#### Document Type

Dissertation

#### Date of Degree

Summer 2010

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Julianna Tymoczko

#### Second Advisor

Frederick Goodman

#### First Committee Member

Charles Frohman

#### Second Committee Member

Jonathan Simon

#### Third Committee Member

Sriram Pemmaraju

#### 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[x_{1},…,x_{n}]. 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 *I _{h}*, 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

*I*is redundant. We give a minimal generating set for this ideal. The second ideal, which we call

_{h}*J*, 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

_{h}*J*, we identify a basis for the quotient

_{h}*R/J*.

_{h}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 *I _{h} ⊂ I_{h'}* and

*J*. We prove that

_{h}⊂ J_{h'}*I*equals

_{h}*J*when

_{h}*h*is maximal. Since

*I*is the ideal generated by the elementary symmetric functions when

_{h}*h*is maximal, the generating set for

*J*forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient

_{h}*R/J*gives another description of the cohomology ring of the full flag variety.

_{h}The generators of the ring *R/J _{h}* 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/J*may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.

_{h}#### Keywords

algebraic geometry, cohomology, combinatorics, commutative ring theory, Grobner basis, symmetric functions

#### Pages

viii, 89 pages

#### Bibliography

Includes bibliographical references (pages 88-89).

#### Copyright

Copyright 2010 Abukuse Mbirika III