Past Algebra Seminar

15 November 2016
Greg Stevenson

 For a group algebra over a self-injective ring
there are two stable categories: the usual one modulo projectives
and a relative one where one works modulo representations
which are free over the coefficient ring.
I'll describe the connection between these two stable categories,
which are "birational" in an appropriate sense.
I'll then make some comments on the specific case
where the coefficient ring is Z/nZ and give a more
precise description of the relative stable category.

8 November 2016
Max Gurevich

What are the irreducible constituents of a smooth representation of a p-adic group that is constructed through parabolic induction? In the case of GL_n this is the study of the multiplicative behaviour of irreducible representations in the Bernstein-Zelevinski ring. Strikingly, the same decomposition problem can be reformulated through various Lie-theoretic settings of type A, such as canonical bases in quantum groups, representations of affine Hecke algebras, quantum affine Lie algebras, or more recently, KLR algebras. While partially touching on some of these phenomena, I will present new results on the problem using mostly classical tools. In particular, we will see how introducing a width invariant to an irreducible representation can circumvent the complexity involved in computations of Kazhdan-Lusztig polynomials.

1 November 2016
J Grabowski

It has long been expected, and is now proved in many important cases, 
that quantum algebras are more rigid than their classical limits. That is, they 
have much smaller automorphism groups. This begs the question of whether this 
broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative 
projective geometry, from which we see that the correct object to study is a 
groupoid, rather than a group, and maps in this groupoid are the replacement 
for automorphisms. I will illustrate this with the example of quantum 
projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

25 October 2016

In this talk, we consider a split connected semisimple group G defined over a global field F. Let A denote the ring of adèles of F and K a maximal compact subgroup of G(A) with the property that the local factors of K are hyperspecial at every non-archimedian place. Our interest is to study a certain subspace of the space of square-integrable functions on the adelic quotient G(F)\G(A). Namely, we want to study functions coming from induced representations from an unramified character of a Borel subgroup and which are K-invariant.

Our goal is to describe how the decomposition of such space can be related with the Plancherel decomposition of a graded affine Hecke algebra (GAHA).

The main ingredients are standard analytic properties of the Dedekind zeta-function as well as known properties of the so-called residue distributions, introduced by Heckman-Opdam in their study of the Plancherel decomposition of a GAHA and a result by M. Reeder on the support of the weight spaces of
the anti-spherical  discrete series representations of affine Hecke algebras. These last ingredients are of a purely local nature.

This talk is based on joint work with V. Heiermann and E. Opdam.

18 October 2016
Lisa Lamberti

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional. 
In this talk, I report on recent developments in the 3-dimensional setting of this theory.

11 October 2016
Petter Bergh

We define categorical matrix factorizations in a suspended additive category, 
with respect to a central element. Such a factorization is a sequence of maps 
which is two-periodic up to suspension, and whose composition equals the 
corresponding coordinate map of the central element. When the category in 
question is that of free modules over a commutative ring, together with the 
identity suspension, then these factorizations are just the classical matrix 
factorizations. We show that the homotopy category of categorical matrix 
factorizations is triangulated, and discuss some possible future directions. 
This is joint work with Dave Jorgensen.

1 June 2016

Torsion in homology are invariants that have received increasing attention over the last twenty years, by the work of Lück, Bergeron, Venkatesh and others. While there are various vanishing results, no one has found a finitely presented group where the torsion in the first homology is exponential over a normal chain with trivial intersection. On the other hand, conjecturally, every 3-manifold group should be an example.

A group is right angled if it can be generated by a list of infinite order elements, such that every element commutes with its neighbors. Many lattices in higher rank Lie groups (like SL(n,Z), n>2) are right angled. We prove that for a right angled group, the torsion in the first homology has subexponential growth for any Farber sequence of subgroups, in particular, any chain of normal subgroups with trivial intersection. We also exhibit right angled cocompact lattices in SL(n,R) (n>2), for which the Congruence Subgroup Property is not known. This is joint work with Nik Nikolov and Tsachik Gelander.

24 May 2016
Dawid Kielak

We will introduce the Thurston norm in the setting of 3-manifold groups, and show how the techniques coming from L2-homology allow us to extend its definition to the setting of free-by-cyclic groups.
We will also look at the relationship between this Thurston norm and the Alexander norm, and the BNS invariants, in particular focusing on the case of ascending HNN extensions of the 2-generated free group.

17 May 2016
Michael Collins

In 1878, Jordan showed that there is a function f on the set of natural numbers such that, if $G$ is a finite subgroup of $GL(n,C)$, then $G$ has an abelian normal subgroup of index at most $f(n)$. Early bounds were given by Frobenius and Schur, and close to optimal bounds were given by Weisfeiler in unpublished work in 1984 using the classification of finite simple groups; about ten years ago I obtained the optimal bounds. Crucially, these are "absolute" bounds; they do not address the wider question of divisibility of orders.

In 1887, Minkowski established a bound for the order of a Sylow p-subgroup of a finite subgroup of GL(n,Z). Recently, Serre asked me whether I could obtain Minkowski-like results for complex linear groups, and posed a very specific question. The answer turns out to be no, but his suggestion is actually quite close to the truth, and I shall address this question in my seminar. The answer addresses the divisibility issue in general, and it turns out that a central technical theorem on the structure of linear groups from my earlier work which there was framed as a replacement theorem can be reinterpreted as an embedding theorem and so can be used to preserve divisibility.