Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties

Abukuse Mbirika III

doi:10.17077/etd.5mgrqru9

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Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties

Dissertation

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Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties

Abukuse Mbirika III

University of Iowa

Doctor of Philosophy (PhD), University of Iowa

Summer 2010

DOI: 10.17077/etd.5mgrqru9

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Abstract

Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties. Let R be the polynomial ring Ζ[x1,…,xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal I_h generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for Jh, we identify a basis for the quotient R/Jh. We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h>h', we have Ih ⊂ Ih' and Jh ⊂ Jh'. We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety. The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator X and a Hessenberg function h. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as h varies, the quotient R/Jh may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.

Mathematics

Combinatorics

algebraic geometry

cohomology

commutative ring theory

Grobner basis

symmetric functions

Details

Title: Subtitle: Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties
Creators: Abukuse Mbirika III - University of Iowa
Contributors: Julianna Tymoczko (Advisor)
Frederick Goodman (Advisor)
Charles Frohman (Committee Member)
Jonathan Simon (Committee Member)
Sriram Pemmaraju (Committee Member)
Resource Type: Dissertation
Degree Awarded: Doctor of Philosophy (PhD), University of Iowa
Degree in: Mathematics
Date degree season: Summer 2010
Publisher: University of Iowa
DOI: 10.17077/etd.5mgrqru9
Number of pages: viii, 89 pages
Language: English
Description bibliographic: Includes bibliographical references (pages 88-89).
Academic Unit: Mathematics
Record Identifier: 9983777100902771

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