Forthcoming events in this series


Tue, 12 Jun 2018
14:15
L4

Decomposition spaces: theory and applications

Andrew Tonks
(Leicester)
Abstract


Decomposition (aka unital 2-Segal) spaces are simplicial ∞-groupoids with a certain exactness property: they take pushouts of active (end-point preserving) along inert (distance preserving) maps in the simplicial category Δ to pullbacks. They encode the information needed for an 'objective' generalisation of the notion of incidence (co)algebra of a poset, and motivating examples include the decomposition spaces for (derived) Hall algebras, the Connes-Kreimer algebra of trees and Schmitt's algebra of graphs. In this talk I will survey recent activity in this area, including some work in progress on a categorification of (Hopf) bialgebroids.
This is joint work with Imma Gálvez and Joachim Kock.
 

Tue, 22 May 2018

14:15 - 15:30
L4

g-algebras and the representations of their invariant subrings.

Anthony Joseph
(Weizmann Institute)
Abstract

Let $\mathfrak g$ be a semisimple Lie algebra.  A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations.  There are several significant examples.  Let $V$ a finite dimensional $\mathfrak g$ module and take  $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on  $V$ . Again take $R=U(\mathfrak g)$.   In all these cases  $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra.  Finally let $T$ denote the subalgebra of invariants of $S$.
 
For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials.  In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules.  In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring,  except for the case $\mathfrak  g =\mathfrak {sl}(2)$.

Tue, 24 Apr 2018

14:15 - 15:15
L4

Short Laws for Finite Groups and Residual Finiteness Growth

Henry Bradford
(Goettingen)
Abstract

 A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

Wed, 07 Mar 2018

10:00 - 12:00
L5

Hall algebras of coherent sheaves on toric varieties over F_1.

Prof. Matt Szczesny
(Boston University)
Abstract

Hall algebras of categories of quiver representations and coherent sheaves

on smooth projective curves over F_q recover interesting

representation-theoretic objects such as quantum groups and their

generalizations. I will define and describe the structure of the Hall

algebra of coherent sheaves on a projective variety over F_1, with P^2 as

the main example. Examples suggest that it should be viewed as a degenerate

q->1 limit of its counterpart over F_q.

Tue, 06 Mar 2018
14:15
L4

Morita equivalence of Peter-Weyl Iwahori algebras

Allen Moy
(Hong Kong University of Science and Technology)
Abstract

The Peter-Weyl idempotent of a parahoric subgroup is the sum of the idempotents of irreducible representations which have a nonzero Iwahori fixed vector. The associated convolution algebra is called a Peter-Weyl Iwahori algebra.  We show any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra.  Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algebra have a natural C*-algebra structure, and the Morita equivalence preserves irreducible hermitian and unitary modules.  Both algebras have another anti-involution denoted as •, and the Morita equivalence preserves irreducible and unitary modules for the • involution.   This work is joint with Dan Barbasch.
 

Tue, 27 Feb 2018
14:15
L4

The regular representations of GL_N over finite local principal ideal rings

Alexander Stasinski
(Durham University)
Abstract

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O$ and maximal ideal $\mathfrak p$. T. Shintani and G. Hill independently introduced a large class of smooth representations of $GL_N(\mathcal O)$, called regular representations. Roughly speaking they correspond to elements in the Lie algebra $M_N(\mathcal O)$ which are regular mod $\mathfrak p$ (i.e, having centraliser of dimension $N$). The study of regular representations of $GL_N(\mathcal O)$ goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of $GL_N(\mathcal O)$ when the residue characteristic of $\mathcal O$ is not $2$.

In this talk I will present a complete construction of all the regular representations of $GL_N(\mathcal O)$. The approach is analogous to, and motivated by, the construction of supercuspidal representations of $GL_N(F)$ due to Bushnell and Kutzko. This is joint work with Shaun Stevens.
 

Tue, 06 Feb 2018
14:15
L4

Dual singularities in exceptional type nilpotent cones

Paul Levy
(University of Lancaster)
Abstract

It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.
It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.
 

