Ottawa-Carleton Joint Algebra Seminar

Deligne’s category Rep\((S_t)\) is a linear monoidal category that interpolates between the representation theory of the symmetric groups. In particular, it is defined even when \(t\) is not a natural number. However, \(t\) is fixed. On the other hand, the Heisenberg category encapsulates the representation theory of all the symmetric groups \(S_n\) (for all natural numbers \(n\)) and the canonical induction and restriction functors between their module categories. It categorifies the Heisenberg algebra and has some deep connections to Hilbert schemes and quiver varieties. In this talk we will describe an embedding of Deligne’s category into the Heisenberg category. We will discuss some nice properties of this embedding and mention how it leads to some natural further directions for research. This is joint work with Samuel Nyobe Likeng and Christopher Ryba.

Given a multiplicity-free action \(V\) of a simple Lie (super)algebra \(\mathfrak{g}\), one can define a distinguished basis for the algebra of \(\mathfrak{g}\)-invariant differential operators on \(V\). The problem of computing the eigenvalues of this basis was first proposed by B. Kostant, and is closely related to the theory of interpolation Jack polynomials and their generalizations. In this talk, we concentrate on an example of similar spirit, associated to the orthosymplectic Lie superalgebras, and compute two formulas for the eigenvalues of the corresponding Capelli operators. Along the way, the Dougall-Ramanujan identity appears in an unexpected fashion. This talk is based on a joint work with Siddhartha Sahi and Vera Serganova.

In representation theory, several families of linear monoidal categories admitting diagrammatic presentations by generators and relations emerged. One of the important questions for such a linear category is to compute bases for every set of morphisms, and to that purpose various computational methods are developed. In this talk, we discuss how these questions can be approached using rewriting theory. We introduce the fundamental properties of termination and confluence for rewriting systems on words in monoids, and explain how one can compute bases in linear categories from these two properties. We illustrate these constructions on a family of algebras defined by Khovanov, Lauda and Rouquier appearing in a process of categorification of a quantum group associated with a symmetrizable Kac-Moody algebra.

The homogeneous coordinate ring \(\mathbb{C}[\text{Gr}(k,n)]\) of the Grassmannian of \(k\)-dimensional subspaces in \(n\) space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmannian cluster category \(C(k,n)\), as shown by Jensen-King-Su in 2016. In this talk we will construct certain combinatorial objects, so-called Pluecker friezes, from the categories \(C(k,n)\). If time permits, we will moreover comment on the infinite construction of the cluster category \(C(2,\infty)\) and certain subcategories of it.

This is a joint work with G.Garkusha. In the talk I will discuss the construction of fibrant replacements for spectra consisting of Thom spaces (suspension spectra of varieties and algebraic cobordism MGL being the motivating examples) that uses the theory of framed correspondences. As a consequence we get a description of the infinite loop space of MGL in terms of Hilbert schemes.

Can one hear the shape of a drum? In other words, to which extent is a Riemannian manifold determined by the spectrum of its Laplace operator? Prasad and Rapinchuk investigated the case of locally symmetric spaces which relates to subtori of algebraic groups and then with non-abelian Galois cohomology. We shall survey results on this algebraic theme by the preceding authors, Beli-G.-Lee, Chernousov-Rapinchuk-Rapinchuk and others.

A Helly graph is a graph in which the metric balls form a Helly family: any pairwise intersecting collection of balls has nonempty total intersection. A Helly group is a group that acts properly and cocompactly on a Helly graph. Helly groups simultaneously generalize hyperbolic, cocompactly cubulated and C(4)-T(4) graphical small cancellation groups while maintaining nice properties, such as biautomaticity. I will show that if a crystallographic group is Helly then its point group preserves an \(L^{\infty}\) metric on \(\mathbb{R}^n\). Thus we will obtain some new nonexamples of Helly groups, including the 3-3-3 Coxeter group, which is a systolic group. This answers a question posed by Chepoi during the recent Simons Semester on Geometric and Analytic Group Theory in Warsaw.

Fix a field \(k\) of characteristic zero. If \(a_1,\ldots,a_n\) are positive integers, the integral domain \(k[X_1,\ldots,X_n] / \langle X_1^{a_1} + X_2^{a_2} + ... X_n^{a_n} \rangle\) is called a Pham-Brieskorn ring. It is conjectured that if \(a_i > 1\) for all \(i\), and \(a_i = 2\) for at most one \(i\), then \(B\) is rigid. (A ring \(B\) is said to be rigid if the only locally nilpotent derivation \(D \colon B \to B\) is the zero derivation.) We give partial results towards the conjecture.

In this talk I will discuss one approach to constructing explicit particular subcategories of motives and give examples coming from curves/abelian varieties.

Diophantine geometry seeks to link properties of rational solutions of a set of equations to the geometric properties of the variety they define. One of the main tools in Diophantine geometry is Diophantine approximation — results bounding how the complexity of a rational point must grow as it approaches a subvariety.
In this talk I will discuss a somewhat recent new measure of the positivity of an ample line bundle along a subscheme, and show how its formal properties give a simple proof of a theorem of Ru-Vojta on Diophantine approximation.

If \(k\) is a field and \(S\) is an inverse semigroup, one may consider the algebra \(kS\) spanned by a copy of \(S\). In 2009 Steinberg realized this algebra as a convolution algebra for a certain topological groupoid \(G\) associated to \(S\). These groupoid convolution algebras have been since used to study Leavitt path algebras, graph \(C^*\)-algebras, and their generalizations. We consider the following question: when are these algebras simple? We come up with necessary and sufficient conditions for simplicity, and show that there is a surprise in the non-Hausdorff case—simplicity can depend on the field \(k\), not just the topological properties of \(G\).