Skew odd orthogonal characters and interpolating Schur polynomials

2025/02/21 by Jing, Naihuan, Li, Zhijun, Wang, Danxia +1
#Combinatorics (math.CO) #FOS: Mathematics #Primary: 05E05 #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Secondary: 17B37

paper · doi:10.48550/arxiv.2502.15586

Abstract

We introduce two vertex operators to realize skew odd orthogonal characters so_λ/μ(x^±) and derive the Cauchy identity for the skew characters via Toeplitz-Hankel-type determinant similar to the Schur functions. The method also gives new proofs of the Jacobi--Trudi identity and Gelfand--Tsetlin patterns for so_λ/μ(x^±). Moreover, combining the vertex operators related to characters of types C,D (\citeBa1996,JN2015) and the new vertex operators related to B-type characters, we obtain three families of symmetric polynomials that interpolate among characters of SO_2n+1(ℂ), SO_2n(ℂ) and Sp_2n(ℂ), Their transition formulas are also explicitly given among symplectic and/or orthogonal characters and odd orthogonal characters.

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