Joint UOttawa/Carleton
Algebra Seminar
Fall 2024
Tuesdays, 4:00-5:00pm, STEM 664
September 17 (UOttawa)
Speaker: Yaolong Shen (Ottawa)
September 24 (UOttawa)
Speaker: Alistair Savage (Ottawa)
Title: The spin Brauer category
Abstract: The Brauer category is a diagrammatic monoidal category describing the representation theory of the orthogonal and symplectic groups. Its endomorphism algebras are Brauer algebras, which replace the group algebra of the symmetric group in the orthogonal and symplectic analogues of Schur-Weyl duality. However, the Brauer category is missing one important piece of the picture—the spin representation. We will introduce a larger category, the spin Brauer category, that remedies this deficiency. This is joint work with Peter McNamara.
October 1 (UOttawa)
Speaker: Rui Xiong (Ottawa)
Title: Fun with Positroid Varieties
Abstract: In this talk, we provide an introduction to the combinatorial structures associated with positroid varieties. In the first part, we concentrate on combinatorics in type A, such as k-Bruhat order, affine bounded permutations, and Grassmannian necklaces. We will discuss the theorem concerning the cohomology classes of positroid varieties. In the second part, we offer a more general discussion of positroid varieties over cominuscule varieties. We will briefly introduce the affine Weyl group and explain its connection to positroid varieties. If time permits, we will also discuss our recent work in this direction.
October 29 (UOttawa, CRM Colloquium)
Speaker: Marcelo Aguiar (Cornell University)
Title: Hopf-Lie theory relative to a hyperplane arrangement
The talk is based on recent and ongoing work with Swapneel Mahajan in which we introduce and develop a theory of Hopf monoids relative to a real hyperplane arrangement. Many concepts from the classical theory of connected Hopf algebras extend to this level, including structure results such as the Cartan-Milnor-Moore theorem. The whole discussion is framed in terms of the Tits monoid of the arrangement, a concept on which the talk will focus.
November 5 (UOttawa)
Speaker: Changlong Zhong (SUNY at Albany)
Title and abstract: Elliptic cohomology and the Fourier-Mukai transform
Recently there has been rapid development in using equivariant elliptic cohomology in the context of enumerative geometry, representation theory, and mathematical physics. In particular, the elliptic stable envelop for symplectic varieties is defined, and a 3d mirror symmetry statement is given by Okounkov. For cotangent bundle of flag varieties, the stable envelop is constructed by Rimanyi-Weber using algebraic geometry, and they are transformed by the newly defined elliptic Demazure-Lusztig (DL) operators. These operators can be thought of as rational sections of certain vector bundle over $A\times A^\vee$, where $A$ is (the spectrum of) the equivariant elliptic cohomology of a point and $A^\vee$ is its dual abelian variety. Classically there is an equivalence between derived category of $A$ and $A^\vee$, defined by the Poincare line bundle, called the Fourier Mukai transform. The module category over the elliptic affine Hecke algebra, defined by Ginzburg-Kapranov-Vasserot, is a subcategory. In this talk I will define an equivalence between this module category and that for the Langlands dual system. This functor is constructed by using algebra over $A\times A^\vee$ determined by the elliptic DL operators. This is joint work with G. Zhao.
November 26 (UOttawa)
Speaker: Henrique Rocha (Carleton University)
Title and abstract: Representations of Lie algebras of vector fields on algebraic
varieties
For a smooth algebraic variety X, we will present our results
on the representation theory of finitely generated modules over the ring
of function of X that has a compatible action of the Lie algebra V of
polynomials vector fields on X. In particular, we show that the
associated representation of V can be sheafified and is a differential
operator of order depending on the rank of the module. The order of the
differential operator provides a natural measure of the complexity of
the representation, with the simplest case being that of D-modules.
Finally, we talk about how to apply these results on gauge modules.