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...
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| Published in: | International mathematics research notices Vol. 2022; no. 9; pp. 6854 - 6902 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Oxford University Press
01-05-2022
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| Online Access: | Get full text |
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| Summary: | 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 $(q,q^2)$. 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 $(q,q^2)$ as well as combinatorial formulas for spherical function values. |
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| ISSN: | 1073-7928 1687-0247 |
| DOI: | 10.1093/imrn/rnaa309 |