Student Algebra Seminar, 2019 Semester 1

Speaker: Joel Gibson

The algebra of symmetric functions is a Hopf algebra which arises naturally from the study of symmetric polynomials. In this talk, I motivate symmetric polynomials as characters of GL_n modules, introduce much of the standard language for talking about symmetric functions, and also identify the algebra of symmetric functions as a kind of "decategorification" of symmetric group representations with the induction product.

References:

• The classic reference for symmetric functions is Chapter I of Symmetric functions and Hall polynomials by Macdonald.
• A very combinatorial approach to Schur functions is developed throughout Young Tableaux by Fulton.
• An amazing concrete development of the algebra of symmetric functions is given in Chapters 1 and 2 of Hopf Algebras in Combinatorics by Grinberg and Reiner.

Speaker: Gaston Burrull

The periodic Hecke module over an affine Hecke algebra was introduced by Lusztig in 1980. It has a canonical basis and corresponding polynomials. These polynomials are related to the Kazhdan-Lusztig polynomials, this relation proves a periodicity property for the last ones.

Let \(V\supset R\supset R^+\supset \Delta\) be a vector space over the reals, a root system, a system of positive roots and the corresponding set of simple roots. Let \(\mathcal{W}\) be the affine Weyl group, \(\mathcal{H}\) be the corresponding Hecke algebra and \(X\) the corresponding set of alcoves. In this talk, I introduce the periodic Hecke module \(\mathcal{P}\) which is a right $\mathcal{H}$-module. I introduce the $\mathcal{H}$-submodule \(\mathcal{P}^0\). A bar involution of \(\mathcal{P}^0\) is introduced. The canonical basis is characterized using the bar operator on \(\mathcal{P}^0\) by a method similar to the one of the ``regular" canonical basis. This canonical basis is indexed by the set of alcoves \(X\). I will focus on baby examples \(\tilde{A}_1\) and \(\tilde{A}_2\). If the time permits I will introduce the spherical (resp. anti-spherical) Hecke modules \(\mathcal{M}\) (resp. \(\mathcal{N}\)). I will state the stability property of the spherical polynomials \(m_{A,B}\) — where \(A,B\) are two alcoves in \(X\) — related to the spherical Hecke module.

References:

• Lusztig, G. (1980). "Hecke algebras and Jantzen's generic decomposition patterns." Advances in Mathematics, 37(2), 121-164. Lusztig here introduces for the first time the periodic Hecke module and provides very useful ways to compute its canonical basis.
• W. Soergel (1997). "Kazhdan-Lusztig polynomials and a combinatoric for tilting modules." Representation Theory of the American Mathematical Society, 1(6), 83-114. This article gives a self-contained treatment of the theory of Kazhdan-Lusztig polynomials with new proofs and notation, this includes the previously mentioned work of Lusztig and the works of Andersen, Kato, Kaneda, and Deodhar in the '80s about the spherical and anti-spherical Hecke module and the relations of them to the periodic and regular modules — This talk follows fairly the notation used in this paper.

Speaker: Joshua Ciappara

We define the automorphism class group of a category, calculate it in a few well-known cases, then discuss two examples where the group isn't known.

References:

• A starter reference for automorphism class groups.
• Questions on MathOverflow about rigidity in the category of schemes and the category of fields.

Speaker: Joseph Baine

Coxeter groups are ubiquitous throughout mathematics, playing a fundamental role in many algebraic, topological, and combinatorial theories. This talk will introduce Coxeter groups and many problems that naturally arise when studying them and their traditional solutions. We will then define Mozes’s Numbers Game and show how many of these natural questions have simple interpretations and solutions in terms of the game.

Speaker: Yeeka Yau

The concept of an automatic group was first introduced in 1986 by William Thurston after he noticed that some geometric properties of the Cayley graph of cocompact discrete hyperbolic groups (proved by J.W. Cannon) could be restated in terms of finite state automata. It turns out that the connections between group theory and automata theory are abundant and that understanding the "automatic structure" of groups tells us some very nice things about some geometric and combinatorial properties of the group. In this talk, we give an introduction to the topic and derive one particularly nice property of automatic groups: that the word problem for these groups can be solved in quadratic time.

References:

• EPSTEIN, D.B, CANNON, J.W, HOLT, D.F ET AL, Word Processing in Groups, James and Bartlett Publishers (1992)
• D.F. Holt, S. Rees and C.E. Rover, Groups, Languages and Automata London Mathematical Society Student Texts, 88, Cambridge Uni. Press, Cambridge, 2017.

Speaker: Giulian Wiggins

We show how to produce convolution algebras from certain fibre products, and show how simple modules of these algebras are classified using this geometric information. This has applications to the classification of modules for Weyl groups and affine Hecke algebras. As an example we will outline how the Weyl group of a semisimple Lie group G can be recovered from a convolution algebra arising from the Steinberg variety.