Equivariant Schubert calculus and applications

Robichaux, Colleen

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https://hdl.handle.net/2142/116152 Copy

Description

Title

Equivariant Schubert calculus and applications

Author(s)

Robichaux, Colleen

Issue Date

2022-06-06

Director of Research (if dissertation) or Advisor (if thesis)

Yong, Alexander

Doctoral Committee Chair(s)

Tolman, Susan

Committee Member(s)

• Di Francesco, Philippe
• Kedem, Rinat

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Schubert calculus
• Combinatorial algorithms
• Equivariant cohomology
• Castelnuovo-Mumford regularity

Abstract

A central problem in algebraic combinatorics is to determine a combinatorial rule to compute the Schubert calculus of generalized flag varieties. For the Grassmannian, a solution is given by the Littlewood–Richardson rule. One strategy for studying these problems is to approach them in the richer setting of equivariant cohomology. Double Schubert polynomials are polynomial representatives of the equivariant cohomology classes in the complete flag variety Fln. Specializing them gives Schubert polynomials, representatives of ordinary cohomology classes in Fln. With A. Adve and A. Yong, we give the first polynomial time algorithm to decide if a monomial coefficient of a Schubert polynomial is zero. We introduce a tableau criterion to determine this vanishing. Further we show that explicitly computing these monomial coefficients is #P-complete. Littlewood–Richardson polynomials control the equivariant Schubert calculus of the Grassmannian. With A. Adve and A. Yong, we show that deciding the vanishing of a Littlewood–Richardson polynomial is polynomial time. This generalizes the work of J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni. With H. Yadav and A. Yong, we study the equivariant Schubert calculus of isotropic flag manifolds. We use Billey’s formula to establish an explicit correspondence between these isotropic structure coefficients. Additionally we introduce shifted edge labeled tableaux that conjecturally compute structure coefficients for a specialization of the equivariant Schubert calculus of the Lagrangian Grassmannian given by D. Anderson-W. Fulton. We prove additional cases of this conjecture. These results work towards finding a rule for computing the equivariant Schubert calculus of the Lagrangian Grassmannian. Double Schubert polynomials also appear as multidegrees of matrix Schubert varieties. The K-polynomials of matrix Schubert varieties are the double Grothendieck polynomials. With J. Rajchgot, Y. Ren, A. St. Dizier, and A. Weigandt, we use these polynomials to derive an explicit combinatorial rule for the Castelnuovo–Mumford regularity of Grassmannian matrix Schubert varieties. With J. Rajchgot and A. Weigandt we generalize these results for certain Kazhdan–Lusztig varieties. We use this to prove a correction of a conjecture of M. Kummini-V. Lakshmibai-P. Sastry-C. S. Seshadri.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Colleen Robichaux

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