Shifted Schur functions II. Binomial formula for characters of classical groups and applications

TL;DR

This paper develops a binomial formula for characters of classical groups, introduces special bases in invariant subalgebras, and explores their relations via a G-equivariant isomorphism, advancing representation theory tools.

Contribution

It derives a Taylor-type expansion for characters of classical groups and constructs distinguished bases in invariant subalgebras, linking them through a special symmetrization.

Findings

Derived a binomial formula for classical group characters.

Identified a basis in Z(g) with shifted Schur functions.

Established a G-equivariant isomorphism between bases.

Abstract

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion for finite-dimensional characters of G (binomial formula) and use it to specify a distinguished linear basis in Z(g). The eigenvalues of the basis elements in highest weight g-modules are certain shifted (or factorial) analogs of Schur functions. We also study an associated homogeneous basis in I(g), the subalgebra of G-invariants in the symmetric algebra S(g). Finally, we show that the both bases are related by a G-equivariant linear isomorphism \sigma: I(g)\to Z(g), called the special symmetrization.