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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 [special characters omitted][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_{ h} is redundant. We give a minimal generating set for this ideal. The second ideal, which we call J_{h}, 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 J_{h}, we identify a basis for the quotient 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_{h} ⊂ J_{h'}. We prove that I_{h} equals J_{h} when h is maximal. Since I_{h} is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for J_{h} forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/J_{h} gives another description of the cohomology ring of the full flag variety.
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_{h} may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.
Advisor: | Tymoczko, Julianna, Goodman, Frederick |
Commitee: | Frohman, Charles, Pemmaraju, Sriram, Simon, Jonathan |
School: | The University of Iowa |
Department: | Mathematics |
School Location: | United States -- Iowa |
Source: | DAI-B 71/10, Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Mathematics |
Keywords: | Algebraic geometry, Cohomology, Combinatorics, Commutative ring theory, Grobner basis, Hessenberg varieties, Symmetric functions |
Publication Number: | 3422240 |
ISBN: | 978-1-124-20949-4 |