Tue, 30 Jan 2018

14:15 - 15:15
L4

2D problems in groups

Nikolay Nikolov
(Oxford University)
Abstract
I will discuss a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem for group presentations.
We can prove the pro-$p$ version of the conjecture, as well as its higher dimensional abstract analogues. Key ingredients are, first a classic result of Wall on the existence of CW complexes with prescribed cellular chain complex, and second, a simple criterion for freeness of modules over group rings. This is joint work with Aditi Kar.
Tue, 28 Nov 2017
14:15
L4

Dirac induction for rational Cherednik algebras

Marcelo De Martino
(Oxford University)
Abstract

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.

Tue, 14 Nov 2017

14:15 - 15:15
L4

Representations of pseudo-reductive groups

Dr David Stewart
(School of Mathematics & Statistics Newcastle University)
Abstract

Pseudo-reductive groups are smooth connected linear algebraic groups over a field k whose k-defined unipotent radical is trivial. If k is perfect then all pseudo-reductive groups are reductive, but if k is imperfect (hence of characteristic p) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by Conrad-Gabber-Prasad has made it possible to reduce the classification of the simple representations of pseudo-reductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.

Tue, 31 Oct 2017
14:15
L4

Multiplicity-free primitive ideals and W-algebras

Alexander Premet
(University of Manchester)
Abstract

In my talk I will explain how to relate 1-dimensional representations of finite W-algebras with multiplicity free primitive ideals of universal enveloping algebras and representations of minimal dimension of the corresponding reduced enveloping algebras (Humphreys' conjecture). I will also mention some open problems in the field.

Tue, 24 Oct 2017

14:15 - 15:15
L4

Dimers with boundary, associated algebras and module categories

Karin Baur
(Graz)
Abstract

Dimer models with boundary were introduced in joint work with King and Marsh as a natural
generalisation of dimers. We use these to derive certain infinite dimensional algebras and
consider idempotent subalgebras w.r.t. the boundary.
The dimer models can be embedded in a surface with boundary. In the disk case, the
maximal CM modules over the boundary algebra are a Frobenius category which
categorifies the cluster structure of the Grassmannian.

 

Tue, 17 Oct 2017

14:15 - 15:15
L4

From classical tilting to 2-term silting

Aslak Buan
(Trondheim)
Abstract

We give a short reminder about central results of classical tilting theory, 
including the Brenner-Butler tilting theorem, and
homological properties of tilted and quasi-tilted algebras. We then discuss 
2-term silting complexes and endomorphism algebras of such objects,
and in particular show that some of these classical results have very natural 
generalizations in this setting.
(joint work with Yu Zhou)

Tue, 16 May 2017
14:15
L4

Cherednik algebras at infinity

Maxim Nazarov
(York University)
Abstract

Heckman introduced N operators on the space of polynomials in N variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these N operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. We compute the limits of our operators at N → ∞ . These limits yield a Lax operator for Macdonald symmetric functions. This is a joint work with Evgeny Sklyanin.

Wed, 03 May 2017

14:00 - 15:00
L3

On finiteness properties of the Johnson filtrations

Mikhail Ershov
(Virginia)
Abstract

Let $A$ denote either the automorphism group of the free group of rank $n$ or the mapping class group of an orientable surface of genus $n$ with at most 1 boundary component, and let $G$ be either the subgroup of IA-automorphisms or the Torelli subgroup of $A$, respectively. I will discuss various finiteness properties of subgroups containing $G_N$, the $N$-th term of the lower central series of $G$, for sufficiently small $N$. In particular, I will explain why
(1) If $n \geq 4N-1$, then any subgroup of G containing $G_N$ (e.g. the $N$-th term of the Johnson filtration) is finitely generated
(2) If $n \geq 8N-3$, then any finite index subgroup of $A$ containing $G_N$ has finite abelianization.
The talk will be based on a joint work with Sue He and a joint work with Tom Church and Andrew Putman

