A characteristic map for the symmetric space of symplectic forms over a finite field
Abstract
The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters . An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur-positivity of skew Macdonald polynomials with parameters as well as combinatorial formulas for spherical function values.
Keywords
Cite
@article{arxiv.1906.05966,
title = {A characteristic map for the symmetric space of symplectic forms over a finite field},
author = {Jimmy He},
journal= {arXiv preprint arXiv:1906.05966},
year = {2021}
}
Comments
Major revision. v1 Theorem 4.7 is incorrect and replaced with v2 Theorem 4.9. The product defined in Section 4.4 is replaced with a module structure. The application to the structure constants for Macdonald polynomials is replaced with an application to the Schur expansion of skew Macdonald Polynomials. The dependence on character sheaves is removed and the connection explained in a new section