# 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 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

**12th November**: Rachel Skipper

**19th November: **Andrea Lucchini

**26th November: **James Taylor

3rd December:

10th December: