Title: Kazhdan-Lusztig polynomials of thagomizer graphs Abstract: To a graph G, one can associate a polynomial with non-negative integer coefficients called the Kazhdan-Lusztig polynomial of G. More generally, you can obtain the Kazhdan-Lusztig polynomial of any matroid, but today we will focus on the specialization to graphs. The Kazhdan-Lusztig theory for matroids was developed in analogy with the classical theory for Coxeter groups, though there are some important differences which I will touch on lightly. In this talk, we will construct the defining recursion for the Kazhdan-Lusztig polynomial of thagomizer graphs and use this obtain a closed form for the coefficients of the polynomial. No prior knowledge of matroids or Kazhdan-Lusztig polynomials will be assumed.

Title: Kazhdan-Lusztig polynomials of thagomizer graphs Abstract: To a graph G, one can associate a polynomial with non-negative integer coefficients called the Kazhdan-Lusztig polynomial of G. More generally, you can obtain the Kazhdan-Lusztig polynomial of any matroid, but today we will focus on the specialization to graphs. The Kazhdan-Lusztig theory for matroids was developed in analogy with the classical theory for Coxeter groups, though there are some important differences which I will touch on lightly. In this talk, we will construct the defining recursion for the Kazhdan-Lusztig polynomial of thagomizer graphs and use this obtain a closed form for the coefficients of the polynomial. No prior knowledge of matroids or Kazhdan-Lusztig polynomials will be assumed.

Title: “Cold atoms, SU(N) symmetry and Young tableaux”



Abstract:  Symmetries play a foundational role in our understanding of physics.

 It is often the case that unexpected symmetries can emerge from unexpected places.

I will discuss a particular example in systems of cold atoms that realize SU(N) symmetry,

where N can be as large as ~10.  I will then describe how these systems can be treated

numerically by using results on the symmetric group dating back to Young.

Title: “Cold atoms, SU(N) symmetry and Young tableaux”



Abstract:  Symmetries play a foundational role in our understanding of physics.

 It is often the case that unexpected symmetries can emerge from unexpected places.

I will discuss a particular example in systems of cold atoms that realize SU(N) symmetry,

where N can be as large as ~10.  I will then describe how these systems can be treated

numerically by using results on the symmetric group dating back to Young.

Discrete CATS Seminar

2pm, Monday Feb 13

POT 745

Speaker: Laura Escobar, UIUC

Discrete CATS Seminar

2pm, Monday Feb 13

POT 745

Speaker: Laura Escobar, UIUC

Discrete CATS Seminar

 

Title: Universal partial words

 

Abstract: A universal word for an alphabet A and a positive integer n is a word containing each of the words of length n over A as a substring exactly once. For instance, de Bruijn sequences are (cyclic) universal words. Universal partial words, introduced by Chen, Kitaev, Mutze, and Sun in 2016, allow for a wild-card character , which can stand for any letter in the alphabet. We settle and strengthen conjectures posed in the same paper where this notion was introduced. For non-binary alphabets, we show that universal partial words have periodic structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for a family of universal partial words over alphabets of even size. This talk is based on joint work with Bennet Goeckner, Corbin Groothius, Brian Kell, Pamela Kirkpatrick, Rachel Kirsch, and Ryan Solava.

 

Discrete CATS Seminar

 

Title: Universal partial words

 

Abstract: A universal word for an alphabet A and a positive integer n is a word containing each of the words of length n over A as a substring exactly once. For instance, de Bruijn sequences are (cyclic) universal words. Universal partial words, introduced by Chen, Kitaev, Mutze, and Sun in 2016, allow for a wild-card character , which can stand for any letter in the alphabet. We settle and strengthen conjectures posed in the same paper where this notion was introduced. For non-binary alphabets, we show that universal partial words have periodic structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for a family of universal partial words over alphabets of even size. This talk is based on joint work with Bennet Goeckner, Corbin Groothius, Brian Kell, Pamela Kirkpatrick, Rachel Kirsch, and Ryan Solava.

 

TITLE:  Toric posets in Coxeter groups, toggle groups, non-crossing partitions, and homomesy.

ABSTRACT:  A toric poset is a cyclic analogue of an ordinary poset that arises as a chamber of a toric graphic hyperplane arrangement. In this talk, I will show how toric posets turn classic concepts and problems on reducibility in Coxeter groups into new problems on cyclic reducibility and conjugacy. After that, I will give a brief introduction to dynamic algebraic combinatorics and the toggle group introduced by Striker and Williams, and show how toric equivalence arises in a number of groups actions involving toggles and the homomesy phenomenon of Propp and Roby.