Crystals for shifted key polynomials
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909979258126336 |
|---|---|
| author | Marberg, Eric Scrimshaw, Travis |
| author_facet | Marberg, Eric Scrimshaw, Travis |
| contents | This article continues our study of $P$- and $Q$-key polynomials, which are (non-symmetric) "partial" Schur $P$- and $Q$-functions as well as "shifted" versions of key polynomials. Our main results provide a crystal interpretation of $P$- and $Q$-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra $\mathfrak{q}_n$. In the $P$-key case, the ambient normal crystals are the $\mathfrak{q}_n$-crystals studied by Grantcharov et al., while in the $Q$-key case, these are replaced by the extended $\mathfrak{q}_n$-crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into $P$- and $Q$-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal $\mathfrak{q}_n$-crystals and Demazure $\mathfrak{gl}_n$-crystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00336 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Crystals for shifted key polynomials Marberg, Eric Scrimshaw, Travis Combinatorics Representation Theory This article continues our study of $P$- and $Q$-key polynomials, which are (non-symmetric) "partial" Schur $P$- and $Q$-functions as well as "shifted" versions of key polynomials. Our main results provide a crystal interpretation of $P$- and $Q$-key polynomials, namely, as the characters of certain connected subcrystals of normal crystals associated to the queer Lie superalgebra $\mathfrak{q}_n$. In the $P$-key case, the ambient normal crystals are the $\mathfrak{q}_n$-crystals studied by Grantcharov et al., while in the $Q$-key case, these are replaced by the extended $\mathfrak{q}_n$-crystals recently introduced by the first author and Tong. Using these constructions, we propose a crystal-theoretic lift of several conjectures about the decomposition of involution Schubert polynomials into $P$- and $Q$-key polynomials. We verify these generalized conjectures in a few special cases. Along the way, we establish some miscellaneous results about normal $\mathfrak{q}_n$-crystals and Demazure $\mathfrak{gl}_n$-crystals. |
| title | Crystals for shifted key polynomials |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2306.00336 |