/β), thus give a short proof of the generalization of the Brent-Krattenthaler-Warnaar identity obtained by Kumari (arXiv:2401.01723).