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
Feburary 25th	Chenchen Zhao, UC Davis
March 4th	Dustin Ross, SFSU
March 11th	Trevor Karn, Texas A&M
March 18th	Spencer Daugherty, University of Colorado Boulder
March 25th	No seminar - Spring Break
April 1st	Hannah Larson, UC Berkeley
April 8th
April 15th
April 22nd
April 29th
May 6th

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