# Algebra Seminar

### Seminars are held on Thursdays at 15:00 on Zoom (jointly with the University of Birmingham)

#### Organisers: Adam Thomas, Inna Capdeboscq and Dmitriy Rumynin.

#### Term 2:

**21st January: **Bettina Eick (Braunschweig)

Title: Nilpotent Associative Algebras and Coclass Theory

This talk surveys the classification of nilpotent associative algebras using the coclass as primary invariant. The talk introduces the coclass and gives an introduction into the topic.

**28th January: **Julianna Tymoczko (Smith College)

Title: Comparing different bases for irreducible symmetric group representations

We describe two different bases for irreducible symmetric group representations: the tableaux basis from combinatorics (and from the geometry of a class of varieties called Springer fibers); and the web basis from knot theory (and from the quantum representations of Lie algebras). We then describe new results comparing the bases, e.g. showing that the change-of-basis matrix is upper-triangular, and sketch some applications to geometry and representation theory. This work is joint with H. Russell, as well as with T. Goldwasser and G. Sun.

**4th February: **Pablo Spiga (Milan-Bicocca)

Title: How vertex-stabilizers grow?

Here we are interested in highly symmetric graphs. (All basic terminology will be given during the talk.) There are various natural ways to “measure” the degree of symmetry of a graph and, in this talk, we look at two possibilities. First, we consider graphs Γ having a group of automorphisms acting transitively on the paths of length s ≥ 1, starting at a given vertex. The larger the value of s is, the more symmetric the graph will be. However, we show that large values of s impose severe restrictions on the structure of Γ and on the size of the stabilizer of a vertex of Γ. This will lead us to the second perspective. We take the size of the stabilizer of a vertex of Γ as a measure of the transitivity. This measure is somehow unbiased among the graphs having the same number of vertices. Again we present some results showing, in some very speciﬁc cases, that nature is not as diverse as one might expect: graphs have either rather small vertex stabilizers or they can be classiﬁed. Finally we give some applications of these investigations: to the enumeration problem of symmetric graphs and to the problem of creating a database of small symmetric graphs.

**11th February: **Matthew Young (Utah)

Title: Green-type theorems for representations of Hall algebras

Associated to a suitably finite abelian category is its Hall algebra, an associative algebra which encodes the first order extension structure of the category. A choice of duality structure on the category defines a module over the Hall algebra which encodes the first order orthogonal/symplectic extension structure. The goal of this talk is to introduce this class of modules and explain their relevance to the theory of quantum groups and Donaldson-Thomas theory. A key gap in our understanding is the lack of result describing the compatibility of the module structure with a natural comodule structure; the analogous result for Hall algebras is called Green’s theorem. I will describe such a Green-type theorem in the simpler setting of categories which are linear over the field with one element, and discuss its potential implications for Donaldson-Thomas theory.

**18th February: **Ulrich Theil (Kaiserslautern)

Title: Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 – the symplectically primitive but complex imprimitive groups – and 10 exceptional groups up to dimension 10, for which it is still open. Recently, we have proven that for all but possibly 39 cases in the remaining infinite series there is no symplectic resolution. We have thereby reduced the classification problem to finitely many open cases. We do not expect any of the remaining cases to admit a symplectic resolution. This is joint work with Gwyn Bellamy and Johannes Schmitt.

**25th February: **Alex Malcolm (Bristol)

Title: Finite simple groups, prime order elements and width

The generation of finite simple groups has been a thriving area of research for many years. Since it was established that each is generated by a pair of elements, many interesting refinements have followed: for instance, determining the existence of generating pairs of prescribed orders.

More recently the notion of width has provided an additional perspective on generation, measuring how efficiently a chosen subset generates a group. For example we may ask, can every element be written as a product of at most 2, or perhaps 3, elements from a fixed conjugacy class? Answering such questions relies on a range of tools involving subgroup structure and character theory.

In this talk we will examine the width of finite simple groups with respect to elements of a fixed prime order. We will report on sharp bounds for particular families, and answer questions concerning Lie-type groups of unbounded rank.

