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

During Spring 2024, the seminar runs on **Thursdays at 2:10pm–3:00pm** in Carver 401. Grad students are especially encouraged to attend.

Topics include:

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

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 \(K\).

Links: Slides

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.

Links: Notes

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.

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.

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.

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.

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.

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.

Links: Slides

This talk will be an introduction to the representation theory of su(d) in the Gelfand-Tsetlin formalism from the point of view of quantum information theory. The tensor product of two irreducible representations splits into the direct sum of certain irreducible representations according to the Clebsch-Gordan d theorem and this process iterates to the tensor product of n representations. Accordingly the tensor product of n Hilbert spaces representing the states of a multipartite quantum system splits. We investigate the question of what entanglement properties quantum state vectors in each of the various subspaces have. Introductory definitions and results of the entanglement theory for quantum states will be given.

An example of a smooth cubic surface is the Clebsch surface \(S\): \[v^3 + w^3 + x^3 + y^3 + z^3 = v + w + x + y + z = 0.\] There are 27 lines on \(S\), that are easy to write down. For nice pictures, see: https://blogs.ams.org/visualinsight/2016/03/01/clebsch-surface/

We'll talk about the 27 lines on a smooth cubic surface and about the geometry and topology of the space \(M\) of all smooth cubic surfaces. We'll describe 10 loops in \(M\) based at \(S\) that conjecturally generate the fundamental group of \(M\), and describe some conjectured relations among these generators encoded by the Petersen graph. One of our main objectives will be to illustrate, via some completely elementary explicit calculations, how the 27 lines on \(S\) get permuted if we go around one of these 10 loops. The group generated by these permutations of the 27 lines (called the "monodromy group") is famously the Weyl group of type E6. We'll try to keep things as elementary as possible. Graduate students are very welcome.

Here is a quote from Ravi Vakil's Algebraic Geometry textbook:

Since the middle of the nineteenth century, geometers have been entranced by the fact that there are 27 lines on every smooth cubic surface, and by the remarkable structure of the configuration of the lines. Their discovery by Cayley and Salmon in 1849 has been called the beginning of modern algebraic geometry, [I. Dolgachev: Luigi Cremona and Cubic surfaces. P. 55]

Decomposing a representation into its constituent parts is a popular activity with long history. A map from a Lie (super)algebra \(\mathfrak{g}\) to an associative (super)algebra \(A\) provides a functor from \(A\)–modules to \(\mathfrak{g}\)–modules via pullback. The *restriction problem* is to decompose \(A\)–module into sums of \(\mathfrak{g}\)–modules. The case when \(A\) is an enveloping algebra is related to *branching rules*. When \(A=U(\mathfrak{g})\otimes U(\mathfrak{g})\) we have *Clebsch-Gordan* type theorems.
When \(\mathfrak{g}\) is reductive, there is a tool known as the *reduction algebra* (or step algebra, or transvector algebra,...) associated to the pair \((\mathfrak{g},A)\). Understanding its modules provides a way to write down not only what summands appear but also explicit intertwining operators.

In this talk I will report on some recent progress for the so-called *differential reduction algebras*, where \(A\) is a tensor product of the enveloping algebra of \(\mathfrak{g}\) with an algebra of differential operators, for the case when \(\mathfrak{g}\) is a symplectic Lie algebra.
The main result is a presentation of the associated reduction algebra. It turns out to belong to a well-known class of algebras called *generalized Weyl algebras*. Therefore its modules are well-known, allowing intertwining operators for certain infinite-dimensional \(\mathfrak{g}\)–modules to be written down.

The case \(\mathfrak{g}=\mathfrak{gl}_n\) has been previously considered in papers by Herlemont, Ogievetsky, and Khoroshkin.

This is based on joint work with Dwight Williams II.

