Algebra and Combinatorics Seminar, Spring 2019

Wednesdays, 3:45–4:35. IES111

Webpages from previous semesters

Organizers:  Emily Peters, Peter Tingley

 January No talks (please attend special colloquia! February 6 No talk 13 No talk 20 Organizational meeting 27 Peter Tingley Quiver varieties and root multiplicities for symmetric Kac-Moody algebras March 6 No talk (spring break) 13 Nicolas Mayers (Lehigh University) Calculating the Index of Poset Algebras 20 Marius Radulescu Vector Bundles on Elliptic Curves 27 April 3 Daniele Rosso (Indiana University Northwest) Quantum Affine Wreath Algebras 10 Aaron Lauve (Loyola) Nakayama's Conjecture for Multiplicity Free Families (POSTPONED in favor of URES Preview) 17 Christopher Drupieski (DePaul University) Some very simple, very explicit examples related to detecting projectivity of modules 24 Peter or Tony?

Directions

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.

Abstracts

February 27: Peter tingley. Quiver varieties and root multiplicities for symmetric Kac-Moody algebras

We discuss combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from a realization of the infinity crystal using quiver varieties. The framework is quite general, but we only work out specifics for one special case. We conjecture that our bound is quite tight, and give both computational evidence and heuristic justification for this conjecture, but unfortunately not a proof.

March 13: Nicholas Mayers. Calculating the Index of Poset Algebras

The index of a Lie algebra is an important algebraic invariant which is usually quite difficult to compute. Recently, there has been a push to find closed-form formulas for the index of families of Lie algebras which are combinatorially defined. Much of this work is related to what are known as Seaweed algebras which are subalgebras of sl(n) which can be parametrized by a pair of compositions of n. In this talk I will discuss some recent work related to finding combinatorial formulas for the index of a different family of Lie algebras which are defined via posets.

March 20: Marius Radulescu. Vector Bundles on Elliptic Curves

In his paper "Vector bundles over an elliptic curve," Michael Atiyah classified indecomposable vector bundles of arbitrary rank and degree on elliptic curves. The set of isomorphic classes of indecomposable vector bundles of fixed rank and degree is isomorphic with the Jacobian of the elliptic curve, which in turn is isomorphic with the curve itself. Based on Atiyah's paper, I will provide a detailed description of rank 2, 3 and 4 vector bundles on an elliptic curve.

April 3: Daniele Rosso. Quantum Affine Wreath Algebras

In this talk I will describe generalizations of Type A affine Hecke algebras, which we call quantum affine wreath algebras, that depend on the choice of a fixed Frobenius (or symmetric) algebra. We obtain results about the structures of these algebras, including a basis and the center, as well as their cyclotomic quotients. This is joint work with Alistair Savage.

April 10: Loyola Undergrads. URES Preview

Loyola's Weekend of Excellence, including the "Undergraduate Research and Engagement Symposium, is this weekend. Come see our students present their work then, or during this "dry run" during the usual Seminar time.

April 17: Christopher Drupieski. Some very simple, very explicit examples related to detecting projectivity of modules

In this talk I will present some very simple, very explicit examples related to detecting whether or not a module over some specific Hopf (super)algebras is or is not projective. I'll then discuss how these examples fit into a bigger picture related to cohomology and some associated algebraic varieties.

April ??: Aaron Lauve. Nakayama's Conjecture for Multiplicity Free Families

Given a partition $$\lambda$$ of $$n$$ and an integer $$p>1$$, we may try to remove, iteratively, all border strips of length $$p$$ from $$\lambda$$. The result is called it's $$p$$-core. In 1940, Nakayama made an interesting conjecture about the symmetric group ring $$FS_n$$ when $$F$$ has characteristic $$p$$: the $$p$$-cores index the blocks! (When $$p>n$$, this reduces to the classic result: partitions index the primitive central idempotents for $$\mathbb CS_n$$.) Nakayama's conjectue was proven by Brauer and Robinson only 7 years after it was announced; by now there many proofs available. In this talk, a mix of survey and speculation, we discuss Murphy's proof, which utilizes the infamous Jucys-Murphy elements. We explain how (we plan) to extend this to many more towers of algebras (what we call multiplicity-free families). This is joint work with Stephen Doty.