AGGITatE 2025: Groups, Representations and Cohomology

AGGITatE 2025 will take place at the University of Essex 4-7th August 2025 and focus on Groups, Representations and Cohomology.  The conference is funded by the London Mathematical Society and the Heilbronn Institute. AGGITatE is an annual workshop taking place at the University of Essex. Its aim is to bring together researchers working in algebraic geometry and algebraic groups. It has run in 2022, 2023 and 2024.

If you have any questions, please write to essexaggitate AT gmail DOT com
(Image: The gate of St John’s Abbey in Colchester, source)

Index

Plenary speakers

Contributed talks speakers

Organisers

Registration

Registration is now closed. If you want to attend, please write to essexaggitate AT gmail DOT com and we will see what we can do.

There is a registration fee of around £40 for all participants, to be paid after we confirm your attendance.

Logistics

Arrival date is Sunday 3rd August. The Conference will start on Monday 4th August in the morning and finish at about 3.00pm on 7th August. We have 12 single ensuite rooms with shared kitchen facilities, booked for 4 nights (Arrival Sunday 3rd, checking out Thursday 7th August). These are prioritised for early-career UK-based participants due to funder’s conditions. If you want to be considered for accommodation, please clearly indicate it in the form.

Transportation: The University of Essex is based in Colchester. Find here a map of the university of Essex. Please, consult the university’s website on how to reach Colchester Campus. The closest railway station is Colchester Hythe, but Colchester North (AKA as Colchester, which is different from Colchester Town) is the one with the best connections. If you use buses, please be aware they can be unreliable, so leave extra time when using them. The closest airport is Stansted and it is best connected to Colchester by bus. Travelling from Gatwick or Stansted by train is not too long (2-3 hours). The cost of a taxi within Colchester is usually £10-15. There are a number of companies (google ‘Colchester taxi’ for a list) that operate independently from each other and the best way to arrange a taxi is to call them directly. It is very difficult to book a taxi at around school ‘rush hour’ or early in the morning, unless arranged in advance.

Wednesday afternoon: we will not have scheduled lectures. However, we have a couple of activities that you can take part on: a visit of Colchester (the oldest recorded town in the United Kingdom, founded by the Romans) or a Disc Golf game. The University of Essex is the home of the first Disc Golf course in England and the eighth oldest in the world. It has been recently redesigned.

Meals: We will have a conference dinner on Tuesday and a barbeque on Wednesday, covered by the conference. We will also provide lunch Monday-Thursday and two tea breaks.

Rooms: The conference will take place in the North Teaching Centre (NTC), in rooms NTC 3.05 (lectures) and NTC 3.07 (registration, coffee breaks and lunch).

Accommodation: If your accommodation is paid by the organisers, we will be in touch with details.
If you are self-funded, the following are good options for accommodation in Colchester. All are in the route of buses 62 and 62B which take you directly to Campus in about 20 minutes. You can also walk to campus in 40-60 minutes, depending on the hotel.

Queries: Please contact essexaggitate AT gmail DOT com

Schedule

Monday 4thTuesday 5thWednesday 6thThursday 7th
09:30Welcome and registrationLinckelmannGreenleesGiannelli
10:00Malle
10:30Tea breakTea breakTea break
11:00Tea breakErdmannBensonLaw
11:30Ruhstorfer
12:00LunchLunchLunch
12:30Lunch
13:00
13:30SemeraroHenke
Kessar		Kessar
(moved to Wednesday)
14:00McDowell
14:30Praderio BovaTea breakActivity
(Disc Golf / Tour)
15:00Tea breakVallejo Rodríguez
15:30Roth
16:00WangGrazian
16:30Jones
EVENINGDinner
(7pm)		Barbecue
 (6:30pm)

Main talks details

  • Dave Benson: “Finite group schemes over symmetric tensor categories”.
    Abstract:     In the first half of the talk, I shall discuss some new incompressible finite symmetric tensor categories in prime characteristic. These appeared in joint work with Etingof and Ostrik, and were constructed using the representation theory of SL(2,k). In the second half, I shall discuss the theory of finite group schemes over a symmetric tensor category, and show how to view the tangent space at the identity as a restricted Lie algebra in a suitable sense.
  • Karin Erdmann: “Schur algebras (S(p, 2p)) and symmetric groups (mathcal{S}_{2p})”
    Abstract:     Let (K) be a field of characteristic (p). This lecture will discuss homological properties of Schur algebras (S(p, 2p)), focussing on the connection with representations of (Kmathcal{S}_{2p}), via Schur Weyl duality. In particular, we determine the global dimension of the Schur algebra, and some other homological dimensions occurring in this context. This is based on joint work with Tiago Cruz.
  • Eugenio Giannelli: “McKay bijections and character degrees”.
    Abstract: The McKay conjecture has been a central question in the representation theory of finite groups for over 60 years. A full proof was very recently achieved through thework of Cabanes and Spaeth. In this talk, I will explore new open problems that arise in connection with the conjecture and present recent results concerning the class of symmetric groups.
  • John Greenlees: “The singularity category of the enhanced group cohomology ring and Gorenstein duality for its Koszul dual“.
    Abstract: For a finite group (G) and a field k of characteristic p, a model for the enhanced group cohomology ring is the ring (C^*(BG;k)) of cochains on  the classifying space (there is a model for this as a commutative ring in the category of spectra so many concepts of commutative algebra can be applied). 