I will introduce Graded Moebius Algebras of a matroid (resp. hyperplane arrangement, lattice, graph). These algebras were integral in the proof of the Dowling-Wilson Top Heavy Conjecture by Braden, Huh, Matherne, Proudfoot and Wang, which states that in a geometric lattice of rank \(r\) (i.e. lattice of flats of a matroids or intersection lattice of an arrangement), the number of elements of rank \(i \le r/2\) is at most the number of elements of rank \(r – i\). In the representable case, when the lattice corresponds to the intersection lattice of a hyperplane arrangement over \(\mathbb{C}\), the graded Moebius algebra is the cohomology ring of something called the Matroid Schubert Variety of the arrangement – a highly coordinatized compactification of the ambient space in a product of projective lines determined by the arrangement. I will present recent work with Adam LaClaire, Matt Mastroeni, and Irena Peeva regarding presentations of the Graded Moebius Algebra, Groebner bases, and the Koszul property. In particular, when \(L\) is the lattice associated to a graphic arrangement of some simple graph \(G\), the graded Moebius algebra is quadratic if and only if \(G\) is chordal and is Koszul if and only if \(G\) is strongly chordal. Along the way, we produce a new characterization of strongly chordal graphs via edge orderings (which I will discuss in the Discrete Math Seminar on 3/22 for anyone interested). This both parallels and breaks from a similar theory for Orlik-Solomon algebras, which describe the cohomology ring of the complement of a hyperplane arrangement.

A differential module is a module equipped with a square-zero endomorphism. Differential modules are natural generalizations of chain complexes and are objects that thus arise throughout algebra and algebraic geometry/topology. In order to understand how classical homological algebraic notions generalize to the setting of differential modules, we explore the relationship between differential modules and free complexes by studying differential modules as deformations of free resolutions. As an application, we gain some insight on "rank conjectures" in commutative algebra and algebraic topology.

I will say a few words about the role of dimensional reduction in fundamental physics. The idea is that complicated phenomena can look simpler when explained as coming from a higher-dimensional setting. Examples include theories of Maxwell, Nordström, Kaluza, Einstein-Yang-Mills, and super KK theories.

Given an associative algebra \(A\) and subalgebra \(\Gamma\), a Harish-Chandra module is an \(A\)–-module which is a direct sum of its generalized weight spaces with respect to \(\Gamma\). Drozd, Futorny, and Ovsienko (1994) used techniques from the representation theory of linear categories to study Harish-Chandra modules when \(\Gamma\) is a generalization of a central subalgebra called a Harish-Chandra subalgebra. We present the new framework of Harish-Chandra block modules, allowing us to drop this assumption, and generalize the results of DFO to this setting. Additionally, we explore the role of the Yoneda embedding in categories of Harish-Chandra-like modules.

Given any semisimple Kac-Moody algebra \(\mathfrak{g}\) and any dominant integral weight \(\lambda\) of \(\mathfrak{g}\), I will generalise the Lie superalgebra of polynomial vector fields on \(n\) odd coordinates. The Lie superalgebra of generalised vector fields has a \(\mathbb{Z}\)–grading where a central extension of \(\mathfrak{g}\) generalises \(\mathfrak{gl}(n)\) at degree \(0\), and the irreducible module with lowest weight \(-\lambda\) is a subspace at degree \(1\), generalising the \(n\)–-dimensional fundamental module of \(\mathfrak{gl}(n)\). Some well known Lie superalgebras appear as special cases, but also new ones that have not been studied before. The construction involves a non-associative generalisation of the Clifford algebra on \(2n\) generators. It can be used to define a generalised Lie derivative, which in turn has been useful in extended gravity theories in physics, where ordinary diffeomorphisms are unified with additional gauge transformations.

Links: Slides

In 2000, Mike Stillman conjectured that the projective dimension of a homogeneous ideal in a standard graded polynomial ring can be bounded just in terms of the number and degrees of its generators. I’ll describe the Ananyan-Hochster principle important to its proof, how to package this up using ultraproducts, and use this to characterize the polynomial rings graded by any abelian group that possess a Stillman bound.

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