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Presenting mirabolic quantum Schur algebras ℳ𝒮v(2,d)
Abstract
Let v be an indeterminate. We give a presentation of the mirabolic quantum Schur algebra ℳ𝒮v(2,d) in terms of generators and defining relations. More precisely, we determine the kernel of the natural epimorphism from Rosso's mirabolic quantum sl2, Umirv(2)=⟨ e,f,k± 1,ℓ⟩, onto ℳ𝒮v(2,d). Setting Pd(k)=∏r=0d(k−v2r−d) and Qd(k)=∏r=1d(k−v2r−d), we prove ℳ𝒮v(2,d)≅ Umirv(2)/ ⟨ Pd(k),(ℓ−1)Qd(k)⟩. We further determine the split Wedderburn decomposition of this quotient and provide an equivalent presentation in terms of weight idempotents.
Related
- Quantum super Schur--Weyl duality and character formulas for mirabolic Hecke algebras
- Combinatorial description of the mirabolic quantum Schur-Weyl duality
- Cellular bases of mirabolic Hecke algebras
- Positivity of the Mirabolic Quantum Groups
- Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma
BibTeX
@misc{chen2026presenting,
title = {{Presenting mirabolic quantum Schur algebras $\mathcalMS_v(2,d)$}},
author = {Hongjia Chen and Jian Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08353},
eprint = {2609.08353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08353},
}
RIS
TY - UNPB
TI - Presenting mirabolic quantum Schur algebras ℳ𝒮_v(2,d)
AU - Hongjia Chen
AU - Jian Chen
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08353
UR - https://arxiv.org/abs/2609.08353
AB - Let v be an indeterminate. We give a presentation of the mirabolic quantum Schur algebra ℳ𝒮_v(2,d) in terms of generators and defining relations. More precisely, we determine the kernel of the natural epimorphism from Rosso's mirabolic quantum sl_2, U^mir_v(2)=⟨ e,f,k^± 1,ℓ⟩, onto ℳ𝒮_v(2,d). Setting P_d(k)=∏_r=0^d(k−v^2r−d) and Q_d(k)=∏_r=1^d(k−v^2r−d), we prove ℳ𝒮_v(2,d)≅ U^mir_v(2)/ ⟨ P_d(k),(ℓ−1)Q_d(k)⟩. We further determine the split Wedderburn decomposition of this quotient and provide an equivalent presentation in terms of weight idempotents.
ER -