    It is known that (C^*(BG;k)) is always Gorenstein (Dwyer-Greenlees-Iyengar) and it is regular if and only if (G) is p-nilpotent. Hence it is natural to ask (joint work with G.Stevenson) about its singularity category. This has been calculated explicitly in a few cases (e.g. in work with Benson, if (G) has cyclic Sylow (p)-subgroup), but the talk is about what can be said more generally. If (G) is a complete intersection in a suitable sense, one may show it is a bounded derived category
    $$D_{sg}(C^*(BG))=D^b(TE)$$
    where (E) is the Koszul dual of (C^*(BG)) and (TE) is a Tate-like localization of (E). Finally, still assuming the complete intersection condition, (E) itself is Gorenstein and (TE) is Brown-Comenetz self-dual.  
  • Ellen Henke: Classifying fusion systems and searching for exotic ones
    Abstract:     Saturated fusion systems generalize important aspects of finite group theory, as each finite group leads to a saturated fusion systems encoding the conjugacy relations between subgroups of a fixed Sylow p-subgroup. There are however saturated fusion systems which do not arise in this way from a finite group and these fusion systems are called exotic. In this talk I will outline how one might want to get an idea about the relative abundance of exotic fusion systems (primarily for the prime 2) following strategies employed in the proof of the classification of finite simple groups. In particular, I will explain how group-like structures called localities might help to classify larger classes of simple fusion systems. Moreover, I will give a rough idea where the critical cases might be. 

    UNFORTUNATELY, ELLEN HENKE’S TALK HAS BEEN CANCELLED DUE TO UNFORSEEN CIRCUMSTANCES.
  • Radha Kessar: “Weight conjectures  for some  classes  of exotic fusion systems”.
    Abstract:      I will  give an introduction to  the weight conjectures for abstract saturated  fusion  systems, explain the connections with  local-global counting in the representation theory of finite groups,  and  report on  some recent papers  which check the validity of  some of these conjectures for  certain classes  of exotic systems.
  • Stacey Law: “Sylow branching coefficients for symmetric groups”.
    Abstract:     One of the central themes in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and those of its local subgroups. In particular, Sylow branching coefficients describe how an irreducible character of G decomposes upon restriction to a Sylow subgroup of G. In this talk, we will present some new results in the case of symmetric groups particularly for p=2 and discuss some connections to probability and combinatorics.
  • Markus Linckelmann: “The source permutation module of a block of a finite group”.
    Abstract:      For G a finte group, k a field of prime characteristic p, and S a Sylow p-subgroup of G, the Sylow permutation module (Ind^G_S(k)) plays a role in a variety of group theoretic and representation theoretic aspects, ranging from Alperin’s weight conjecture to statistical considerations of S-S-double cosets in G. The Sylow permutation module breaks up along the block decomposition of the group algebra kG, but the resulting block components are not block invariants. We introduce a summand of the block component, which we call source permutation module, and show that it is a block invariant. Besides some general structural properties of the source permutation module of a block, we show that well-known results relating the self-injectivity of the endomorphism algebra of the Sylow permutation to Alperin’s weight conjecture carry over to the source permutation module. We calculate this module in various cases, such as certain blocks with cyclic or Klein four defect, and, prompted by a question raised in a talk by Persi Diaconis, for some blocks of symmetric groups. This is joint work with Radha Kessar.
  • Gunter Malle: “Subnormalisers and picky elements”.
    Abstract:     Motivated by new yet unpublished local-global conjectures by A.~Moret’o and N.~Rizo for character values of finite groups, we investigate the concepts of picky elements and of subnormalisers of p-elements of a finite group. We also report on partial classifications for finite simple groups of Lie type as well as for simple algebraic groups, leading to the verification of the conjectures at least in some special cases.
  • Lucas Ruhstorfer: “Towards the inductive Galois-McKay condition”
    Abstract:     Navarro’s refinement of the McKay conjecture proposes that there exists a McKay bijection which is equivariant with respect to the action of certain Galois automorphisms.

