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1.
Gedeon, Katie.
* Kazhdan*-

Degree: PhD, Department of Mathematics, 2018, University of Oregon

URL: http://hdl.handle.net/1794/23913

The Kazhdan-Lusztig polynomial of a matroid M, denoted P_{M}( t ), was recently defined by Elias, Proudfoot, and Wakefield. These polynomials are analogous to the classical Kazhdan-Lusztig polynomials associated with Coxeter groups. For example, in both cases there is a purely combinatorial recursive definition. Furthermore, in the classical setting, if the Coxeter group is a Weyl group then the Kazhdan-Lusztig polynomial is a Poincare polynomial for the intersection cohomology of a particular variety; in the matroid setting, if M is a realizable matroid then the Kazhdan-Lusztig polynomial is also the intersection cohomology Poincare polynomial of a variety corresponding to M. (Though there are several analogies between the two types of polynomials, the theory is quite different.)
Here we compute the Kazhdan-Lusztig polynomials of several graphical matroids, including thagomizer graphs, the complete bipartite graph K_{2,n}, and (conjecturally) fan graphs. Additionally, we investigate a conjecture by the author, Proudfoot, and Young on the real-rootedness for Kazhdan-Lusztig polynomials of these matroids as well as a conjecture on the interlacing behavior of these roots. We also show that the Kazhdan-Lusztig polynomials of uniform matroids of rank n − 1 on n elements are real-rooted.
This dissertation includes both previously published and unpublished co-authored material.
*Advisors/Committee Members: Proudfoot, Nicholas (advisor).*

Subjects/Keywords: Kazhdan-Lusztig polynomials; Matroid theory; real-rootedness

…laid out the analogy between this new *theory* and the classical *theory*
of *Kazhdan*-*Lusztig*… …Matroids and Their *Kazhdan*-*Lusztig* Polynomials . . . . . . .
5
2.2.
Equivariant Matroids and… …The Equivariant *Kazhdan*-*Lusztig*
Polynomial… …30
ROOTS OF *KAZHDAN*-*LUSZTIG* POLYNOMIALS OF
MATROIDS… …xii
LIST OF TABLES
Table
Page
4.1. *Kazhdan*-*Lusztig* polynomials of some uniform matroids…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Gedeon, K. (2018). Kazhdan-Lusztig Polynomials of Matroids and Their Roots. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/23913

Chicago Manual of Style (16^{th} Edition):

Gedeon, Katie. “Kazhdan-Lusztig Polynomials of Matroids and Their Roots.” 2018. Doctoral Dissertation, University of Oregon. Accessed December 05, 2020. http://hdl.handle.net/1794/23913.

MLA Handbook (7^{th} Edition):

Gedeon, Katie. “Kazhdan-Lusztig Polynomials of Matroids and Their Roots.” 2018. Web. 05 Dec 2020.

Vancouver:

Gedeon K. Kazhdan-Lusztig Polynomials of Matroids and Their Roots. [Internet] [Doctoral dissertation]. University of Oregon; 2018. [cited 2020 Dec 05]. Available from: http://hdl.handle.net/1794/23913.

Council of Science Editors:

Gedeon K. Kazhdan-Lusztig Polynomials of Matroids and Their Roots. [Doctoral Dissertation]. University of Oregon; 2018. Available from: http://hdl.handle.net/1794/23913

University of North Texas

2.
Alhaddad, Shemsi I.
Generic Algebras and *Kazhdan*-*Lusztig* *Theory* for Monomial Groups.

Degree: 2006, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc5235/

The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials.
*Advisors/Committee Members: Douglass, Matthew, Bator, Elizabeth M., Brozovic, Douglas, Shepler, Anne, Thiem, Nathanial.*

Subjects/Keywords: Hecke algebras.; Kazhdan-Lusztig polynomials.; Coxeter groups.; Hecke algebra; Kazhdan-Lusztig theory; monomial groups

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Alhaddad, S. I. (2006). Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc5235/

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Alhaddad, Shemsi I. “Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups.” 2006. Thesis, University of North Texas. Accessed December 05, 2020. https://digital.library.unt.edu/ark:/67531/metadc5235/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Alhaddad, Shemsi I. “Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups.” 2006. Web. 05 Dec 2020.

Vancouver:

Alhaddad SI. Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups. [Internet] [Thesis]. University of North Texas; 2006. [cited 2020 Dec 05]. Available from: https://digital.library.unt.edu/ark:/67531/metadc5235/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Alhaddad SI. Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups. [Thesis]. University of North Texas; 2006. Available from: https://digital.library.unt.edu/ark:/67531/metadc5235/

Not specified: Masters Thesis or Doctoral Dissertation

3. Xu, Tianyuan. On the Subregular J-ring of Coxeter Systems.

Degree: PhD, Department of Mathematics, 2017, University of Oregon

URL: http://hdl.handle.net/1794/22741

Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke
algebra associated to (W, S) via Kazhdan-Lusztig polynomials by Lusztig. We study
a subalgebra J_C of J corresponding to the subregular cell C of W . We prove a
factorization theorem that allows us to compute products in J_C without inputs
from Kazhdan-Lusztig theory, then discuss two applications of this result. First, we
describe J_C in terms of the Coxeter diagram of (W, S) in the case (W, S) is simply-
laced, and deduce more connections between the diagram and J_C in some other
cases. Second, we prove that for certain specific Coxeter systems, some subalgebras
of J_C are free fusion rings, thereby connecting the algebras to compact quantum
groups arising in operator algebra theory.
*Advisors/Committee Members: Ostrik, Victor (advisor).*

Subjects/Keywords: Coxeter groups; Fusion categories; Hecke algebras; Kazhdan-Lusztig theory; Partition quantum groups; Tensor categories

…8
8
12
15
17
III. HECKE ALGEBRAS
25
3.1. Hecke Algebras and Their *Kazhdan*-*Lusztig* Bases… …25
3.2. *Kazhdan*-*Lusztig* Cells . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3. The… …*Kazhdan*-*Lusztig*
polynomials, *Lusztig* constructed the asymptotic Hecke algebra J of (W, S… …constants of the *Kazhdan*-*Lusztig* basis of the Hecke algebra H of
(W, S). The non… …*Kazhdan*-*Lusztig* cell E
of W , the subgroup
JE = ⊕w∈E Ztw
of J is a subalgebra of J and also a…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Xu, T. (2017). On the Subregular J-ring of Coxeter Systems. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/22741

Chicago Manual of Style (16^{th} Edition):

Xu, Tianyuan. “On the Subregular J-ring of Coxeter Systems.” 2017. Doctoral Dissertation, University of Oregon. Accessed December 05, 2020. http://hdl.handle.net/1794/22741.

MLA Handbook (7^{th} Edition):

Xu, Tianyuan. “On the Subregular J-ring of Coxeter Systems.” 2017. Web. 05 Dec 2020.

Vancouver:

Xu T. On the Subregular J-ring of Coxeter Systems. [Internet] [Doctoral dissertation]. University of Oregon; 2017. [cited 2020 Dec 05]. Available from: http://hdl.handle.net/1794/22741.

Council of Science Editors:

Xu T. On the Subregular J-ring of Coxeter Systems. [Doctoral Dissertation]. University of Oregon; 2017. Available from: http://hdl.handle.net/1794/22741