Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917308047294464 |
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| author | Wan, Jinkui |
| author_facet | Wan, Jinkui |
| contents | We first introduce a new presentation for the mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$ over an arbitrary commutative ring $R$ and derive a new basis. Based on this presentation, specializing to the case of $\mathscr{H}_n(q)$ over the field $\mathbb{C}(q)$, we construct a basis for the cocenter of $\mathscr{H}_n(q)$, which facilitates the definition of its character table. We further establish a Schur--Weyl duality between $\mathscr{H}_n(q)$ and the quantum group $U_q(\mathfrak{gl}_r)$. As an application, we obtain Frobenius character formulas for the irreducible characters of $\mathscr{H}_n(q)$ within the ring of symmetric functions. Finally, we derive a recursive Murnaghan--Nakayama rule for the computation of the character table. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas Wan, Jinkui Representation Theory Combinatorics Quantum Algebra We first introduce a new presentation for the mirabolic Hecke algebra $\mathscr{H}_{n,R}(q)$ over an arbitrary commutative ring $R$ and derive a new basis. Based on this presentation, specializing to the case of $\mathscr{H}_n(q)$ over the field $\mathbb{C}(q)$, we construct a basis for the cocenter of $\mathscr{H}_n(q)$, which facilitates the definition of its character table. We further establish a Schur--Weyl duality between $\mathscr{H}_n(q)$ and the quantum group $U_q(\mathfrak{gl}_r)$. As an application, we obtain Frobenius character formulas for the irreducible characters of $\mathscr{H}_n(q)$ within the ring of symmetric functions. Finally, we derive a recursive Murnaghan--Nakayama rule for the computation of the character table. |
| title | Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas |
| topic | Representation Theory Combinatorics Quantum Algebra |
| url | https://arxiv.org/abs/2603.00603 |