Past Junior Algebra and Representation Theory Seminar

22 October 2021
14:00
James Timmins
Abstract

The Krull dimension is an ideal-theoretic invariant of an algebra. It has an important meaning in algebraic geometry: the Krull dimension of a commutative algebra is equal to the dimension of the corresponding affine variety/scheme. In my talk I'll explain how this idea can be transformed into a tool for measuring non-commutative rings. I'll illustrate this with important examples and techniques, and describe what is known for Iwasawa algebras of compact $p$-adic Lie groups.

  • Junior Algebra and Representation Theory Seminar
18 June 2021
14:00
Mike Daas
Abstract

It is a well-known fact that Boolean rings, those rings in which $x^2 = x$ for all $x$, are necessarily commutative. There is a short and completely elementary proof of this. One may wonder what the situation is for rings in which $x^n = x$ for all $x$, where $n > 2$ is some positive integer. Jacobson and Herstein proved a very general theorem regarding these rings, and the proof follows a widely applicable strategy that can often be used to reduce questions about general rings to more manageable ones. We discuss this strategy, but will also focus on a different approach: can we also find ''elementary'' proofs of some special cases of the theorem? We treat a number of these explicit computations, among which a few new results.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
11 June 2021
14:00
Duncan Laurie
Abstract

The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.

Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.

In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.

Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.

Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:

  •  Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
  •  Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$

The majority of my talk will then aim to address each of these points.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
4 June 2021
14:00
Kamilla Rekvényi
Abstract

Let $G$ be a group acting transitively on a finite set $\Omega$. Then $G$ acts on $\Omega \times \Omega$ componentwise. Define the orbitals to be the orbits of $G$ on $\Omega \times \Omega$. The diagonal orbital is the orbital of the form $\Delta = \{(\alpha, \alpha) \mid \alpha \in \Omega \}$. The others are called non-diagonal orbitals. Let $\Gamma$ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set $\Omega$ and edge set $(\alpha,\beta) \in \Gamma$ with $\alpha, \beta \in \Omega$. If the action of $G$ on $\Omega$ is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups. 

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
21 May 2021
14:00
Finn Wiersig
Abstract

How to calculate the minimal number of summands of a nonzero polynomial in a given polynomial ideal? In this talk, we first discuss the roots of this question in computational algebra. Afterwards, we switch to the viewpoint of a commutative algebraist. In particular, we see that classical tools from this field, such as primary decomposition or the Castelnuovo–Mumford regularity, fail to provide a solution to this problem. Finally, we discuss a concrete example: A standard determinantal ideal generated by $t$-minors does not contain any polynomials with fewer than $t!/2$ terms.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
14 May 2021
14:00
Nadav Gropper
Abstract

Anabelian geometry asks how much we can say about a variety from its fundamental group. In 1997 Shinichi Mochizuki, using p-adic hodge theory, proved a fundamental anabelian result for the case of p-adic fields. In my talk I will discuss representation theoretical data which can be reconstructed from an absolute Galois group of a field, and also types of representations that cannot be constructed solely from a Galois group. I will also sketch how these types of ideas can potentially give many new results about p-adic Galois representations.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
7 May 2021
14:00
Robert Spencer
Abstract

The Temperley-Lieb algebra is a diagrammatic algebra - defined on a basis of "string diagrams" with multiplication given by "joining the diagrams together".  It first arose as an algebra of operators in statistical mechanics but quickly found application in knot theory (where Jones used it to define his famed polynomial) and the representation theory of $sl_2$.  From the outset, the representation theory of the Temperley-Lieb algebra itself has been of interest from a physics viewpoint and in characteristic zero it is well understood.  In this talk we will explore the representation theory over mixed characteristic (i.e. over positive characteristic fields and specialised at a root of unity).  This gentle introduction will take the listener through the beautifully fractal-like structure of the algebras and their cell modules with plenty of examples.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
12 March 2021
14:00
Tom Zielinski
Abstract

Introduced by Fomin and Zelevinsky in 2002, cluster algebras have become ubiquitous in algebra, combinatorics and geometry. In this talk, I'll introduce the notion of a cluster algebra and present the approach of Kang-Kashiwara-Kim-Oh to categorify a large class of them arising from quantum groups. Time allowing, I will explain some recent developments related to the coherent Satake category.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar
5 March 2021
14:00
Miriam Norris
Abstract

For $G$ a finite group, $p$ a prime and $(K, \mathcal{O}_K, k)$ a $p$-modular system the group ring $\mathcal{O}_K G$ is an $\mathcal{O}_k$-order in the $K$-algebra $KG.$ Graduated $\mathcal{O}_K$-orders are a particularly nice class of $\mathcal{O}_K$-orders first introduced by Zassenhaus. In this talk will see that an $\mathcal{O}_K$-order $\Lambda$ in a split $K$-algebra $A$ is graduated if the decomposition numbers for the regular $A$-module are no greater than $1$. Furthermore will see that graduated orders can be described (not uniquely) by a tuple $n$ and a matrix $M$ called the exponant matrix. Finding a suitable $n$ and $M$ for a graduated order $\Lambda$ in the $K$-algebra $A$ provides a parameterisation of the $\Lambda$-lattices inside the regular $A$-module. Understanding the $\mathcal{O}_K G$-lattices inside representations of certain groups $G$ is of interest to those involved in the Langlands programme as well as of independent interest to algebraists.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Algebra and Representation Theory Seminar

Pages