The Algebraic Combinatorics Seminar

A Friday afternoon research seminar

In reverse-chronological order. On dates marked with * the seminar will meet in the Library instead (third floor).

Date Speaker Title (click titles for abstract) ~~23 Apr. 2010~~No seminar No Seminar 26 Mar. 2010 Sonya Berg A quantum algorithm for the quantum Schur transform The quantum Schur transform is a unitary implementation of $q$-deformed Schur-Weyl duality in type $A$. I'll present an efficient quantum algorithm for its computation, and explain its relationship to RSK algorithms. (Note the double use of the word quantum: one for $q$-deformed algebras and one for quantum computation.)19 Mar. 2010 Mike Zabrocki A discussion on $k$-Schur functions and cores This will be a discussion on the relationship between $k$-Schur functions and cores. It will begin with a presentation of the definition of $k$-Schur functions and cores, for those who are not familiar with these objects.12 Mar. 2010 Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics 5 Mar. 2010 Chris Berg and

Franco SaliolaPrimitive orthogonal idempoitents for the 0-Hecke algebra We will look at constructing primitive orthogonal idempotents for the 0-Hecke algebra. We have some candidates, but we are not sure that they are orthogonal. Chris will begin by explaining how to construct these idempotents, and convince us that they are indeed idempotents. Then Franco will present an argument---that proves orthogonality of a set of idempotents for a different algebra---that we hope to adapt to this algebra.17 Feb. 2010

*special day, 3:15pm*Benjamin Steinberg The representation theory of finite semigroups In recent years the representation theory of certain classes of finite semigroups have been used successfully to study hyperplane chamber random walks and other Markov chains. In this talk we give a survey of semigroup representation theory intended for people working in algebraic combinatorics. We focus on the following aspects:If time permits we will discuss some applications to probability and automata theory.

- Construction of the irreducible representations
- The character table
- The radical and triangularizability
12 Feb. 2010 Ton Dieker Interlacings, representation theory, and the interchange process on weighted graphs (lecture notes) A central question in the theory of card shuffling is how quickly a deck of cards becomes 'well-shuffled' given a shuffling rule. Using basic tools from the representation theory of the symmetric group, I will discuss a probabilistic card shuffling model known as the 'interchange process'. A 1992 conjecture by Aldous and Diaconis about this model has recently been resolved (see here) and I will indicate how my work has been involved with this.5 Feb. 2010 Chris Berg and Mike Zabrocki Exploring $(q,t)$-Catalan numbers and Cores We will continue to explore the bijections from the previous seminar. The aim is to understand various operations on cores and different ways to compute the statistics in the new settings.29 Jan. 2010 Chris Berg $(q,t)$-Catalan numbers and Cores Anderson gave a bijection between Dyck paths and $n$-cores which are also $(n+1)$-cores. I will describe this bijection and explain how to calculate the dinv and area statistics on these cores. More recently, Vazirani and Fishel gave a bijection between $n$ and $nm+1$ cores and shi arrangements. I will define statistics on these cores in an attempt to give a formula for the Catalan Fuss polynomials.~~22 Jan. 2010~~No Seminar No Seminar No Seminar15 Jan. 2010 Nantel Bergeron Radical of Weakly Ordered Semigroup Algebras We define the notion of weakly ordered semigroups. For this class of semigroups, we compute the radical of the semigroup algebras. This generalizes some results on left regular bands and on $0$- Hecke algebras. One open problem is to give a construction of the minimal idempotent for the $0$-Hecke algebra [Our hope is to use generalize (for Weakly Ordered Semigroup Algebras) the technique presented by Franco for left regular bands].11 Dec. 2009 Anouk Bergeron-Brlek Words on non-commutative invariants of the hyperoctahedral group Consider the hyperoctahedral group $B_n$ and let $V$ be a vector space which has a $B_n$-module structure. We present a general combinatorial method to decompose the tensor algebra $T(V)$ on $V$ into irreducible modules in terms of words in a particular Cayley graph of $B_n$. We make explicit the example of $V$ being the geometric module (corresponding to the action of $B_n$ as a reflection group) and give combinatorial interpretations for the graded dimensions and the number of free generators of the subalgebra $T(V)^{B_n}$ of invariants of $B_n$, in term of those words.~~04 Dec. 2009~~No Seminar No Seminar No Seminar27 Nov. 2009 Franco Saliola Random Walks on Hyperplane Arrangements I will present a new derivation of the main results of the Bidigare-Hanlon-Rockmore theory of random walks on hyperplane arrangements. The approach is to use an idea from the work of Ken Brown: consider the probability distribution as an element in a semigroup algebra and use algebraic techniques to study the random walk. I will introduce a recursive construction of orthogonal idempotents and explain how this construction produces orthogonal idempotents decomposing the probability element.~~20 Nov. 2009~~No Seminar No Seminar No Seminar16 Nov. 2009 Igor Pak, UCLA Random standard Young tableaux I will discuss the problem of generating random standard Young tableaux of a given shape. I will then define and analyze a weighted hook walk, which is a multivariable deformation of the usual hook walk. Finally, I will show how this weighted deformation gives a new bijective proof of the hook length formula for the number of standard Young tableaux. This is joint work with Ionut Ciocan-Fontanine and Matjaz Konvalinka.13 Nov. 2009 Mike Zabrocki Decomposing $GL_n$-modules into $S_n$-modules III: Progress I will present some recent progress on the problem of decomposing $GL_n$-modules into $S_n$-modules.06 Nov. 2009 Franco Saliola Symmetric Functions Computations with Sage

