## Mathematics## Seminar |

## Rocky Mountain Algebraic Combinatorics Seminar

### Upcoming Seminars Schedule

### A Potpourri of Projective 2-Designs

Emily King

Colorado State University

Zauner conjectured that for all *d* there exists a set of *d*^{2} equiangular
vectors in *C*^{d} (SICs or Gabor equiangular tight frames). Another important
open problem concerns the existence of *d*+1 so-called mutually unbiased bases
(MUBs) in *C*^{d} (maximal sets of MUBs) for any *d* not equal to a prime power.
Both classes of objects are complex projective 2-designs. In this talk, the
basics of (weighted) projective 2-designs will be discussed, including their
recent generalization to finite fields. A collection of new results will be
presented.

*H*

^{d}determines an equi-isoclinic tight fusion frame of

*d*(2

*d*−1) subspaces of

*R*

^{d(2d+1)}of dimension 3. This is joint work with Joey Iverson and Dustin Mixon.

### Representation theoretic Hopf algebras and q-chromatic symmetric functions

Lucas Gagnon

UC Boulder

Hopf algebras are a nice way to incorporate algebraic tools into the study
of combinatorial objects with recursive structure, and in some cases these
combinatorial Hopf structures also encode representation theoretic
information. This talk will use a new example to explain this general idea.
On the combinatorial side is the chromatic quasisymmetric function, a
q-analogue of Stanley’s chromatic symmetric function of a graph; this
function can be realized as the image of a homomorphism between Hopf algebra
structures on graphs and (quasi-)symmetric functions. On the representation
theory side is a collection of GL_{n}(F_{q})-modules that arise in the study of
the maximal unipotent upper triangular subgroup of GL_{n}(F_{q}). I will show
that the GL_{n}(F_{q})-modules are, up to a canonical equivalence, a
representation-theoretic realization of some chromatic quasisymmetric
functions. Along the way I will define a new Hopf algebra of class
functions on the unipotent upper triangular groups and use lots of pictures
and examples.

Weber 223

4–6 pm

Friday, May 6, 2022

(Refreshments 3:30–4 pm)

Colorado State University

This is a joint Denver U / UC Boulder / U of Wyoming / CSU seminar that meets biweekly. Anyone interested is welcome to join us at a local restaurant for dinner after the talks.