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

TL;DR

This paper derives explicit formulas for decomposing classical group characters into smaller rank subgroups, revealing connections to discrete splines and extending prior results from unitary groups to symplectic and orthogonal groups.

Contribution

It provides new determinantal formulas for character decompositions of classical groups, including simplified piecewise polynomial expressions linked to discrete splines.

Findings

Explicit formulas for character restriction coefficients

Piecewise polynomial formulas for special Jacobi cases

Extension of previous unitary group results to symplectic and orthogonal groups

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.