The Algebraic Combinatorics Seminar
Scheduled for Fridays at 3:15 PM
Nantel Bergeron and Mike Zabrocki

#### Schedule

In reverse-chronological order.

Date Speaker Title (click titles for abstract)
30 May 2014 Holly Heglin thesis work on k-Schur functions in type C
23 May 2014 Cesar Ceballos The diameter of type D associahedra and the non-leaving-face property
We prove that the graph diameter of the n-dimensional associahedron of type D is precisely 2n-2 for all n. Furthermore, we show that all type ABCD associahedra have the non-leaving-face property, that is, any minimal path connecting two vertices in the graph of the polytope stays in the minimal face containing both. This is joint work with Vincent Pilaud.
4 April 2014 Tom Denton symmetric function graph invariants
28 March 2014 Tom Denton fast fourier transform on the symmetric group.
21 March 2014 Allejandro Morales (UQAM) Combinatorics of diagrams of permutations
In his study of the totally nonnegative Grassmannian, Postnikov introduced several combinatorial objects linked to a Grassmannian permutation $w=w_{\lambda}$. We study the connection between these objects when $w$ is no longer required to be Grassmannian. These objects include regions in the inversion hyperplane arrangement of $w$, rook placements on the complement of the diagram of $w$, Le"-fillings of the diagram of $w$, and permutations below $w$ in the strong Bruhat order. We show that for any fixed permutation $w$ the number of regions equals the number of rook placements and the number of certain fillings related to Le"-fillings. Then thanks to a conjecture of Postnikov, settled by Hultman-Linusson-Shareshian-Sj\"ostrand, we relate this number of regions/placements/fillings and one of its $q$-analogues to the number of permutations below $w$ in the Bruhat order. This last relation settles part of a conjecture with Klein and Lewis. This is joint work with Joel Lewis.
14 March 2014 Eric Ens Consistent colorings of polytopes.
We call a coloring of the facets of a polytope P consistent if each member of the automorphism group of P acts as a bijection on groups of facets of the same colour. I will give some examples with the Platonic solids as well as some results on {4,4} polytopes on a torus. As well as some questions I want answers to.
7 March 2014 Farid Aliniaeifard sums of idempotents in superclasses
28 February 2014 Nantel Bergeron left Pieri rule for the immaculate basis.
Finishing the proof.
14 February 2014 Cesar Ceballos and Tom Denton recent progress on the rational Catalan problems
7 February 2014 Nantel Bergeron More on Immaculate function for NSym
31 January 2014 Nantel Bergeron Problem I have been working on.
17 January 2014 Tom Denton recent reading on applying representation theory to machine learning problems.
10 January 2014 Tom Denton Informal meeting
discuss what's been moving forward over break and directions forward for the Winter term. Additionally, unless anyone else wants the space to talk, I will probably be giving some update next Friday on a proof which shows that the (a,b)-Catalan skew-length statistic matches with the dinv statistic in the classical case.
29 November 2013 Anna Bertiger (U. of Waterloo) Multiplying Schubert Polynomials, Pipe Dreams and the Fomin-Kirillov Algebra
A longstanding problem in algebraic combinatorics is that of providing a manifestly positive formula for multiplying two Schubert classes in the cohomology ring of the full flag manifold. One open sub-problem to this problem is multiplying any Schubert polynomial by any Schur polynomial (Schurs are Schuberts for permutations with exactly one descent). I'll give an overview of the current state of affairs as I see it with respect to this problem and talk about what open conjectures, and a few theorems, are relevant. The key players will be pipe dreams for computing Schubert polynomials to represent Schubert classes, and the Fomin-Kirillov algebra a non-commutative algebra which mimics the covering relations of the Bruhat order. I intend for the talk to be friendly, with many definitions explained, including those of pipe dreams and the Fomin-Kirillov algebra.
22 November 2013 Nantel Bergeron quotient of the polynomial ring in $n$ variables by the ideal generated by quasi-symmetric functions
He previously showed that this quotient has Catalan dimension. We'll be interested to investigate whether similar quotients can be used to obtain (a,b)-Catalan numbers.
15 November 2013 Cesar Ceballos Ants, determinants, and the sign function
Cesar will talk about his current work with Nantel Bergeron and Jean-Philippe Labbé.
8 November 2013 Tom Denton progress made on (a,b)-cores during Chris Hanusa's visit
1 November 2013 Trueman MacHenry Relation between Fibonacci sequence with homogeneous symmetric functions
Trueman discusses work relating Fibonacci and Lucas numbers to the homogeneous and power symmetric functions. The hook-Schur functions (another free algebraic basis of Sym) also made an appearance. The hope is to relate these objects to the ongoing study of the q-Fibolan numbers.
25 October 2013 Nantel Bergeron How to discover beautiful math in Fibo-Land
Nantel discusses developments in the q-Fibolan numbers, arising from q-deforming the Fibolans and adjusting the combinatorics to match the q-enumeration.
18 October 2013 Nantel Bergeron Fibolan
Nantel Bergeron presents on the 'Fibolan' numbers recently discovered by Sagan and Savage. These are a replacement of the usual integers with Fibonacci numbers in the formula for binomial coefficients. Sagan and Savage have developed a combinatorial object that these numbers count.
4 October 2013 Cesar Ceballos and Tom Denton lattice paths and a,b-cores
27 September 2013 Nantel Bergeron The Saturation Conjecture and NSym
Nantel presented an approach to the saturation conjecture (which is no longer a conjecture in Sym) for NSym, using a polytope realized from the immaculate basis. He also talked about the dual quasi-Schur functions, and possibilities for creating a polytope for the relevant composition tabeleaux, which could then be used to attack the saturaation conjecture for that basis.
13 September 2013 Cesar Ceballos and tom denton Quonting Quores
We present the lattice path model for computing statistics on simultaneous core partitions. The lattice path model allows us to quickly compute the skew length statistic, conjectured to be the q-statistic for simultaneous (a,b)-cores when a and b are relatively prime. We demonstrate that the skew length is symmetric in a and b.
6 September 2013 tom denton and Mike Zabrocki Organizational Session
We presented problems that we're interested in working on this year.

Notes.

• Unless otherwise indicated, the seminar meets on Fridays at 3:15 pm in Room 210 of the Fields Institute. If the date is marked by *, then we will meet in the Stewart Library instead.
• If you are interested in speaking at the seminar, contact Nantel Bergeron.
• You may also be interested in the Applied Algebra Seminar (Monday afternoons at York University).

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

Year Topic
2012-2013 NSym and the Immaculate Basis
2011-2012 k-Schur functions and affine permutations
2010-2011 Littlewood Richardson rule k-Schur functions.
2009-2010 Idempotents and weakly ordered semigroups. (q,t) Catalan Numbers.
2008-2009 Littlewood-Richardson Rule, Shifted Tableaux and P-Schur functions
2007-2008 Open problems around k-Schur functions and non-commutative symmetric functions
2006-2007 Open problems
2005-2006 Cluster Algebras and Quivers
Spring 2005 Formal languages and analytic classes of functions
Fall 2004 (Quasi-) Symmetric functions in noncommutative variables and applications
Winter 2003 Crystal Bases and Representation Theory, Super-algebras, etc.
Fall 2003 Quasi-Symmetric functions and applications
Fall 2002 Crystal Bases and Representation Theory