Parabolic restrictions and double deformations of weight multiplicities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917065974087680 |
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| author | Lecouvey, Cédric |
| author_facet | Lecouvey, Cédric |
| contents | We introduce some (p,q)-deformations of the weight multiplicities for the representations of any simple Lie algebra g over the complex numbers. This is done by associating the indeterminate q to the positive roots of a parabolic subsystem of g and the indeterminate p to the remaining positive roots. When p=q, we just recover the usual Lusztig analogues of weight multiplicities. We then study the positivity of the coefficients in these double deformations. In particular, the positivity holds when p=1 in which case the polynomials have a natural algebraic interpretation in terms of a parabolic Brylinski filtration. For the parabolic restriction from type C to type A, this positivity result was conjectured by Lee. We also establish this positivity, in any finite type and for any p, for a stabilized version of our double deformation. In addition, we study the double deformation obtained by replacing the pair (p,q) by (p+1,q+1), show it has nonnegative coefficients and admits a combinatorial description in terms of crystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10003 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parabolic restrictions and double deformations of weight multiplicities Lecouvey, Cédric Combinatorics Representation Theory We introduce some (p,q)-deformations of the weight multiplicities for the representations of any simple Lie algebra g over the complex numbers. This is done by associating the indeterminate q to the positive roots of a parabolic subsystem of g and the indeterminate p to the remaining positive roots. When p=q, we just recover the usual Lusztig analogues of weight multiplicities. We then study the positivity of the coefficients in these double deformations. In particular, the positivity holds when p=1 in which case the polynomials have a natural algebraic interpretation in terms of a parabolic Brylinski filtration. For the parabolic restriction from type C to type A, this positivity result was conjectured by Lee. We also establish this positivity, in any finite type and for any p, for a stabilized version of our double deformation. In addition, we study the double deformation obtained by replacing the pair (p,q) by (p+1,q+1), show it has nonnegative coefficients and admits a combinatorial description in terms of crystals. |
| title | Parabolic restrictions and double deformations of weight multiplicities |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2412.10003 |