Tue, 02 May 2017
14:15
L4

Representations of p-adic groups via geometric invariant theory

Beth Romano
(Cambridge University)
Abstract

Let G be a split reductive group over a finite extension k of Q_p. Reeder and Yu have given a new construction of supercuspidal representations of G(k) using geometric invariant theory. Their construction is uniform for all p but requires as input stable vectors in certain representations coming from Moy-Prasad filtrations. In joint work, Jessica Fintzen and I have classified the representations of this kind which contain stable vectors; as a corollary, the construction of Reeder-Yu gives new representations when p is small. In my talk, I will give an overview of this work, as well as explicit examples for the case when G = G_2. For these examples, I will explicitly describe the locus of all stable vectors, as well as the Langlands parameters which correspond under the local Langlands correspondence to the representations of G(k). 

Tue, 07 Mar 2017
14:15
L4

The rationality of blocks of quasi-simple finite groups

Niamh Farrell
(City University London)
Abstract

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Morita Frobenius numbers were introduced by Kessar in 2004 in the context of Donovan’s Conjecture in block theory. I will present the latest results of a project in which we aim to calculate the Morita Frobenius numbers of the blocks of quasi-simple finite groups. I will also discuss the importance of a recent result of Bonnafe-Dat-Rouquier for our methods, and explain the relationship between Morita Frobenius numbers and Donovan’s Conjecture. 

Tue, 28 Feb 2017
14:15
L4

Sklyanin algebras are minimal surfaces

Sue Sierra
(University of Edinburgh)
Abstract

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is:  what are the minimal models within a birational class?  It is not even clear a priori what the correct definition is of a minimal model in this context.

We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.
This is joint work in progress with Dan Rogalski and Toby Stafford.

 

Tue, 21 Feb 2017

14:15 - 15:15
L4

Growth, generation, and conjectures of Gowers and Viola

Aner Shalev
(Hebrew University of Jerusalem)
Abstract

I will discuss recent results in finite simple groups. These include growth, generation (with a number theoretic flavour), and conjectures of Gowers and Viola on mixing and complexity whose proof requires representation theory as a main tool.
 

Tue, 07 Feb 2017
14:15
L4

Modular W-algebras and reduced enveloping algebras

Simon Goodwin
(University of Birmingham)
Abstract

We give an overview of joint work with Lewis Topley on modular W-algebras. In particular, we outline the classification 1-dimensional modules for modular W-algebras for gl_n, which in turn this leads to a classification of minimal dimensional modules for reduced enveloping algebras for gl_n.

Tue, 24 Jan 2017

14:15 - 15:15
L4

An Euler-Poincare formula for a depth zero Bernstein projector

Allen Moy
(Hong Kong University of Science and Technology)
Abstract


Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an
Euler-Poincare formula for the r-depth Bernstein projector. We establish an Euler-Poincare formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P.  This work is joint with Dan Barbasch and Dan Ciubotaru.
 

Tue, 17 Jan 2017

14:15 - 15:15
L4

Endo-parameters and the Local Langlands Correspondence for classical groups

Shaun Stevens
(University of East Anglia)
Abstract

The local Langlands correspondence for classical groups gives a natural finite-to-one map between certain representations of p-adic classical groups and certain self-dual representations of the absolute Weil group of a p-adic field (and more). On both sides of the correspondence, the description of the representations involves a ``wild part'' of more arithmetic nature and a ``tame part'' of more geometric nature, and the notion of endo-parameter (due to Bushnell--Henniart for general linear groups) is designed to describe the ``wild part'' of the Langlands correspondence. I will explain what this means and the connection with representations of affine Hecke algebras. This is joint work with Blondel--Henniart, with Lust, and with Kurinczuk--Skodlerack.

Tue, 15 Nov 2016

14:15 - 15:15
L4

Representations of finite groups over self-injective rings

Greg Stevenson
(Bielefeld)
Abstract

 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.

Tue, 08 Nov 2016
14:15
L4

Decomposition rules for representations of p-adic groups

Max Gurevich
(Weizmann Institute)
Abstract


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.