—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Spring 2015

Wednesdays 3:40-4:30 BVM hall 11th floor seminar room

Organizers:  Aaron Lauve, Peter Tingley

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)

Parking is available on-campus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.

Given an arbitrary field F of characteristic 0, we study Lie bialgebra structures on sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure, the classical double is isomorphic to the tensor product of sl(n,F) with either F[e]/e^2, F+F, or a quadratic field extension of F. In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n,F). In the second and third cases, a Belavin-Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The Belavin-Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed. For the Cremmer-Gervais r-matrix in sl(3), we also construct a natural map of sets between the total Belavin-Drinfeld twisted cohomology set and the Brauer group of the field F.

Feb 11: Peter Tingley (Loyola), Crystals, PBW bases, and related combinatorics.

Kashiwara's crystals are combinatorial objects used to study complex simple Lie groups and their highest weight representations. A crystal consists of an underlying set, which roughly indexes a basis for a representation, along with combinatorial operations related to the Chevalley generators. Here we are interested in types A and D, where the crystals are realized using Young tableaux and Kashiwara-Nakashima tableaux respectively. However the combinatorics is not always enough, and one would like to understand an actual basis corresponding to the crystal. This can be done using Lusztig's theory of PBW bases, and so there is a unique structure-preserving bijection between the PBW basis and the combinatorially constructed crystal. We describe that bijection explicitly. In type A the answer was to some extent known to experts, but in type D we believe it is new. I will explain as much of this as I can without assuming any prior knowledge of crystals or PBW bases, mainly through examples. This is joint work with two Loyola students (John Claxton and Adam Shultze), and also with Ben Salisbury.

Feb 25: Joshua Hallam (Michigan State), Increasing Forests

Let G be a graph with vertex set {1,2,...,n}. We say a subtree of G is increasing if the labels along any path starting at the minimum vertex increase. Moreover, we call a subforest of G increasing if each of its components is an increasing tree. We will show that the generating function for increasing forests of graph always factors with nonnegative integer roots. We also give a characterization of when the generating function is equal to the graph's chromatic polynomial. We finish by generalizing this work to simplicial complexes.
This joint work with Jeremy Martin and Bruce Sagan.

March 11: Shamil Shakirov (UC Berkeley), On Macdonald deformation of Reshetikhin-Turaev TQFT

The central quantity in the Reshetikhin-Turaev approach to TQFT is the 6j-symbol of the quantum group U_q( sl_N ). This quantity satisfies several important algebraic identities, such as the Pentagon identity or the Yang-Baxter equation, which stand behind the topological invariance of Reshetikhin-Turaev state sums. Most of these identities do not admit local deformations, except for one, which is sometimes called the Tetrahedral Duality. We will discuss this deformation, its relation to Macdonald theory, and conjectural relation to categorification of the Reshetikhin-Turaev TQFT.

April 1: Peter Tingley (Loyola), Gelfand-Tsetlin bases and PBW bases.

Gelfand-Tsetlin bases are part of the history leading to the modern theory of canonical bases. The idea is that, if you have a family of nested semi-simple algebras A_1,...A_n such that, for any irreducible representation of any A_k, its restriction to A_{k-1} breaks up as a multiplicity-free sum of irreducible representations of A_{n-1}, then there is a unique (up to rescaling) basis for any irreducible representation of A_n with the property that it respects the decomposition into representations of any A_k.

Probably the most important example of this phenomenon is the algebras sl(n) with the usual embeddings. I'll discuss that case in detail, explaining how it related to various combinatorial and algebraic things we've seen in the seminar, specifically Young tableaux and PBW bases.

I'll then talk a bit about the other classical Lie algebras. The restrictions of so(n) to so(n-1) are multiplicity free, so that case basically works, but there are complications. For sp(n) the restrictions are not multiplicity free, but some things still work.

April 8: Andrew Carroll (dePaul), Moduli spaces for Schur-tame algebras

Over the last 40 years there have been a number of attempts to characterize the complexity of the category of modules over a (finite-dimensional associative) algebra by means of invariant theory. More precisely, we seek to determine whether an algebra is tame or wild by considering rings of invariant functions, and the geometry of their moduli spaces. Such characterizations have been demonstrated in the context of path algebras of quivers, but attempts to generalize to bound quiver algebras have met resistance. I will describe these efforts, their success, their failures, and attempts to address these failures by defining the notion of Schur representation type. This is ongoing work with Calin Chindris.

April 22: Stefan Catoiu (DePaul), Generalized Trigonometric Hopf Algebras and Fermat's Equation

It is well-known that the Pythagorean equation x^2+y^2=z^2 has infinitely many integer solutions. Fermat's Last Theorem says that the equation x^n+y^n=z^n has no non-trivial solutions when n>2. Therefore, despite its similar look, this is not an appropriate generalization of the Pythagorean equation. Using Hopf algebras and generalized trigonometry we produce the appropriate degree n generalization that has infinitely many solutions. These are parametrized in the same way as the ones for the Pythagorean equation.

April 29: Christophe Hohlweg (U. du Quebec, Montreal), Artin-Tits Braid groups, low elements and weak order in Coxeter groups

In this talk we will explain that the question of solving the conjugacy problem in the context of a general Artin-Tits Braid group reveals strong connections between the weak order of a Coxeter system (W,S), inversion sets of elements of W and small roots. Small roots are the main ingredient introduced by Brink and Howlett in order to build a 'canonical automaton' that recognizes the language of reduced words of elements of W over S. From small roots and inversion sets, we define a new finite class of elements in W called 'low elements.' These low elements are the key to prove that the smallest subset of W containing S, closed under join (for the right weak order) and suffix is finite, and by ricochet that finitely generated Artin-Tits groups have a finite Garside family. Low elements seem rich in further applications in the study of infinite Coxeter groups, which will discuss if time allows. This is based on joint works with Patrick Dehornoy and Matthew Dyer.