**4th March: **Niamh Farrell (Hannover)

Title: Decomposition matrices of quasi-isolated blocks of exceptional groups

I will discuss ongoing work in the calculation of the decomposition matrices of finite groups of Lie type. Much of what is known in this well-established field relates to unipotent blocks. I will talk about adapting the unipotent methods to the case of quasi-isolated blocks of exceptional groups.

**11th March: **Daniel Tubbenhauer (Bonn)

Title: SL2 and fractals

In this friendly introduction to modular representation theory of reductive groups I will explain in what sense fractal patterns arise when working in finite characteristic, with the emphasis on representations of SL2 .

**18th March: **Mikhail Belolipetsky (IMPA, Rio de Janeiro)

Title: Growth of lattices in semisimple Lie groups

A discrete subgroup G of a Lie group H is called a lattice if the quotient space H/G has finite volume. By a classical theorem of Bieberbach we know that the group of isometries of an n-dimensional Euclidean space has only finitely many different types of lattices. The situation is different for the semisimple Lie groups H. Here the total number of lattices is infinite and we can study its growth rate with respect to the covolume. This topic has been a subject of our joint work with A. Lubotzky for a number of years. In the talk I will discuss our work and some other more recent related results.

**25th March: **Stefano Scalese (Manchester)

Title: Irreducible representations of restricted Lie algebras

In this talk, we recall the notions of [p]-mapping and restricted Lie algebras over an algebraically closed field in positive characteristic. We will present the first Kac-Weisfeiler conjecture (KW1), which gives a bound on the maximal dimension of the irreducible representations of finite-dimensional restricted Lie algebras. We will explain what we already know about KW1 and our current progress, as we aim to prove KW1 for Lie algebras of algebraic groups with some assumptions on the characteristic of the field.

#### Term 1:

**1st October:** Oliver Dudas (IMJ-PRG)

Title: Macdonald polynomials and decomposition numbers for finite unitary groups.

(work in progress with R. Rouquier) I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. These matrices encode how ordinary representations decompose when they are reduced to a field with positive characteristic \ell. There is an algorithm to compute them for GL(n,q) when \ell is large enough, but finding these matrices for other groups of Lie type is a very challenging problem.

In this talk I will focus on the finite general unitary group GU(n,q). I will first explain how one can produce a “natural” self-equivalence in the case of GL(n,q) coming from the topology of the Hilbert scheme of the complex plane . The combinatorial part of this equivalence is related to Macdonald’s theory of symmetric functions and gives (q,t)-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence, which encodes the relation between GU(n,q) and GL(n,-q).

**8th October:** Filippo Ambrosio (Padova)

Title: Birational sheets of conjugacy classes in reductive groups

If G is an algebraic group acting on a variety X, the sheets of X are the irreducible components of subsets of elements of X with equidimensional G-orbits. For G complex connected reductive, the sheets for the adjoint action of G on its Lie algebra g were studied by Borho and Kraft in 1979. More recently, Losev has introduced finitely-many subsets of g consisting of equidimensional orbits, called birational sheets: their definition is not as immediate as the one of a sheet, but birational sheets behave better in geometric and representation-theoretic terms. Indeed, birational sheets are disjoint, unibranch varieties with smooth normalization, while this is not true for sheets, in general. Moreover, the G-module structure of the ring of functions C[O] does not change as the orbit O varies in a birational sheet. In this seminar, we define an analogue of birational sheets of conjugacy classes in G: we start by recalling Lusztig-Spaltenstein induction of conjugacy classes in terms of the so-called Springer generalized map and analyse its interplay with birationality. With this tools, we give a definition of birational sheets of G in the case that the derived subgroup of G is simply connected. We conclude with an overview of the main features of these varieties, which mirror some of the properties enjoyed by the objects defined by Losev.

