Castelnuovo–Mumford regularity of matrix Schubert varieties

• Oliver Pechenik ^[1] ; David E Speyer ^[2] ; Anna Weigandt ^[3]
  1. [1] University of Waterloo

     University of Waterloo

     Canadá

     
  2. [2] University of Michigan–Ann Arbor

     University of Michigan–Ann Arbor

     City of Ann Arbor, Estados Unidos

     
  3. [3] University of Minnesota

     University of Minnesota

     City of Minneapolis, Estados Unidos

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-44
• Idioma: inglés
• DOI: 10.1007/s00029-024-00959-x
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• Resumen

  • 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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