Castelnuovo–Mumford regularity of matrix Schubert varieties

Oliver Pechenik, David E Speyer, Anna Weigandt
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Abstract

Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order.

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矩阵舒伯特变体的卡斯特诺沃-芒福德正则性
矩阵舒伯特变种是完整旗变种的舒伯特微积分中出现的仿射变种。我们给出了矩阵舒伯特变的卡斯特诺沃-芒福德正则性公式,回答了珍娜-拉奇戈特(Jenna Rajchgot)的一个问题。我们按照她提出的策略研究格罗内迪克多项式的最高阶同调部分,我们称之为卡斯特诺沃-芒福德多项式。除了正则公式外,我们还得到了所有卡斯特诺沃-蒙福德多项式及其前导项的度数公式,以及两个卡斯特诺沃-蒙福德多项式在标量倍数以内一致时的完整描述。格罗登第克多项式的度数是一种新的置换统计量,我们称之为拉吉哥特指数;我们发展了拉吉哥特指数的性质,并将其与主要指数和弱序联系起来。
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