Our seminar views algebraic combinatorics broadly, explicitly 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.
Our local audience consists principally of combinatorics faculty and grad students. 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 our Friday faculty seminar mailing list.
If you would like to attend virtually, please email an organizer for the Zoom link and/or to be added to the weekly mailing list.
| Date |
Speaker |
Title |
| Jan 13 | Gabriel Frieden (UQAM) |
Crystal invariant theory and geometric RSK.
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| | The original problem of classical invariant theory was to describe the invariants of SLm acting on a polynomial ring in an m × n matrix of variables. One way to solve this problem is to consider the polynomial ring as a GLm × GLn representation, and decompose this representation into its irreducible components.
Berenstein and Kazhdan's theory of geometric crystals gives rise to two families of rational actions on the space of m × n complex matrices, which we view as "crystallized versions" of the usual GLm and GLn actions. We describe the invariants of these two families of actions. Our main tool is Noumi and Yamada's geometric lifting (or de-tropicalization) of the RSK correspondence, which is analogous to the above-mentioned irreducible decomposition in the classical setting. This is joint work with Ben Brubaker, Pasha Pylyavskyy, and Travis Scrimshaw.
I will not assume prior knowledge of crystals or geometric crystals; the basic building blocks of these theories will be introduced through examples. |
| Jan 20 | Cesar Cuenca (Harvard) | Weighted set-partitions in random matrix theory.
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| | We explain how perfect matchings and set-partitions (as well as their weighted versions) manifest in Law of Large Numbers from Random Matrix Theory. One of the examples discussed is a deformation of the operation of "free convolution" from the theory of Free Probability. At the end, we discuss open questions in probability motivated from the combinatorial point of view. No advanced knowledge of probability is needed to understand the theorems. The new results from this presentation are joint work with Florent Benaych-Georges and Vadim Gorin. |
| Jan 27 | Sean Griffin (UC Davis) | Springer fibers and the Delta Conjecture at t=0.
|
| | Springer fibers are a family of varieties that have remarkable connections to combinatorics and representation theory. Springer used them to geometrically construct all of the irreducible representations of the symmetric group (Specht modules). Moreover, they give a geometric meaning to Hall-Littlewood symmetric functions. In this talk, I will introduce a generalization of Springer fibers called Δ-Springer varieties, a special case of which gives a new geometric meaning to the expression in the Delta Conjecture at t=0. We’ll then use these varieties to geometrically construct induced versions of the Specht modules. This is joint work with Jake Levinson and Alexander Woo. |
| Feb 3 | Colin Defant (Princeton) | Semidistrim lattices
|
| | This talk will introduce semidistrim lattices, which generalize semidistributive lattices and trim lattices; these two families, in turn, generalize distributive lattices. We will discuss structural, topological, and dynamical properties of semidistrim lattices. In particular, we will see how one can define a certain bijective operator on a semidistrim lattice called rowmotion; this definition unifies the definition that Barnard gave for semidistributive lattices and the definition that Thomas and Williams gave for trim lattices. Somewhat surprisingly, rowmotion for semidistrim lattices is intimately connected with a noninvertible operator called pop-stack sorting, which can be defined for any lattice. This talk is based on joint work with Nathan Williams. |
| Feb 10 | Maciej Dołęga (IMPAN) | A curious identity between the orthogonal Brezin—Gross—Witten integral and Schur symmetric functions via b-deformed monotone Hurwitz numbers |
| |
This talk is intended for an algebraic combinatorial community and no prior knowledge is required. All the difficult words (Hurwitz numbers, KP hierarchy, HCIZ and BGW integrals, Jack symmetric functions, the b-conjecture) will be explained and gently introduced.
The monotone Hurwitz numbers can be understood combinatorially as cardinalities of monotone transitive walks in the Cayley graph of the symmetric group. They share many beautiful properties with ordinary Hurwitz numbers, but one of their most interesting properties is that their generating function coincides with the topological expansion of the celebrated Harish-Chandra—Itzykson—Zuber integral (in the case of double numbers) and the Brezin—Gross—Witten integral (in the case of single numbers). Using standard tools from algebraic combinatorics, one can express this generating function in terms of Schur symmetric functions and prove that it is a solution of the infinite system of partial differential equations called the KP hierarchy.
Inspired by a mysterious conjecture of Goulden and Jackson which connects generating function of Jack symmetric functions with enumeration of combinatorial maps, we define (following joint work with Chapuy) the generating function of b-deformed monotone Hurwitz numbers by replacing Schur symmetric functions by their one-parameter deformation — Jack symmetric functions. We show that it has an explicit combinatorial interpretation, which gives a topological expansion of the β-HCIZ integral. Finally, we show that for b=1 this generating function has a very interesting structure — it is a solution of the infinite system of Partial Differential Equations called the BKP hierarchy. We prove it by finding an explicit expansion in Schur symmetric functions, which surprisingly involves dimensions of the irreducible representations of the orthogonal group. As an application, we deduce an explicit Pfaffian formula for the Brezin—Gross—Witten integral over the orthogonal group. This is joint work with Valentin Bonzom and Guillaume Chapuy.
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| Feb 17 | | Social hour |
| Feb 24 | Reading week | no seminar |
| Mar 3 | Federico Castillo (U. Católica de Chile) |
Lineup polytopes and exclusion principles
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| |
The set of all possible spectra of 1-reduced density operators for systems of N particles on a d-dimensional Hilbert space is a polytope called hypersimplex and this is related to Pauli's exclusion principle. If the spectrum of the original density operators is fixed, the set of spectra (ordered decreasingly) of 1-reduced density operators is also a polytope. A theoretical description of this polytope using inequalities was provided by Klyachko in the early 2000's.
