# Algebra Seminar

This seminar is held on Tuesdays at 4pm in 210 Deady.

### Fall Quarter, 2018

- October 2,
**Justin Sawon** (UNC)

Lagrangian fibrations by Prym varieties
**Abstract**: The Hitchin systems are Lagrangian fibrations on moduli spaces of Higgs bundles. Their compact counterparts are Lagrangian fibrations on compact holomorphic symplectic manifolds, such as the integrable systems of Beauville-Mukai, Debarre, Arbarello-Ferretti-SaccĂ , Markushevich-Tikhomirov, and Matteini. The GL-Hitchin system and the Beauville-Mukai system are both fibrations by Jacobians of curves. Thus they are isomorphic to their own dual fibrations. Moreover, Donagi-Ein-Lazarsfeld showed that the Beauville-Mukai system can be degenerated to a compactification of the GL-Hitchin system. The other Hitchin systems and the other compact integrable systems mentioned above are fibrations by Prym varieties. In this talk, we explore these Prym fibrations and the relations between them. In particular, we describe degenerations of some compact examples to compactifications of the Sp-Hitchin systems. We also describe dual fibrations of certain Lagrangian fibrations by Prym varieties.

- October 8,
**Ivan Loseu** (Northeastern)

**Special Time:** 3pm in 210 Deady

Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras
**Abstract**: We are interested in two numerical invariants of primitive ideals (=the annihilators of irreducible modules) in the universal enveloping algebras.

The first invariant, the Goldie rank, is classical, it measures how many zero divisors the quotient by the ideal has. The second invariant, the dimension of the finite dimensional irreducible representation of a W-algebra corresponding to the primitive ideal, is more recent. It is expected (and is known in many important cases) that the latter invariant is computable via a Kazhdan-Lusztig type formula. Also the two invariants are known to be very closely related thanks to the work of Premet and the speaker. In this talk we discuss the joint work of Ivan Panin and the speaker, arXiv:1802.05651, where we find a lower bound on the ratio of the dimension of the W-algebra module by the Goldie rank. This lower bound is the index of a suitable equivariant Azumaya algebra on a cover of the nilpotent orbit corresponding to the ideal. We explain how to compute the index in the elementary representation theoretic terms. We hope that our lower bound is exact at least for the classical Lie algebras. No prior knowledge of primitive ideals, W-algebras or Azumaya algebras is required.

- October 9,
**Rahul Singh** (Northeastern)

The Conormal Variety of a Schubert Variety
**Abstract**: Let N be the conormal variety of a Schubert variety X.

In this talk, we discuss the geometry of N in two cases, when X is cominuscule, and when X is a divisor.

In particular, we present a resolution of singularities and a system of defining equations for N, and also describe certain cases when N is normal, Cohen-Macaulay, and Frobenius split.

Time permitting, we will also illustrate the close relationship between N and orbital varieties, and discuss consequences of the above constructions for orbital varieties.

- October 16,
**Dick Hain** (Duke)

Hodge Theory and the Goldman–Turaev Lie Bialgebra
**Abstract**: The Goldman–Turaev Lie bialgebra of a compact oriented surface X is the free Z-module generated by the conjugacy classes in the fundamental group of X — equivalently, by the closed geodesics in X when X is hyperbolic. The bracket, constructed by Goldman in the 1980s, is related to the Poisson geometry of the moduli space of GL_N bundles over X. The cobracket was defined by Turaev (1970s and 1990s). In its most refined version, it requires a framing of X.

The Kashiwara–Vergne problem is, in some sense, a vast generalization of the Baker-Campbell-Hausdorff formula. Solutions in all genera were recently constructed by Alexeev, Kawazumi, Kuno and Naef using classical and elliptic associators. Solutions of the KV problem in each genus form a torsor under a group, referred to as the Galois group of the problem. Determining this group is an important problem. It should be related to relative completions of mapping class groups and to what is called the “motivic Lie algebra” — the free Lie algebra generated by the set {s_3, s_5, s_7, s_9, ….} indexed by odd integers > 1.

In this talk, I will construct the Goldman bracket and Turaev cobracket, using elementary surface topology, and then recall the KV problem. I will then explain how one can use Hodge theory to reprove the existence of solutions of the KV problem for all framed surfaces. This approach yields a large and relatively explicit Galois group which is closely related to the motivic Lie algebra and to relative completions of mapping class groups.

- October 23,
**Jen Berg** (Rice)

Odd ordered transcendental obstructions to the Hasse principle on K3 surfaces
**Abstract**: K3 surfaces are 2-dimensional analogues of elliptic curves, but lack a group structure. Moreover, they need not have rational points. However, in 2009 Skorobogatov conjectured that the Brauer group (a torsion abelian group which encodes reciprocity laws) should account for all failures of the local-to-global principle for rational points on K3 surfaces. In this talk I will describe the geometric origin of certain 3-torsion classes in the Brauer group of a K3 surface. We utilize this geometric description to show that these classes can in fact obstruct the existence of rational points. This is joint work with Tony

Varilly-Alvarado.

- November 6,
**Nicolle S Gonzalez** (U. Southern California)

Categorical Bernstein Operators and the Boson-Fermion correspondence
**Abstract**: Bernstein operators are vertex operators that create and annhilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov’s Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence.

- November 13,
**Victor Ostrik** (UO)

Negligible and non-negligible tilting modules
**Abstract**: Tilting modules form an interesting family of representations of a reductive group over a field of positive characteristic p. One big open problem is compute their characters and, in particular, dimensions. In this talk we discuss how to compute their dimensions modulo p. This is a joint work with Pavel Etingof.

- November 20,
**Zach Hamaker** (Michigan)

Involution Coxeter combinatorics
**Abstract**: The combinatorics of Coxeter groups has long been a rich area of study with important applications to representation theory and geometry. Many of the key ideas in this realm have natural analogues when we restrict our attention to involutions in Coxeter groups. Based on pioneering work of Richardson and Springer on K-orbits, I will survey many results translated through the lens of involution words, which are the natural analog of reduced words for involutions. This will include new formulas for equivariant cohomology representatives of some spherical varieties and several combinatorial results awaiting geometric or representation-theoretic explanations. These results only scratch the surface, and many open problems remain! This is joint work with Eric Marberg and Brendan Pawlowski.

- November 27, Algebra classes organizational meeting
- November 29,
**Lucia Morotti** (Leibniz Universitat Hannover)

**Special Time:** 4pm in 210 Deady

Irreducible tensor products of representations of symmetric and related groups
**Abstract**: Given two irreducible representations of dimension greater than 1, their tensor product is in general not irreducible. For example, for symmetric groups such irreducible tensor products are only possible in characteristic 2. For alternating groups and for covering groups of symmetric and alternating groups however there are examples of such irreducible tensor products in arbitrary characteristic.

In this talk I will present a (partial) characterisation of irreducible tensor products of symmetric, alternating groups and their covering groups.

- December 19,
**Amit Hazi** (University of Leeds)

**Previous years**: 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005