This is the website for the Algebraic and Enumerative Combinatorics Seminar at the University of Waterloo. We view algebraic combinatorics broadly, explictly including algebraic enumeration and related asymptotic and bijective combinatorics as well as algebraic combinatorics as it appears in pure algebra and in applications outside mathematics.
We begin with a pre-seminar which is designed to get participants up to speed on useful and interesting background for the talk. It will be at the level for beginning grad students. Then there will be a short coffee break followed by the seminar itself.
Our audience consists principally of combinatorics faculty and grad students. Seminar talks are 50 minutes with questions following.
If you are speaking, we need your abstract at least a week in advance so it can make the deadline for the Friday math faculty seminar mailing.
Spring 2026
Usual location and time: 1:30 pre-seminar, 2:30 seminar, both in MC 5479.
Tuesday, May 5 in MC 6029: Siddhartha Sahi, Hypergeometric functions of matrix argument Click here for abstract
In a widely circulated manuscript from the 1990s, I.G. Macdonald introduced certain higher-rank analogs of the classical hypergeometric functions $_pF_q$, which are expressed as explicit series in Jack and Macdonald polynomials in one and two sets of variables. For special choices of parameters, these series reduce to the hypergeometric functions of matrix argument introduced earlier by C. Herz and A.T. James, which have numerous applications in number theory, multivariate statistics, signal processing, and random matrix theory.
The classical hypergeometric functions are solutions to the hypergeometric differential equation. Macdonald raised the problem of providing an analogous characterization for higher-rank functions by means of differential equations. Over the years, this problem was solved for a small number of cases where p and q are at most 3. However, as the operators become increasingly complicated, the general problem remained open for 40 years. In this talk, we will present a complete solution. This is joint work with Hong Chen.
May 21: Kaveh Mousavand, Left modularity and extremality of some (finite and infinite) lattices via representation theory Click here for abstract
Motivated by the representation theory of finite-dimensional algebras, we recently investigated the notions of left modularity and extremality in (completely) semidistributive lattices. For lattices of torsion classes, we obtain a simultaneous characterization of left modularity and extremality in terms of the behavior of certain indecomposable modules, called bricks. Our results extend the classical theory beyond the realm of finite lattices, while remaining within the framework of (completely) semidistributive lattices. Time permitting, I will also discuss extensions of these results to arbitrary infinite lattices that are completely semidistributive and weakly atomic. This talk is based on recent joint work with Sota Asai, Osamu Iyama, and Charles Paquette.
During the pre-seminar, after reviewing some basic notions from the representation theory of finite-dimensional algebras through the language of quiver representations, I will recall the classical notion of directedness and compare it with our new generalization, called brick-directedness. I will then discuss some basic properties of torsion classes and compare the notions of splitting and brick-splitting torsion pairs. No prior background in the representation theory of algebras will be assumed.
May 28: Sergio Alejandro Fernandez de Soto Guerrero, New combinatorial possibilities to describe (quotients of) positroids Click here for abstract
Positroids were introduced by Postnikov in 2006 as a special class of matroids with nice combinatorial properties. Since 2008, starting with Suho Ho, several attempts have been made to describe the poset of quotients for this class of matroids in a combinatorial way. However, these descriptions are incomplete and always come from the same perspective. That is why we will explore new combinatorial objects and the context in which they arise (magic, polytopes, and antisymmetric algebras) to see if it is possible to describe this poset.
June 4 in MC 6460: Theodore Morrison, Satisfiability thresholds of linear equations over a commutative ring Click here for abstract
The satisfiability threshold of a random constraint satisfaction problem (CSP) is the density of constraints at which a random CSP instance transitions from being satisfiable to unsatisfiable with high probability. Much of the research on well known CSPs, including the $k$-SAT problem, $k$-XORSAT problem, hypergraph colouring, and systems of linear equations, has focused on determining satisfiability thresholds.
In this talk we consider systems of linear equations over finite commutative rings as CSPs, and build on the work of Ayre, Coja-Oghlan, Gao, and Müller, who determined the satisfiability threshold for random linear equations over a finite field. We determine when the satisfiability threshold is linear in the number of variables, and show that any linear threshold over a principal ideal ring coincides with the (unique) linear threshold over fields. We also determine the satisfiability threshold for some examples of non-principal ideal rings.
This is joint work with Jane Gao.
June 8 (Monday) and 9 (Tuesday) in MC 6029: Watch party for AlCoVE 2026, from 10am to 6pm.
June 11 in MC 6460: Kevin Purbhoo, The hook length formula massacree Click here for abstract
Around 1900 Young and Frobenius (independently, and through very different techniques) obtained a formula for the dimensions of the irreducible representations of the symmetric group. Some 53 years later, Frame, Robinson and Thrall noticed that the Young-Frobenius formula simplified into the now famous hook length formula. Nowadays there are many proofs, but the hook length formula remains something of a mystery, as if some deeper understanding lies just out of reach. One aspect of this mystery is that none of the proofs seem to indicate how one might come up with the formula in the first place, other than just guessing.
I will attempt to answer that question. It is an improbable tale that meanders through scenes of Young symmetrizers, Schur-Weyl duality, Weyl algebras, elementary combinatorics, and Plücker relations. All because Google's AI gave me a very obviously wrong answer when I was trying to find out the square of a Young symmetrizer.
June 18 in MC 6460: Scott Neville
June 25 in MC 6460: Mike Cummings
July 2: Jerónimo Valencia-Porras, Type C multiline queues and the open-boundary TASEP