DISCRETE CATS SEMINAR

Discrete CATS Seminar

Speaker:  Steven Karp, Notre Dame

Title:  q-Whittaker functions, finite fields, and Jordan forms

Abstract:

The q-Whittaker symmetric function associated to an integer partition is a q-analogue of the Schur symmetric function. Its coefficients in the monomial basis enumerate partial flags compatible with a nilpotent endomorphism over the finite field of size 1/q. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the q-Burge correspondence. In the q -> 0 limit, we recover a known description of the classical Burge correspondence (also called column RSK). We use the q-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver.

This is joint work with Hugh Thomas.

Date:
Location:
745 POT
Event Series:

Discrete CATS Seminar

Part I of DOUBLE HEADER!

745 POT; 1-1:45 pm

 

Speaker:  Daniel Tamayo, Université Paris-Saclay

Title:  On some recent combinatorial properties of permutree congruences

of the weak order

Abstract:

Since the work of Nathan Reading in 2004, the field of lattice

quotients of the weak order has received plenty of attention on the

combinatorial, algebraic, and geometric fronts. More recently, Viviane

Pons and Vincent Pilaud defined permutrees which are combinatorial

objects with nice combinatorial properties that describe a special

family of lattice congruences. In this talk we will give a brief

introduction into the world of (permutree) lattice congruences, how they

lead to structures such as the Tamari and boolean lattice, followed by

connections to pattern avoidance, automata and some examples of sorting

algorithms.

Daniel Tamayo is visiting Martha Yip.

Date:
Location:
745 POT
Event Series:

Discrete CATS Seminar

Qualifying Exam

Speaker:  Williem Rizer, University of Kentucky

Title:  Combinatorics of the Positroidal Stratification of the Totally Nonnegative Grassmannian

Abstract:

The Grassmannian has been an object of much interest in algebra, geometry, and combinatorics. We can decompose the Grassmannian into matroid strata, in which each element of the stratum has the same set of nonzero Plücker coordinates corresponding to the bases of a matroid. If we restrict to the elements of the Grassmannian with nonngeative coordinates, the corresponding matroids are called positroids. Postnikov revealed a family of combinatorial objects that can be used to parameterize positroidal cells. Recently, various authors have studied objects (X-diagrams and LACD colored permutations) that are in correspondence with this family, though the bijections exhibited do not commute with one another. In this talk we discuss all of these objects, the bijections between them, the information they reveal about positroids, and the possible connections and generalizations we can make to fold these newer objects neatly into the family.

 

Date:
Location:
745 POT

Discrete CATS Seminar

Qualifying Exam

Speaker:  Williem Rizer, University of Kentucky

Title:  Combinatorics of the Positroidal Stratification of the Totally Nonnegative Grassmannian

Abstract:

The Grassmannian has been an object of much interest in algebra, geometry, and combinatorics. We can decompose the Grassmannian into matroid strata, in which each element of the stratum has the same set of nonzero Plücker coordinates corresponding to the bases of a matroid. If we restrict to the elements of the Grassmannian with nonngeative coordinates, the corresponding matroids are called positroids. Postnikov revealed a family of combinatorial objects that can be used to parameterize positroidal cells. Recently, various authors have studied objects (X-diagrams and LACD colored permutations) that are in correspondence with this family, though the bijections exhibited do not commute with one another. In this talk we discuss all of these objects, the bijections between them, the information they reveal about positroids, and the possible connections and generalizations we can make to fold these newer objects neatly into the family.

 

Date:
Location:
745 POT

KOI Combinatorics Lectures

Presenting the KOI Combinatorics Lectures.

http://www.ms.uky.edu/~readdy/KOI

Friday, March 31, 2023

03:15 - 03:50 pm Coffee/Tea

04:00 - 05:00 pm Mihai Ciucu, Colloquium, Cruciform regions and a conjecture of Di Francesco, CB 214

Saturday, April 1, 2023 (CB 114)

09:00 - 10:00 am, Arrival/Registration/Meet and Greet

09:59 - 10:00 am Welcome, Welcome speech

10:00 - 11:00 am, Lei Xue, A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds

11:00 - 11:30 am, Coffee Break

11:30 - 12:30 pm, Eric Katz, Models of matroids

12:30 - 02:30 pm, Lunch Break

02:30 - 03:00 pm, Problem Session, run by Saúl A. Blanco

03:00 - 03:30 pm, Tea time and the One Picture/One Theorem Poster Session

04:00 - 05:00 pm, Richard Ehrenborg, Sharing pizza in n dimensions

06:00 - 08:00 pm, Conference Dinner

Date:
Location:
CB 114

KOI Combinatorics Lectures

Presenting the KOI Combinatorics Lectures.