Adapting and enhancing tools from discrete geometry and combinatorics (symmetric polytopes, sweep polytopes, and the Gale order), we obtained such necessary inequalities explicitly, that are also valid for arbitrarily large N and d.
This approach leads to a new class of polytopes called lineup polytopes.
This is joint work with physicists Jean Philippe Labbe, Julia Liebert, Eva Philippe, Arnau Padrol, and Christian Schilling.
|
| Mar 10 | Josh Swanson (USC) | Type B q-Stirling numbers
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| |
The Stirling numbers of the first and second kind are classical objects in enumerative combinatorics which count the number of permutations or set partitions with a given number of blocks or cycles, respectively. Carlitz and Gould introduced q-analogues of the Stirling numbers of the first and second kinds, which have been further studied by many authors including Gessel, Garsia, Remmel, Wilson, and others, particularly in relation to certain statistics on ordered set partitions. Separately, type B analogues of the Stirling numbers of the first and second kind arise from the study of the intersection lattice of the type B hyperplane arrangement. We combine the two directions and introduce new type B q-analogues of the Stirling numbers of the first and second kinds. We will discuss connections between these new q-analogues and generating functions identities, inversion and major index-style statistics on type B set partitions, and aspects of super coinvariant algebras which provided the original motivation for the definition. This is joint work with Bruce Sagan. |
| Mar 17 | Alex McDonough (UC Davis) | A multijection of cokernels |
| |
I discovered an intriguing linear algebra relationship which I call a multijection. I used this construction to solve an open problem about higher-dimensional sandpile groups, but I think that it has more to say. In this talk, I will focus on sharing the most general version of the multijection that I know of, which involves a family of beautiful periodic tilings. This talk uses mostly linear algebra, so it should be accessible to a general math audience. |
| Mar 24 | Greta Panova (USC) | Sorting probabilities for Young diagrams and beyond |
| |
Sorting probability for a partially ordered set P is defined as the min |Pr[x<y] - Pr[y<x]| going over all pairs of elements x,y in P, where Pr[x<y] is the probability that in a uniformly random linear extension (extension to total order) x appears before y.
The celebrated 1/3-2/3 conjecture states that for every poset the sorting probability is at most 1/3, i.e. there are two elements x and y, such that 1/3 ≤ Pr[x<y] ≤ 2/3. The asymptotic extension of this conjecture states that the sorting probability goes to 0 as the width (maximal antichain) of the poset grows to infinity. We will prove the last conjecture for Young diagrams, where the linear extensions are Standard Young Tableaux.
Beyond SYTs, these conjectures bring out a variety of poset inequalities, which have connections to both algebra as in group actions and probability as in random walks.
Based on joint works with Swee Hong Chan and Igor Pak. |
| Mar 31 | Laura Colmenarejo (NCSU) | Multiplying quantum Schubert polynomials using combinatorics |
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Schubert polynomials are a very interesting family of polynomials in algebraic geometry due to their relation with the cohomology of the flag variety. Moreover, they are also very interesting from a combinatorial point of view because they can be considered generalizations of Schur functions. In this talk, we will talk about how to multiply a Schubert polynomial by a Schur function indexed by a hook and how we can extend this multiplication to the quantum world. This is a current work with C. Benedetti, N. Bergeron, F. Saliola, and F. Sottile. |
| Apr 7 | Farid Aliniaeifard (UBC) | Modular relations between chromatic symmetric functions |
| |
In 1995, Stanley introduced the chromatic symmetric functions. The study of chromatic symmetric functions of graphs inspired two main research directions. The first research direction is to prove the Stanley-Stembridge conjecture: if a poset is (3+1)-free, then the chromatic symmetric function of its incomparability graph is e-positive, i.e., a nonnegative linear combination of elementary symmetric functions. The second research direction is to determine whether two non-isomorphic trees can have the same chromatic symmetric function. In this talk, we present several modular relations between chromatic symmetric functions and apply them to show that the Stanley-Stembridge conjecture is true for several new families of graphs. Moreover, using the modular relations, we give an algorithm to write the chromatic symmetric functions of trees in terms of the chromatic symmetric functions of paths. (Joint work with Victor Wang and Stephanie van Willigenburg). |
| Apr 14 | Darij Grinberg (Drexel) | The one-sided cycle shuffles in the symmetric group algebra |
| |
Elements in the group algebra of a symmetric group Sn are known to
have an interpretation in terms of card shuffling. I will discuss a
new family of such elements, recently constructed by Nadia Lafreni\egravere:
Given a positive integer n, we define n elements t1, t2, ..., tn in
the group algebra of Sn by ti = the sum of the cycles (i), (i, i+1), (i, i+1, i+2), ..., (i, i+1, ..., n), where the cycle (i) is the identity permutation. The first of them,
t1, is known as the top-to-random shuffle and has been studied by
Diaconis, Fill, Pitman (among others).
The n elements t1, t2, ..., tn do not commute. However, we show
that they can be simultaneously triangularized in an appropriate basis
of the group algebra (the "descent-destroying basis"). As a
consequence, any rational linear combination of these n elements has
rational eigenvalues. The maximum number of possible distinct
eigenvalues turns out to be the Fibonacci number fn+1, and
underlying this fact is a filtration of the group algebra connected to
"lacunar subsets" (i.e., subsets containing no consecutive integers).
This talk will include an overview of other families (both well-known
and exotic) of elements of these group algebras. I will also briefly
discuss the probabilistic meaning of these elements as well as some
tempting conjectures.
This is joint work with Nadia Lafrenière. |
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