**15th October:** Noelia Rizo (University of the Basque Country)

Title: On the number of characters in the principal p-block

(Joint work with A. Schaeffer Fry and Carolina Vallejo) Let G be a finite group, let p be a prime number and let B be a p-block of G with defect group D. Studying the structure of D by means of the knowledge of some aspects of B is a main area in character theory of finite groups. Let k(B) be the number of irreducible characters in the p-block B. It is well-known that k(B)=1 if, and only if, D is trivial. It is also true that k(B)=2 if, and only if, D \cong C_2. For blocks B with k(B)=3 it is conjectured that D \cong C_3. In this talk we restrict our attention to the principal p-block of G, B_0(G), that is, the p-block containing the trivial character of G. It is well known that the defect groups of the principal block of G are exactly the Sylow p-subgroups of G. In this case it is even true that k(B_0(G))=3 if, and only if, D \cong C_3. Recently, Koshitani and Sakurai have shown that k(B_0(G))=4 implies that D in {C_2 x C_2, C_4, C_5}. In this work we go one step further and analyse the isomorphism classes of Sylow p-subgroups of groups G for which B_0(G) has exactly 5 irreducible characters. In particular we show that if k(B_0)=5, then D in {C_5, C_7, D_8, Q_8}.

**22nd October**: Pham Tiep (Rutgers)

Title: Character bounds for finite groups of Lie type

We will discuss new bounds on character values for finite groups of Lie type, obtained in recent work of the speaker and collaborators. Some applications of these character bounds will be also described.

**29th October:** Katrin Tent (Münster)

Title: Defining R and G(R)

In joint work with Segal we use the fact that for Chevalley groups G(R) of rank at least 2 over a ring R the root subgroups are (nearly always) the double centralizer of a corresponding root element to show for many important classes of rings and fields that R and G(R) are bi-interpretable. For such groups it then follows that the group G(R) is finitely axiomatizable in the appropriate class of groups provided R is finitely axiomatizable in the corresponding class of rings.

**5th November: **Anne Moreau (Paris-Saclay)

Title: Nilpotent orbits arising from admissible affine vertex algebras

In this talk, I will give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.

This is a joint work with Tomoyuki Arakawa and Jethro Van Ekeren.

**12th November**: Rachel Skipper (Ohio)

Title: Generating lamplighter groups with bireversible automata

This is a joint work with Benjamin Steinberg.

**19th November: **Andrea Lucchini (Padova)

Title: Intersections of maximal subgroups in finite groups

We will investigate a series of questions in group theory which, although with different motivations, are all related with the study of lattice M(G) consisting of the subgroups of a finite group that can be obtained as intersection of maximal subgroups.

**26th November: **James Taylor (Oxford)

Title: Real Representations of C_2 - Graded Groups

A Real representation of a C_2-graded group H < G (H an index two subgroup) is a complex representation of H with an action of the other coset G \ H (“odd” elements) satisfying appropriate algebraic coherence conditions. In this talk I will present three such Real representation theories. In these, each odd element acts as an antilinear operator, a bilinear form or a sesquilinear form (equivalently a linear map to V from the conjugate, the dual, or the conjugate dual of V) respectively. I will describe how these theories are related, how representations in each are classified, and how the first generalises the classical representation theory of H over the real numbers – retaining much of its beauty and subtlety. This is recent joint work with D.Rumynin.

**3rd December:** Horacia Guerra (Newcastle)

Title: The classification of the simple restricted modules of the non-restricted, non-graded Hamiltonian Lie algebra H(2; (1,1); Φ(1))

We classify the simple restricted modules for the minimal p-envelope of the non-graded, non-restricted Hamiltonian Lie algebra H(2; (1,1); Φ(1)) over an algebraically closed field k of characteristic p at least 5. We also give the restrictions of these modules to a subalgebra isomorphic to the first Witt Algebra, a result stated in S. Herpel and D. Stewart, Selecta Mathematica with an incomplete proof.

**10th December:** Chris Bowman (Kent)

Title: Tautological p-Kazhdan Lusztig Theory for cyclotomic Hecke algebras

We discuss a new explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson’s diagrammatic endomorphism algebras of Bott–Samelson bimodules. This allows us to deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. This allows us to give an elementary and explicit proof of the main theorem of Riche–Williamson’s recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky–Plaza’s categorical blob conjecture.

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