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Grobner bases via linkage for classes of generalized determinantal ideals

Date

2022-04-12

Journal Title

Journal ISSN

Volume Title

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ORCID

0000-0003-1201-754X

Type

Thesis

Degree Level

Doctoral

Abstract

Grobner bases are an important tool for working with ideals in polynomial rings. They have both computational and theoretical importance. In this dissertation, we produce Grobner bases for some families of generalized determinantal ideals. Our main contribution is a Grobner basis for Schubert patch ideals. Schubert patch ideals are prime defining ideals of open patches of Schubert varieties in the type A flag variety. We adapt E. Gorla, J. Migliore, and U. Nagel's "Grobner basis via linkage" technique to prove a conjecture of A. Yong, namely, the essential minors of every Schubert patch ideal form a Grobner basis. Using the same approach, we recover the result of A. Woo and A. Yong that the essential minors of a Kazhdan-Lusztig ideal (and hence, essential minors of a Schubert determinantal ideal) form a Grobner basis with respect to an appropriate term order. In addition, with respect to the standard grading, we show that homogeneous Schubert patch ideals, homogeneous Kazhdan-Lusztig ideals and Schubert determinantal ideals are glicci. In the last chapter of this dissertation, we briefly discuss some future directions.

Description

Keywords

Grobner basis, Schubert patch ideal, Kazhdan-Lusztig ideal, Schubert determinantal ideal, Schubert varieties, Glicci, Vertex decomposable

Citation

Degree

Doctor of Philosophy (Ph.D.)

Department

Mathematics and Statistics

Program

Mathematics

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