发表机构
North Carolina State University; Hefei University of Technology（北卡罗来纳州立大学; 合肥工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立了mirabolic Hecke代数的量子超Schur–Weyl对偶，推导了其不可约特征标的相关公式，还构造了超mirabolic RSK双射，并建立了其与q-rook幺半群代数的参数反转同构以转移相关结果。
AI 中文摘要
我们建立了参数q为不定元的mirabolic Hecke代数\boldsymbol{H_n^{\text{mir}}(q)}的量子超Schur–Weyl对偶，并确定了对应的双模分解。mirabolic作用的中心化子被证明是齐次量子超Schur代数的直和。我们描述了张量空间零化子及其得到的忠实商，当非零时，该零化子由钩子条件排除的最小矩形对应的显式谱幂等元生成。从量子超Schur–Weyl对偶出发，我们推导了mirabolic Hecke代数的超Frobenius公式，该公式进而得到了不可约特征标的Murnaghan–Nakayama规则和Regev型公式。我们还构造了从带有额外偶字母的超字母表中的字到钩子半标准插入表auxiliary的超mirabolic RSK双射，以及记录对，用于区分额外字母的位置，该对应关系给出了\boldsymbol{H_n^{\text{mir}}(q)}不可约特征标的Roichman公式。最后，我们建立了\boldsymbol{H_n^{\text{mir}}(q^{-1})}与q-rook幺半群代数\boldsymbol{R_n(q)}之间的参数反转同构，并利用它在两个方向上转移对偶、零化子和特征标公式，转移后的张量分解为\boldsymbol{R_n(q)}的钩子和两行特征标和提供了表示论解释。
英文摘要
We establish a quantum super Schur--Weyl duality for the mirabolic Hecke algebra \(H_n^{\mathrm{mir}}(q)\) 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 \(H_n^{\mathrm{mir}}(q)\). Finally, we establish a parameter-inversion isomorphism between \(H_n^{\mathrm{mir}}(q^{-1})\) 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 \(R_n(q)\).
Comments56 pages
