Quantum super Schur--Weyl duality and character formulas for mirabolic Hecke algebras
Abstract
We establish a quantum super Schur--Weyl duality for the mirabolic Hecke algebra Hnmir(q) with q indeterminate and determine the corresponding bimodule decomposition. The centralizer of the mirabolic action is shown as a direct sum of homogeneous quantum Schur superalgebras. We describe the tensor-space annihilator and the resulting faithful quotient. When nonzero, the annihilator is generated by an explicit spectral idempotent attached to the smallest rectangle excluded by the hook condition. From the quantum super Schur--Weyl duality, we derive a super Frobenius formula for mirabolic Hecke algebras. This formula then yields a Murnaghan--Nakayama rule and Regev-type formulas for the irreducible characters. We also construct a super mirabolic RSK bijection from words in a super alphabet with an additional even letter to hook semistandard insertion tableaux together with recording pairs that distinguish the positions of the additional letter. This correspondence yields a Roichman formula for irreducible characters of Hnmir(q). Finally, we establish a parameter-inversion isomorphism between Hnmir(qâ1) and the q-rook monoid algebra Rn(q) and use it to transport the duality, annihilators, and character formulas in both directions. The transported tensor decomposition provides a representation-theoretic interpretation of hook and two-row character sums for Rn(q).
Related
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BibTeX
@misc{jing2026quantum,
title = {{Quantum super Schur--Weyl duality and character formulas for mirabolic Hecke algebras}},
author = {Naihuan Jing and Ning Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11101},
eprint = {2610.11101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11101},
}
RIS
TY - UNPB
TI - Quantum super Schur--Weyl duality and character formulas for mirabolic Hecke algebras
AU - Naihuan Jing
AU - Ning Liu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11101
UR - https://arxiv.org/abs/2610.11101
AB - We establish a quantum super Schur--Weyl duality for the mirabolic Hecke algebra H_n^mir(q) with q indeterminate and determine the corresponding bimodule decomposition. The centralizer of the mirabolic action is shown as a direct sum of homogeneous quantum Schur superalgebras. We describe the tensor-space annihilator and the resulting faithful quotient. When nonzero, the annihilator is generated by an explicit spectral idempotent attached to the smallest rectangle excluded by the hook condition. From the quantum super Schur--Weyl duality, we derive a super Frobenius formula for mirabolic Hecke algebras. This formula then yields a Murnaghan--Nakayama rule and Regev-type formulas for the irreducible characters. We also construct a super mirabolic RSK bijection from words in a super alphabet with an additional even letter to hook semistandard insertion tableaux together with recording pairs that distinguish the positions of the additional letter. This correspondence yields a Roichman formula for irreducible characters of H_n^mir(q). Finally, we establish a parameter-inversion isomorphism between H_n^mir(q^â1) and the q-rook monoid algebra R_n(q) and use it to transport the duality, annihilators, and character formulas in both directions. The transported tensor decomposition provides a representation-theoretic interpretation of hook and two-row character sums for R_n(q).
ER -