http://www.ms.uky.edu/~readdy/KOI

Friday, March 31, 2023

03:15 - 03:50 pm Coffee/Tea

04:00 - 05:00 pm Mihai Ciucu, Colloquium, Cruciform regions and a conjecture of Di Francesco, CB 214

Saturday, April 1, 2023 (CB 114)

09:00 - 10:00 am, Arrival/Registration/Meet and Greet

09:59 - 10:00 am Welcome, Welcome speech

10:00 - 11:00 am, Lei Xue, A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds

11:00 - 11:30 am, Coffee Break

11:30 - 12:30 pm, Eric Katz, Models of matroids

12:30 - 02:30 pm, Lunch Break

02:30 - 03:00 pm, Problem Session, run by Saúl A. Blanco

03:00 - 03:30 pm, Tea time and the One Picture/One Theorem Poster Session

04:00 - 05:00 pm, Richard Ehrenborg, Sharing pizza in n dimensions

06:00 - 08:00 pm, Conference Dinner

Date:
Location:
CB 114

Discrete CATS Seminar

Speaker:  Galen Dorpalen-Barry, Ruhr-Universität Bochum

Title:   The Poincaré-extended ab-index

Abstract:

Motivated by a conjecture of Maglione-Voll from group theory, we introduce and study the Poincaré-extended ab-index. This polynomial generalizes both the ab-index and the Poincaré polynomial.  For posets admitting R-labelings, we prove that the coefficients are nonnegative and give a combinatorial description of the coefficients. This proves Maglione-Voll's conjecture as well as a conjecture of the Kühne-Maglione. We also recover, generalize, and unify results from Billera-Ehrenborg-Readdy, Ehrenborg, and Saliola-Thomas.



This is joint work with Joshua Maglione and Christian Stump.

We will be meeting in 745 POT and the speaker will be live from Germany via Zoom.

 

Our website:  https://www.ms.uky.edu/~readdy/Seminar

Date:
Location:
745 POT
Event Series:

Discrete CATS & Algebra Seminar

Speaker:  Ana Garcia Elsener, Universidad Nacional de Mar del Plata

Title:  skew-Brauer graph algebras

Abstract:

Brauer graph algebras are defined by combinatorial data based on graphs:

Underlying every Brauer graph algebra is a finite graph, the Brauer graph, equipped with with a cyclic orientation of the edges at every vertex and a multiplicity function. This combinatorial data encodes much of the representation theory of Brauer graph algebras and is part of the reason for the ongoing interest in this class of algebras. A known result by Schroll states that Brauer graph algebras, with multiplicity function one, give rise to all possible trivial extensions for gentle algebras. On the other hand, Geiss and de la Peña studied a generalization of gentle algebras called skew-gentle algebras.

In our ongoing project we establish the right definition of skew-Brauer graph algebra in such a way that the result by Schroll can be enunciated in this context. That is, A is a skew-Brauer graph algebra with multiplicity function equal to one if and only if it is the trivial extension of a skew-gentle algebra. Moreover, the family of skew-Brauer graph algebras with arbitrary multiplicity function generalizes the family of Brauer graph algebras with arbitrary multiplicity function.

(Joint work with Victoria Guazzelli from Universidad Nacional de Mar del Plata, Argentina, and Yadira Valdivieso Diaz from Universidad de Puebla, México)

Ana Garcia Elsener is visiting Khrystyna Serhiyenko.

Date:
Location:
745 POT

Discrete CATS & Algebra Seminar

Speaker:  Ana Garcia Elsener, Universidad Nacional de Mar del Plata

Title:  skew-Brauer graph algebras

Abstract:

Brauer graph algebras are defined by combinatorial data based on graphs:

Underlying every Brauer graph algebra is a finite graph, the Brauer graph, equipped with with a cyclic orientation of the edges at every vertex and a multiplicity function. This combinatorial data encodes much of the representation theory of Brauer graph algebras and is part of the reason for the ongoing interest in this class of algebras. A known result by Schroll states that Brauer graph algebras, with multiplicity function one, give rise to all possible trivial extensions for gentle algebras. On the other hand, Geiss and de la Peña studied a generalization of gentle algebras called skew-gentle algebras.

In our ongoing project we establish the right definition of skew-Brauer graph algebra in such a way that the result by Schroll can be enunciated in this context. That is, A is a skew-Brauer graph algebra with multiplicity function equal to one if and only if it is the trivial extension of a skew-gentle algebra. Moreover, the family of skew-Brauer graph algebras with arbitrary multiplicity function generalizes the family of Brauer graph algebras with arbitrary multiplicity function.

(Joint work with Victoria Guazzelli from Universidad Nacional de Mar del Plata, Argentina, and Yadira Valdivieso Diaz from Universidad de Puebla, México)

Ana Garcia Elsener is visiting Khrystyna Serhiyenko.

Date:
Location:
745 POT