The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar has been running since 1997. The topics of talks have typically been any mixture of algebra with any other field: combinatorics, geometry, topology, physics, etc. Further down this page you will find links to the seminar webpages for previous years. The audience usually consists of 6–12 people, including several graduate students and post-docs. For this reason, speakers are encouraged to devote a portion of their talk to the suggestion of open problems and the directions for research in their area. If you are interested in speaking at the seminar, contact Nantel Bergeron.

You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

Dates are listed in reverse-chronological order. Unless otherwise indicated, all talks will take place at 2:30pm in N638 Ross Building (York University).

Date Speaker Title (click titles for abstract) 6 March 2012 Louis-Francois Preville-Ratelle

(UQaM)Enumeration of Combinatorial Structures Associated to the Trivariate Diagonal Coinvariant Spaces We will explain from scratch what are the multivariate diagonal coinvariant spaces. We will then show some combinatorial structures associated to them. Our main contribution is the enumeration of some new structures that strongly seem to be associated to the trivariate spaces. If time permits, we will also speak of some open problems related to them. This talk is accessible to all.13 Feb. 2012 Nathan Williams

(Minnesota)A Generalization of Suter's Surprising Cyclic Symmetry and an Associated CSP In 2002, R. Suter found a surprising cyclic symmetry of order (k+1) in a subposet of Young's lattice. In 2012, C. Berg and M. Zabrocki generalized this result geometrically to the subposet of k-Young's lattice consisting of those (k+1)-cores contained in a product of (m-1) rectangles. Thanks to H. Thomas' recent proof of my Dendrodistinctivity conjecture, we now know that there is an equivariant bijection from these posets under their cyclic action to words of length (k+1) that sum to (m-1) mod m under rotation. This bijection allows us to completely understand the orbit structure of this cyclic symmetry through the cyclic sieving phenomenon.21 Nov. 2011 Cesar Ceballos

(Berlin)Subword complexes, cluster complexes, and generalized multi-associahedra We present a new family of simplicial complexes called multi-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra to arbitrary finite Coxeter groups. We present combinatorial and geometric properties of these objects, as well as open problems and conjectures. This is joint work with Jean-Philippe LabbĂ© and Christian Stump14 Nov. 2011 Trueman Machenry

(York University)A TALE OF TWO SEQUENCES This work deals with the Ring of symmetric polynomials Λ and can be viewed in two ways. Either as applications using Λ to model or represent other mathematical systems, or as a new look at the internal structure of Λ. In the first category, it is shown that there are faithful local representations of the groups of multiplicative arithmetic functions and of the groups of additive arithmetic functions inside Λ. These representations can be used to prove new results in number theory and to introduce new classification systems there. Similarly one can show that a given algebraic number field is completely determined by structures with in Λ, producing new results for number fields. A description of Polya's counting theorem as well as an algorithm for constructing the table of irreducible characters of the Symmetric group S(n) is given using only the two sequences of the title . Fundamental to these results is the fact that all linear recursions can be represented inside of Λ. The generating polynomials for these linear recursions are called core polynomials. In the second category, certain substructures of Λ, whose importance may have been overlooked before, are singled out. The companion matrix of the core polynomial generates a matrix called the infinite companion matrix which is closely linked to the submodule of weighted isobaric polynomials, whose elements form linear recursive sequences inside Λ. The derivatives of the core polynomials determine matrices whose determinants are the discriminants of the core polynomial and which induce a gadget called the infinite difference matrix, whose elements lie in Λ. Also new operators are introduced into the ring: formal logarithmic and exponential operators as well as formal trigonometric functions. These operators have interesting features in Λ as well as applications in, for example, arithmetic number theory.7 Nov. 2011 Tom Denton Promotion Operators and Dual Equivalence The promotion operator is a combinatorial function on standard tableaux which, for rectangular tableaux, implements the rotational automorphism of the affine dynkin diagram. This is useful in giving a combinatorial affine structure on crystal bases indexed by rectangular tableaux, or tensor products thereof. We will discuss the use of the promotion operator in the crystal basis theory, and also examine how it interacts with dual equivalence graphs on standard tableaux.31 Oct. 2011 Viviane Pon

(Marne la vallee, France)Mixed product of two kinds of 0-Hecke algebra generators We study two families of generators of 0-Hecke algebra that define two bases. Their change of basis is directly linked to the Bruhat order on permutations and generators can be seen as two kinds of sorting operators. Although the relations between the bases are simply expressed, general description of mixed products of elements from both families is an open problem. To a permutation and an integer k, we associate a specific mixed product that has interesting interpretation in terms of Grothendieck and key polynomials. We show that the expansions of this product over the two bases can be expressed as sums over the Bruhat order and the k-Bruhat order.24 Oct. 2011 Zhi Chen

(York)A Plethysm formula on the induced linear character from U_n( F_q) into GL_n( F_q) We give a plethysm formula which involves the study on the characteristic of the induced linear character from the uniponent upper-triangular matrices U_n(F_q) into the general linear group over finite field F_q. It turns out to be a multiple of a twisted version of the Hall-Littlewood symmetric functions P_n(y,t). A recurrence relation is also given so that it is easy to carry out the computation.17 Oct. 2011 Oded Yacobi

(U of Toronto)Polynomial functors and categorifications of Fock space In order to motivate "higher represenation theory", we begin by discussing Kleshchev's socle branching rules for modular representations of the symmetric groups, and the LLT categorification of the basic representation of affine sl_n. Via Schur-Weyl duality, this leads us to introduce the category P of strict polynomial functors. We will then describe how P naturally categorifies the commuting actions of affine sl_n and the Heisenberg algebra on Fock space. This is joint work with Jiuzu Hong10 Oct. 2011

NO SEMINARThanksgiving 3 Oct. 2011

2:00 PM SPECIAL TIMEChris Berg

(LaCIM, UQAM)k-Schur functions in non-commutative variables. Lam showed that the $k$-Schur functions model Schubert classes for the homology of the affine Grassmanian in type A. This was done by constructing an isomorphism between the homology and a subalgebra of the nil Coxeter algebra. Explicit formulas for the Schubert classes in the nil Coxeter algebra have not been constructed. I will speak on how this generalizes to a definition of $k$-Schur functions in all types, and how to get explicit formulas in certain cases.

Below you will find links to the seminar webpages for previous years.