Ninth variation of classical group characters of type A-D and Littlewood identities

TL;DR

This paper generalizes classical Lie group characters and Schur functions using arbitrary polynomial sequences, deriving new identities and extending Macdonald's ninth variation to symplectic and orthogonal cases.

Contribution

It introduces a broad generalization of classical characters and Schur functions, including new identities and extensions of Macdonald's ninth variation.

Findings

Derived Littlewood-type identities for generalized characters

Established new Jacobi-Trudi identities for factorial and rational Schur functions

Extended Macdonald's ninth variation to symplectic and orthogonal characters

Abstract

We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. In this extension, the factorial powers are replaced with an arbitrary sequence of polynomials, as in Sergeev-Veselov's generalised Schur functions and Okada's generalised Schur P- and Q-functions. We also offer a similar generalisation for the rational Schur functions. We derive Littlewood-type identities for our generalisations. These identities allow us to give new (unflagged) Jacobi-Trudi identities for the Foley-King factorial characters and for rational versions of the factorial Schur functions. We also propose an extension of the original Macdonald's ninth variation of Schur functions to the case of symplectic and orthogonal characters, which helps us prove N\"agelsbach-Kostka identities.