The UC Berkeley Combinatorics Seminar

DATE	SPEAKER	TITLE (click to show abstract)
January 28th	John Shareshian, Washington University in St. Louis	Prime divisors of binomial coefficients, invariable generation of finite simple groups, and noncontractibility of order complexes of coset posets I will begin by discussing a problem on binomial coefficients. We fix an integer n>1 and consider the set BC(n) of nontrivial binomial coefficients {{n} \choose {k}}, 1 \leq k \leq n-1. It follows quickly from a theorem of Kummer that the gcd of these binomial coefficients is larger than 1 if and only if n is a prime power. Given this, we aim to partition BC(n) into as few subsets as possible so that the gcd of the elements of each subset is larger than 1. We know of no n for which we cannot partition BC(n) into at most two such subsets. After explaining what is known about the partitioning problem, I will point out its relation to invariable generation of alternating groups by Sylow subgroups. Two subsets X,Y of a group G are said to generate G ivariably if g^{-1}Xg and h^{-1}Yh together generate G for every pair (x,y) of elements of G. In joint work with Bob Guralnick and Russ Woodroofe, we study invariable generation of finite simple groups by various pairs of subsets. In particular, we show that every finite sinple group is generated invariably by a cyclic group and a p-group. Using this result and Smith Theory, we show that if G is any finite group and C(G) is the poset of all cosets of all proper subgroups of G, ordered by inclusion, then the order complex of C(G) has nontrivial reduced rational homology and therefore cannot be contractible.
Feburary 4th	Patty Commins, SL Math	The combinatorial representation theory of algebras coming from a special class of semigroups Left regular bands (or LRBs) are a special family of finite, noncommutative semigroups which arise surprisingly frequently in algebraic combinatorics and discrete geometry. The representation theory of their semigroup algebras is rich but tractable and has close connections to poset topology. Many of the LRBs in the literature come equipped with natural symmetry groups. In such cases, one can study the invariant subalgebra of the semigroup algebra. Solomons descent algebra arises as one such invariant subalgebra. In this talk, we will discuss joint work with Benjamin Steinberg which approaches understanding the representation theory of these invariant subalgebras through group-equivariant poset topology.
Feburary 11th	Anne Schilling, UC Davis	q-deformations of the Tsetlin library The Tsetlin library is a random shuffling process on permutations of $n$ letters, where each letter $i$ can be interpreted as a book; book i is brought to the front of the bookshelf with an assigned probability $x_i$ . We define a $q$-deformation of the Tsetlin library by replacing the symmetric group action on permutations by the action of the type $A$ Iwahori-Hecke algebra. We compute the stationary distribution and spectrum of this Markov chain by relating it to a Markov chain on complete flags over the finite field vector space $F_q^n$ and applying techniques from semigroup theory. We also generalize the $q$ -Tsetlin library to words (with repeated letters), and compute its stationary distribution and spectrum. This is based on work with Arvind Ayyer, Sarah Brauner and Jan de Gier (https://arxiv.org/abs/2601.21195)
Feburary 18th	Christian Gaetz, UC Berkeley	Combinatorial invariance for the coefficient of q in Kazhdan-Lusztig polynomials I will describe joint work with Grant Barkley and Thomas Lam in which we study the Combinatorial Invariance Conjecture (CIC), which asserts that Kazhdan-Lusztig polynomials depend only on the combinatorics of Bruhat order. We prove the combinatorial invariance of the coefficient of q in KL polynomials for arbitrary Coxeter groups. As a result, we obtain the CIC for Bruhat intervals of length at most 6. We also prove the Gabber-Joseph conjecture for the second-highest Ext group of a pair of Verma modules, as well as the combinatorial invariance of the dimension of this group.
Feburary 25th	Chenchen Zhao, UC Davis	Properties of plactic monoid centralizers Let u be a word over the positive integers and consider its centralizer C(u) in the plactic monoid. Motivated by questions from crystal theory, Sagan and Wilson conjectured a stability phenomenon: as we take higher and higher powers of a word u, the collection of words that commute with it eventually stops changing. In this talk, I will explain this stabilization phenomenon and present results proving it for several natural families of words, including words built from 1s and 2s and permutations. I will also discuss related counting questions that arise from studying these commuting words. This is joint work with Bruce Sagan.
March 4th	Dustin Ross, SFSU	A matroidal twist on Brion’s Formula In the 1980’s, Michel Brion proved a beautiful formula describing the Laurent polynomial of lattice points of a rational polytope as a sum of the rational generating functions of lattice points in the vertex cones of the polytope. For any generalized permutohedron P in n-space and any matroid M on n elements, I will describe a way to modify each vertex summand in Brion’s formula for P by the data of M, yielding a Laurent polynomial Q_M(P) that depends on both P and M. Our motivating question is: In what ways does Q_M(P) behave like the Laurent polynomial of lattice points in P? I will discuss some of the interesting properties of Q_M(P), including how it can be used to compute Euler characteristics of matroids, shedding light on duality and positivity properties for these Euler characteristics. This is ongoing work with Matt Beck and Carly Klivans.
March 11th	Trevor Karn, Texas A&M	Invariant subalgebras of Orlik–Solomon algebras in type A We provide a presentation in terms of generators and relations for the invariant subring of the Orlik–Solomon algebra of the rank-n type A reflection arrangement under the action of the symmetric group of permutations of {1,2,...,n}. Our results may be interpreted as a presentation of the cohomology ring of the configuration space of red points and one blue point in the plane. While the result is algebraic with a topological interpretation, techniques utilized in the proof are very combinatorial.
March 18th	Spencer Daugherty, University of Colorado Boulder
March 25th	No seminar - Spring Break
April 1st	Hannah Larson, UC Berkeley
April 8th	Joshua Greene, Boston College	Lattices problems from low-dimensional topology Several interesting problems in knot theory, 3-manifold topology, and 4-manifold topology boil down to lattice embedding problems. The basic kind of lattice embedding problem that comes up is this: you have a positive (semi-)definite symmetric integer matrix, oftentimes the Laplacian of a planar graph, and you want to know whether it is the Gram matrix of some collection of integer vectors in some low-rank Euclidean space. I'll spend a little time on the topological motivation and techniques, a lot of time on the variety of lattice embedding problems that come up, and a little more time on how some of them are solved.
April 15th	John Lentfer, UC Berkeley
April 22nd
April 29th	Hanna Mularczyk, MIT
May 6th

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