Spring 2019

The Graduate Algebra Symposium is a workshop organized by and for graduate students, the goal of which is to bring together students from several universities in the area to discuss research in algebra by graduate students. The symposium gives graduate students the opportunity to present their research to others in their field, gain experience giving conference-style talks, and network with local researchers. The Symposium is a collaborative effort by the University of North Texas, University of Texas at Arlington, Texas A&M University, Oklahoma State University, and the University of Oklahoma.

The Spring 2019 Symposium will be held at the **University of Oklahoma** in Norman, on **Saturday March 2nd.** Bagels & coffee will be provided in the morning, as well as pizza for lunch.

10:30-11am

OU

The Hilbert series derived from a finite dimensional representation of the complex circle $\mathbb{C}^{\times}$ for the purpose of identitifying the weight vectors with Gorenstein and non-Gorenstein invariant algebras. In this talk, I will present an explicit formula for the Hilbert series in terms of partial Schur polynomials, as well as the first few coefficients in the series.

11:10-11:40pm

TAMU

The Hochschild cohomology of the tensor product of algebras is fairly well understood. However, the Hochschild cohomology of the twisted tensor product of algebras is not. In this talk we will work towards a better understanding of it: we will introduce the notion of twisted tensor product, we will see the necessary conditions for the bar resolution to be compatible with it, and that it indeed satisfies them. We will conclude with some future applications of these results.

1:00-1:30am

OU

Following a paper from 2004 by Halverson and Ram, we will construct $\mathbb{C}A_k(n)$, the so-called Partition Algebras. We first construct the Partition Monoid $A_k$ which view partitions of a set of size $2k$ as diagrams with edges between vertices in the same partition. Once these associative algebras are defined, we will study certain subalgebras and define some interesting maps. Using the trace map, we will note conditions under which $\mathbb{C}A_k(n)$ is semisimple. Lastly, we will construct the ”Specht Modules” of $\mathbb{C}A_k(n)$, named to suggest the importance of the irreducible modules of the group algebra of the symmetric group $S_k$.

1:40-2:10pm

UTA

In the talk, I give a few examples of module where the Cartan subalgebra acts freely. Moreover, I will mention the idea of weighting functor, which was introduce by Olivier Mathieu. Weighting functor is a useful tool to study objects in category of $U(g)$-module.

2:20-2:50pm

OU

In 2007, Roberts and Schmidt had a satisfactory local new- and oldform theory for $\text{GSp}(4)$ with trivial central character, in which they considered the vectors fixed by the paramodular groups $K(p^n)$. In this talk, we consider the vectors fixed by the Klingen subgroup of level $p^2$. We determine the dimensions of the spaces of these invariant vectors for all irreducible, admissible representations of $\text{GSp}(4)$ over a $p$-adic field.

3:00-3:30pm

UTA

In this talk, we will discuss the class of algebras that is ideal in studying non-commutative algebraic geometry. Specifically we're going to talk about the Gorenstein Condition. This is the property that imposes some homological symmetry onto AS-regular algebras and is said to be the key condition in the definition.

3:40-4:10pm

OU

Partial conjugations play a key role in the study of the automorphism group of a right-angled Coxeter group. A partial conjugation by a vertex $v$ is an automorphism which conjugates only the elements of certain subsets of the vertex set of the defining graph by $v$. One can better these subsets by studying $\text{Sep}(v,w)$, the set of vertices which "separate" $v$ and $w$, i.e. vertices which can conjugate $v$ and not $w$, or vice-versa. We prove some nice properties of separating sets, in particular when the separating sets are minimal with respect to inclusion.

Parking will be available for free on Saturday in the Elm Street Parking Garage, directly across Elm street from the Physical Science Center. It is suggested that you park there, though there are other (read: paid) options across campus.

Rebekah Aduddell (UTA), Chelsea Drescher (UNT), Naomi Krawzik (UNT), and Jordan Wiebe (OU).