Building Bridges Between \( \mathbb{F}_1 \)-Geometry, Combinatorics, and Representation Theory

This page is devoted to the special session on "Building Bridges Between \( \mathbb{F}_1 \)-Geometry, Combinatorics, and Representation Theory" at the AMS Northeast Fall 2023 Sectional Meeting. The meeting was held at the University at Buffalo on the 9th and 10th of September, 2023. You can access the official program on the AMS website. This page includes some information on the talks given, including slides (or external links to slides) where available.

My co-organizers, Jaiung Jun (SUNY New Paltz) and Alex Sistko (Manhattan College), and I would like to thank all participants and guests at our session. It was great to get to meet you, and we hope to see you again soon! We'd also like to thank the folks who shared their slides with us (making this page possible).

This session will highlight recent advances in \( \mathbb{F}_1 \)-geometry and combinatorics and their connections to representation theoretic problems. Combinatorial methods often provide a new set of tools and ideas to solve various problems. This direction leads to natural connections to \( \mathbb{F}_1 \)-geometry by which combinatorial structures can be considered as algebro-geometric objects. This session aims to explore both its algebraic and combinatorial aspects of the theory and their interactions.

Speaker

Jacob Van Grinsven (University of Iowa)

Abstract

For \( H \), a Hopf algebra over a field \( k \), an module algebra is both a \( k \)–algebra and a representation of \( H \) with action on products determined by the coproduct of \( H \). We will discuss the data that defines semisimple \( U_q(\mathfrak{g}) \)–module algebras (where \( \mathfrak{g} \) is a semisimple Lie algebra) and discuss filtered actions on path algebras as well.

Slides

PDF.

Speaker

Uly Alvarez (The University of Alabama)

Abstract

Given a quiver, we can define the cluster variables of a cluster algebra. If the cluster algebra arises from a triangulation of a surface, we can associate to each cluster variable a quiver Grassmannian and a snake graph.

It is an open problem to find cell decompositions of quiver Grassmannians associated to cluster variables. We initiate an approach to this problem by giving an explicit description to each cell in the case of Kronecker quivers using the perfect matchings of the associated snake graph.

Speaker

Cristian Lenart (State University of New York at Albany)

Abstract

This talk is a survey of a combinatorial model in the representation theory of semisimple and affine Lie algebras, which I developed in several joint papers. The model is uniform across all Lie types, and is called the alcove model, as it is based on the affine Weyl group and the related alcove setup. In particular, we realize the corresponding Kashiwara crystals, which are colored directed graphs encoding information about certain representations (highest weight modules and Kirillov-Reshetikhin modules). Related applications of the alcove model are: combinatorial formulas for Hall-Littlewood and Macdonald polynomials (which generalize the irreducible characters of simple Lie algebras), and combinatorial multiplication formulas in the \( K \)-theory of flag varieties.

Slides

PDF.

Speaker

Harpreet Singh Bedi (Alfred University)

Abstract

In this talk we present canonical differential forms over \( F_1 \) by using fibered categories. The addition operation is completely avoided in the base category, but we manage to show de Rham cohomology as well as Leibniz rule without using blueprints.

Slides

PDF.

Speaker

Cody Gilbert (University of Iowa)

Abstract

We prove irreducible components of moduli spaces of semistable representations of skewed-gentle algebras, and more generally, clannish algebras, are isomorphic to products of projective spaces. This is achieved by showing irreducible components of varieties of representations of clannish algebras can be viewed as irreducible components of skewed-gentle algebras, which we show are always normal. The main theorem generalizes an analogous result for moduli of representations of special biserial algebras proven by Carroll-Chindris-Kinser-Weyman.

Slides

PDF.

Speaker

Tong Jin (Georgia Institute of Technology)

Abstract

We generalize Baker-Bowler’s theory of matroids over tracts to orthogonal matroids, define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets, and orthogonal vector sets, and establish basic properties on functoriality, duality, and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we prove that an orthogonal matroid is regular if and only if it is representable over \( \mathbb{F}_2 \) and \( \mathbb{F}_3 \), which was originally shown by Geelen, and that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over \( \mathbb{F}_3 \) and \( \mathbb{F}_4 \).

Slides

Link to PDF.

Speaker

Zachery Peterson (University of Kentucky)

Abstract

An \( SL_k \)-frieze is a bi-infinite array of integers where adjacent entries satisfy the diamond rule. \( SL_2 \)-friezes were first studied by Conway and Coxeter. Later, these were extended to infinite matrix-like structures called tilings. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and \( SL_2 \)-tilings. We extend this result to higher dimensions \( k \) by constructing a bijection between \( SL_k \)-tilings and certain pairs of bi-infinite strips of vectors in \( \mathbb{Z}^k \) called paths. This is accomplished by relating them to Plücker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and partial positivity for \( SL_k \)-friezes.

Slides

PDF.