CanaDAM 2013 Université Memorial de Terre-Neuve, 10 - 13 juin 2013 www.smc.math.ca//2013f

Algebraic Combinatorics
Responsable et prÃ©sident: Steph Van Willigenburg (University of British Columbia)
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DREW ARMSTRONG, University of Miami
Maximal Chains of Parabolic Subgroups  [PDF]

Erd\"os, Guy and Moon (1973) observed that the number of maximal chains of partitions of the set $\{1,2,\ldots,n\}$ is equal to $n!(n-1)!/2^{n-1}$. The proof is easy and it is based on this fact: $$\binom{n}{2}\binom{n-1}{2}\cdots \binom{2}{2}=\frac{n!(n-1)!}{2^{n-1}}.$$ I will generalize this result to a finite Coxeter group $W$ of rank $r$. Specifically, I will show that the number of maximal chains of parabolic subgroups of $W$ is equal to $r!|W|/2^r$. (Taking the symmetric group $W=S_n$ with rank $r=n-1$ recovers the classical result.) The same method can be used to count chambers of Shi hyperplane arrangements by the dimension of their recession cone.

Algorithms of making linebundles on cotangent bundles of complete homogeneous spaces more positive.  [PDF]

Let $X$ be the cotangent bundle of a complete homogeneous space of a complex reductive group $G$ and $L$ a linebundle on $X$. Any cohomology group $H^i(X,L)$ has the structure of a graded $AG$-module, where $A$ is the ring of global regular functions on $X$.

We have developed methods to modify pairs $(X,L)$ to similar pairs $(X',L')$ without changing $A$ and without changing some other property like the Euler characteristic (respectively $H^0$, or without changing any $H^i$) but where $L'$ is strictly more positive than $L$. Modification depends on the property to be preserved.

Joint work with Ascah-Coallier and Jauffret.

ROSA ORELLANA, Dartmouth College
The quasi-partition algebra  [PDF]

We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra of the symmetric group. We construct a basis for $QP_k(n)$, give a formula for its dimension in terms of the Bell numbers, and describe a set of generators. In addition, we study its representation theory, giving dimensions and indexing sets of irreducible representations. We also provide the Bratteli diagram for the tower of quasi-partition algebras (constructed by letting $k$ range over the positive integers).

Joint work with Z. Daugherty

BRENDON RHOADES, University of California, San Diego
Extending the parking space  [PDF] [SLIDES]

A length $n$ sequence $(a_1, \dots, a_n)$ of positive integers is called a {\em parking function of size $n$} if its nondecreasing rearrangement $(b_1 \leq \dots \leq b_n)$ satisfies $b_i \leq i$ for all $i$. We prove that the action of the symmetric group $S_n$ on the vector space spanned by size $n$ parking functions extends to an action of $S_{n+1}$. We construct this extended module explicitly as a graded $S_{n+1}$ representation and describe its restricted graded Frobenius character. This is joint with Andrew Berget at the University of Washington.

HUGH THOMAS, University of New Brunswick
A reflection group perspective on c-vectors  [PDF]

$c$-vectors are part of the framework of a cluster algebra. Though late-comers in cluster algebra theory, they play a powerful role: it was shown by Nakanishi and Zelevinsky that many facts about cluster algebras follow by an elementary argument once it is known that each $c$-vector has all entries either non-negative or non-positive. I will give a classification of the collections of vectors that can appear as the $c$-vector of a cluster in a skew-symmetrizable cluster algebra starting from an acyclic seed, in terms of reflection groups. This is joint work with David Speyer, arXiv:1203.0277.