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The center $$Z_n(q)$$ of the integral group algebra of the general linear group $$GL_n(q)$$ over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras $$\mathscr{G}_n(q)$$ are independent of $$n$$, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras $$\mathscr{G}_n(q)$$ for all $$n$$. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author.
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Embeddings Among Quantum Affine sl_n
We establish an explicit embedding of a quantum affine sl_n into a quantum affine sl_{n+1} . This embedding serves as a common generalization of two natural, but seemingly unrelated embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine sl_n and gl_n. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the em- bedding on the idempotent version with that on the quantum affine Schur algebra level. A gl_n-variant of the embedding is also established.
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- Award ID(s):
- 1801915
- PAR ID:
- 10446731
- Date Published:
- Journal Name:
- Acta Mathematica Sinica, English Series
- ISSN:
- 1439-8516
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10114-023-2073-2
