# Algebra and Geometry Seminar

## Info

Welcome to the Algebra and Geometry Seminar at Iowa State University, organized by Jonas Hartwig, Jason McCullough, and Tathagata Basak.

During Fall 2023, the seminar runs on Thursdays at 2:10pm–3:00pm (plus optional discussion) in Carver 401. Grad students are especially encouraged to attend.

Topics include:

• associative, commutative, and Lie algebras;
• algebraic, differential, and hyperbolic geometry;
• representations theory and tensor categories;
• mathematics of quantum theory and fundamental physics.

## Talks

### September 7, 2023

#### The symmetric group and its action on a ring of multivariate polynomials — with applications to Galois theory

##### Christian Roettger (ISU)

Let $$S$$ be the ring of polynomials in $$n$$ variables over the ground field $$K$$, of characteristic zero. S has the subring $$B$$ of symmetric functions. So we can consider $$S$$ as a $$B$$–module. In fact, $$S$$ is a free B-module of rank $$n!$$ ($$n$$ factorial).

Now $$S$$ carries a $$\textrm{Sym}(n)$$–action compatible with the $$B$$–module structure, so $$S$$ is a module over the group ring $$B[\textrm{Sym}(n)]$$, which itself is also a free $$B$$–module of rank $$n!$$.

Conjecture: $$S$$ is isomorphic to $$B[\textrm{Sym}(n)]$$ as $$B[\textrm{Sym}(n)]$$–modules.

We will show some evidence for this conjecture, eg that it is true if we replace the rings $$S$$ and $$B$$ by their respective fields of fractions, and also in the case $$n=3$$.

Then we will discuss applications to Galois theory. If the conjecture is true, then we can draw conclusions about the subrings of $$S$$ which are invariant under a subgroup of $$\textrm{Sym}(n)$$. This can give information about the fixed field of a subgroup in a finite Galois extension of the ground field K, eg universal expressions for generators.

The approach generalizes the well-known theorem that the splitting field of a polynomial $$f$$ over $$K$$ has a group contained in the alternating group iff the discriminant of $$f$$ is a square in $$K$$.

### September 14, 2023

#### Groups like the braid group

##### Tathagata Basak (ISU)

Given two braids in $$n$$ strands, you can compose them by stacking one below the other. This gives the $$n$$ strand braid group $$\mathrm{Br}_n$$. This group appears in knot theory, geometric topology, quantum groups...

The braid group $$\mathrm{Br}_n$$ can be generalized in different ways. One generalization comes from looking at $$\mathrm{Br}_n$$ as the "fundamental group of the discriminant complement of type $$A_n$$". Another generalization comes from looking at $$\mathrm{Br}_n$$ as the "fundamental group of the configuration space of n points in the plane". The more general "discriminant complements" and "configuration spaces" also appear in many places in algebra and geometry.

We shall begin with an introduction to the braid group $$\mathrm{Br}_n$$ and try to explain some of these generalizations. Time permitting, we may mention a strange isomorphism between a certain discriminant complement of complex dimension 9 and the configuration space of 12 points on the 2-sphere stemming from works of Deligne-Mostow and Thurston.

We'll try to keep prerequisites to a minimum. In particular, we will try to explain the statements with quotation marks above. Graduate students are very welcome.

### September 21, 2023

#### Lengths Of Local Cohomology Using Some Surprising Hilbert Kunz Functions

##### Jenny Kenkel (Grinnell College)

Local cohomology modules were originally invented to answer a question about unique factorization domains, but they're worthy of study in their own right, due to their beautiful properties and many equivalent definitions. I will give one such definition, and then transform a question about a local cohomology module into a question about the vector space dimensions of $$k[x_1,\ldots,x_n]/(x_1^{k_1},\ldots,x_n^{k_n})$$ where $$k$$ is some field, and elements in that ring killed by $$x_1 + \cdots + x_n$$. I will use refinements of functions introduced by Han and Monsky. This project is joint work with Mel Hochster.

### September 28, October 5, and October 12, 2023

#### Quiver Representations

##### Shlomo Gelaki (ISU)

I will give two talks in which I will discuss the basic theory of quivers and their representations, state the famous theorem of Gabriel and present the main ideas and tools used to prove it (e.g., Dynkin diagrams and geometry of orbits).

The talks will be aimed for graduate students.

### October 26, 2023

#### The Geometric Algebra

##### Jonas Hartwig (ISU)

I will give an introduction to the Clifford algebra, sometimes known in physics as the geometric algebra. We will see how it arises using the square root of the Laplace operator. We explore this noncommutative algebra and its matrix representations, and connections to groups of rotations. The second goal is to study the symmetries of the Dirac equation, and define what a spinor is.

### November 2, 2023

#### Koszulity, supersolvability, and Stirling representations

##### Ayah Almousa (U Minnesota)

The intimate relationship between the homological aspects of a Koszul algebra and its corresponding Koszul dual counterpart is a powerful tool in commutative and noncommutative algebra. In this talk we investigate the consequences of this link in the case where a group acts on the algebras. The motivating example will be the Orlik-Solomon and Varchenko-Gel'fand algebras for the braid arrangement with the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality. This is joint work with Vic Reiner and Sheila Sundaram.

### November 9, 2023

#### Lie Superalgebras and some Connections to Physics

##### Jonas Hartwig (ISU)

Continuous symmetries are common in many parts of mathematics and physics. The mathematical language dealing with such symmetries is known as Lie Theory after the Norwegian mathematician Sophus Lie. A group which is also a manifold is called a Lie group. Due to their self-symmetry, they are largely determined by their local structure, encoded in a so-called Lie algebra. A generalization of these local symmetry algebras to Lie superalgebras plays an important role in many applications. In this talk we will look at some small examples of Lie superalgebras and indicate how they are connected to differential equations appearing in physics, namely the Dirac equation and Maxwell’s equations, which govern the behavior of the electron-positron field and the photon field, respectively. This is based on discussions with D. Williams and E. Dolecheck.

### November 16, 2023

#### The Koszul property for algebras associated to matroids

##### Jason McCullough (ISU)

Given a central, essential complex hyperplane arrangement $$A = {H_1,\ldots,H_t}$$ in $$\mathbb{C}^n$$, there are several natural algebras measuring the topology of arrangement. Notably, the cohomology ring of the complement is given by the Orlik-Solomon algebra. More recently, other algebras associated to arrangements (or more generally matroids) were used in the resolution of long-standing conjectures in combinatorics: (1) Adiprasito, Huh, and Katz used Chow rings of matroids to prove the Heron-Rota-Welsh Conjecture regarding the log-concavity of coefficients of the characteristic polynomial of a matroid, and (2) Braden, Huh, Matherne, Proudfoot, and Wang additionally used augmented Chow rings of matroids and graded Moebius algebras of lattices to prove the Dowling-Wilson Top Heavy Conjecture describing the number of elements of a given rank in a geometric lattice. This talk will survey what is known about the Koszul property for these 4 algebras. I will start with a brief intro to Koszul algebras and matroids. This is joint work with Matt Mastroeni and Irena Peeva.