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

Abstract

We study a spectral problem related to the finite-dimensional characters of the groups $Sp(2N)$, $SO(2N+1)$, and $SO(2N)$, which form the classical series $C$, $B$, and $D$, respectively. The irreducible characters of these three series are given by $N$-variate symmetric polynomials. The spectral problem in question consists in the decomposition of the characters after their restriction to the subgroups of the same type but smaller rank $K<N$. The main result of the paper is the derivation of explicit determinantal formulas for the coefficients in this decomposition. In fact, we first compute these coefficients in a greater generality -- for the multivariate symmetric Jacobi polynomials depending on two continuous parameters. Next, we show that the formulas can be drastically simplified for the three special cases of Jacobi polynomials corresponding to the $C$-$B$-$D$ characters. In particular, we show that then the coefficients are given by piecewise polynomial functions. This is where a link with discrete splines arises. In type $A$ (that is, for the characters of the unitary groups $U(N)$), similar results were earlier obtained by Alexei Borodin and the author [Adv. Math., 2012], and then reproved by another method by Leonid Petrov [Moscow Math. J., 2014]. The case of the symplectic and orthogonal characters is more intricate.