slides (pdfs), introductory worksheet, worksheet on symmetric functionsSage is a free, open-source mathematics software system developed by research mathematicians for research purposes. This talk will begin with a discussion of the Sage project and then proceed to survey some of the functionality and tools for algebraic combinatorics research. The main focus will be how to use Sage for computations with symmetric functions. If you want to bring a personal computer and work along with the demonstration, you can download and install the software from the Sage website sagemath.org. I will use version 4.2 for the demonstration.

For more information on Sage, visit the project website sagemath.org. Much of the combinatorics functionality is developed by a very active group of algebraic combinatorics researchers, whose progress can be followed on the Sage-Combinat website combinat.sagemath.org.

30 Oct. 2009 Open Session Working seminar / Problem session This meeting will be a working seminar / problem session. The goal is to decide on a theme for the seminar. Mike will begin by revisiting the problem he introduced in his earlier talks, and others are invited to present any problems they are interested in. Some come armed with ideas!23 Oct. 2009* Franco Saliola Random-to-random shuffles and commuting families of matrices II We will continue our investigation of families of matrices with rows and columns indexed by permutations and with entries given by certain statistics on permutations. For those who missed the first talk, this talk will begin by quickly recalling the necessary definitions and tools. We will then proceed to investigate properties of the matrices (commutation, eigenvalues, eigenspaces, ...).

This is joint work with Victor Reiner and Volkmar Welker.~~16 Oct. 2009~~No Seminar York University's Fall Reading Week TBA09 Oct. 2009* Franco Saliola Random-to-random shuffles and commuting families of matrices We will investigate a family of matrices with rows and columns indexed by permutations and with entries that count the number of increasing subsequences appearing in the permutations. These matrices are related to the inversion statistic on permutations, the Varchenko matrix for the reflection arrangement of the symmetric group, the linear-ordering polytope, and the transition matrix for the random-to-random shuffle. We will explore the connections with random walks on hyperplane arrangements and with the representation theory of the symmetric group, explaining how these can be used to study the eigenvalues and eigenspaces of the matrices.

This is joint work with Victor Reiner and Volkmar Welker.02 Oct. 2009 Mike Zabrocki Decomposing $GL_n$-modules into $S_n$-modules II (notes) We will continue from where we left off the previous week.25 Sep. 2009 Mike Zabrocki Decomposing $GL_n$-modules into $S_n$-modules I The irreducible polynomial representations of $GL_n$ are indexed by partitions of length less than or equal to $n$ and their characters are given by the Schur polynomials. The group of permutation matrices (the symmetric group, $S_n$) is a natural subgroup of the group $GL_n$ and the generating function for their characters are given by the Schur functions. We know that these two pictures are related by Schur-Weyl duality, but an interesting combinatorial question to ask is how an irreducible $GL_n$ module decomposes into irreducible $S_n$ modules when it is restricted to the subgroup of permutation matrices.

Today we will look at how symmetric functions can be used to determine how the $GL_n$ modules decompose when restricted to $S_n$. The goal will be to eventually arrive at a combinatorial formula.

** About the seminar. **
Every year we pick a new topic to explore.