Characters of classical groups, Schur-type functions and discrete splines

Abstract

We study a spectral problem related to finite-dimensional characters of the groups $Sp(2N)$, $SO(2N+1)$ and $SO(2N)$, which form the classical series $\mathcal{C}$, $\mathcal{B}$ and $\mathcal{D}$, respectively. Irreducible characters of these three series are given by $N$-variate symmetric polynomials. The spectral problem in question consists in the decomposition of characters after their restriction to subgroups of the same type but smaller rank $K<N$. The main result of the paper is the derivation of explicit determinantal formulae for the coefficients of the expansion. In fact, first we compute these coefficients in greater generality - for the multivariate symmetric Jacobi polynomials depending on two continuous parameters. Next, we show that the formulae are drastically simplified in the three special cases of Jacobi polynomials corresponding to characters of the series $\mathcal{C}$, $\mathcal{B}$ and $\mathcal{D}$. In particular, we show that then these coefficients are given by piecewise polynomial functions. This is where a link with discrete splines arises. For characters of the series $\mathcal{A}$ (that is, of the unitary groups $U(N)$) similar results were obtained previously by Borodin and this author [5], and then reproved by Petrov [39] by another method. The case of symplectic and orthogonal characters is more intricate. Bibliography: 58 titles.