## Research

Here is a list of research projects that I have been involved in. If you are an undergraduate interested in working on a research project with me, or in talking about research in math more generally, I encourage you to email me or stop by my office (VH 556).

#### Enumerating Linear Systems on Graphs with Forrest Glebe and David Perkinson

The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G, and linear equivalence of divisors is determined by the discrete Laplacian operator for G. As in the case of Riemann surfaces, we are interested in the complete linear system |D| of a divisor D---the collection of nonnegative divisors linearly equivalent to D. Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G. We interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G. If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. Finally, we generalize our results to a model based on integral M-matrices.

Enumerating Linear Systems on Graphs, 2019. Sarah Brauner, Forrest Glebe, David Perkinson.

Our extended abstract, accepted to be presented as a poster at FPSAC 2019: Enumerating Linear Systems on Graphs (extended abstract.) Here's the poster.

Slides from a talk on this paper at the AMS Special Session on Divisors and Chip-Firing at University of Connecticut, Hartford in April 2019.

#### Representations of the Symmetric Group on the Free LAnKewith Tamar Friedmann

In 2011, Tamar Friedmann defined a generalization of a Lie algebra; this generalization, called a Lie Algebra of the nth kind (LAnKe) is a vector space with an n-linear commutator. One can define the multilinear component of the free LAnKe in an analogous way to the multilinear component of the free Lie algebra. I have been working with Friedmann on understanding the representations induced by the action of symmetric group on the multilinear component of the free LAnKe. This generalizes the very well-studied and elegant representations Lie(k).

As part of our study of the these representations, we managed to simplify a dual straightening algorithm which gives a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We were able to apply this result to the representation of the symmetric group on the multi-linear component of the free LAnKe with 2n-1 generators. Check out our paper below for details!

One Garnir to rule them all: on Specht modules and the CataLAnKe theorem, 2018. Sarah Brauner, Tamar Friedmann. (Submitted).

Slides from a talk on this paper at the AMS Special Session on Combinatorics in Algebra and Algebraic Geometry at University of Michigan, October 2018.

#### The Structure of Partial Orders in the Face of Lower BoundsUndergraduate senior thesis

One way of modeling a sorting algorithm is as a sequence of posets, where in each step of the algorithm (a.k.a term in the sequence) we "compare" two incomparable elements. Under the direction of Josh Grochow at the Santa Fe Institute (now at UC Boulder), I studied structural aspects of such a model for sorting. In particular, I explored the relationship between a poset's automorphism group, its set of linear extensions, and the classical information-theoretic bound for sorting. I continued this work in my senior thesis at Reed College, under the supervision of Angelica Osorno. In my thesis, I proved several new results linking the size and structure of a poset's automorphism group to its set of linear extensions.

The final draft of my senior thesis: The Structure of Partial Orders in the Face of Lower Bounds, 2016. Sarah Brauner.