    Similar to the McKay conjecture, this has been reduced by Navarro, Späth, and Vallejo to the verification of a stronger condition on quasi-simple groups, known as the inductive Galois-McKay conditions. I will introduce these inductive conditions and present some recent progress on their verification.
  • Jason Semeraro “Character values for spetses”
    Abstract:     Many representation-theoretic data — Hecke algebras, unipotent degrees, Fourier matrices, etc. — associated to a finite reductive group can be constructed purely in terms of its Weyl group, a real reflection group. Generalising to some complex reflection groups, the associated data is remarkable, satisfying a wealth of desirable properties all motivated by the group case. This lead Michel Broué, Gunter Malle and Jean Michel to speculate that there is an associated unifying object which they termed the “spets”. But what is a spets? Recently, methods from algebraic topology have broadened the remit of the BMM programme to include unipotent p-blocks, their p-fusion systems and characters, which may bring us closer to answering this question. After recalling some of these constructions, I will explain how one can even, in some cases, define character *values* on the p-elements of the associated defect group, extending well-known formulae for reductive groups. These values appear to behave just like those for actual finite groups, and I will provide conjectures, results and examples consistent with that viewpoint.

    This is all joint work with Radha Kessar and Gunter Malle.
  • Carolina Vallejo Rodriguez: “On the properties of conductors of characters of degree prime to p”.
    Abstract: For an irreducible character of a finite group, the conductor is the smallest positive integer such that all of its values lie in the corresponding cyclotomic field. In this talk, I will discuss some interesting features of the p-part of character conductors, specifically for characters whose degree is prime to p.

    This talk is based on joint work with G. Malle and J. M. Martínez

Contributed talks details

  • Valentina Grazian: “The ongoing quest for exotic fusion systems”
    Abstract: Saturated fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number. For example, it is always possible to define the saturated fusion system realized by a finite group G on one of its Sylow p-subgroups S: this is the category where the objects are the subgroups of S and the morphisms are the restrictions of conjugation maps induced by the elements of G. An active research direction in the theory of fusion systems consists in the understanding of the ones that are not realized by a finite group, called exotic, especially for odd primes (this question was suggested by Oliver in [1]). Many of the known examples of exotic fusion systems are now captured by two key theorems: The classification of fusion systems on p-groups having a maximal subgroup that is abelian ([2],[5],[6]) and the classification of fusion systems on p-groups of maximal nilpotency class ([3]). The next step consists in generalizing these two theorems in order to frame more exotic fusion systems into unified contexts. In this talk we will present recent results in this direction and a new family of exotic fusion systems (described in the preprint [4]). This is a joint work with J. Lynd, B. Oliver, C. Parker, J. Semeraro and M. van Beek.

    References:
    [1] M. Aschbacher; R. Kessar; B. Oliver, Fusion systems in algebra and topology, London Math. Soc. Lecture Note Ser., 391 Cambridge University Press, Cambridge (2011). vi+320 pp.
    [2] D. Craven; B. Oliver; J. Semeraro, Reduced fusion systems over p-groups with abelian subgroup of index p: II, Adv. Math. 322 (2017), 201–268.
    [3] V. Grazian; C. Parker, Saturated Fusion Systems on p-Groups of Maximal Class, Mem. Amer. Math. Soc. 307 (2025), no. 1549, v+115 pp.
    [4] V. Grazian; C. Parker; J. Semeraro; M. van Beek, Fusion systems related to polynomial representations of SL2(q), arXiv preprint: 2502.20873 (2025).
    [5] B. Oliver, Simple fusion systems over p-groups with abelian subgroup of index p: I, J. Algebra 398 (2014), 527–541.
    [6] B. Oliver; A. Ruiz, Reduced fusion systems over p-groups with abelian subgroup of index p: III, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1187–1239.
  • Adam Jones: “Smooth, modular representations of p-adic SL_3 and its pro-p Iwahori-Hecke algebra”.
    Abstract:     The canonical relationship between the category of smooth representations of G=SL_n(K), for K a p-adic field, and the category of modules over the corresponding pro-p Iwahori-Hecke algebra H is well understood in characteristic 0, and it is fundamental to the proof of the classical local Langlands correspondence. In characteristic p, this relationship is very mysterious, and the behaviour of supersingular representations likely makes a full understanding impossible. When G=SL_2(K), Schneider and Ollivier used torsion theory to define a category of H-modules that is equivalent to the corresponding category of G-representations, and they proved that this category encompasses all non-supersingular representations. Their proof utilised the Bruhat-Tits building of SL_n(K), which is a tree when n=2, but for n>2 becomes immeasurably complex. I will present results which utilise the methods of Schneider-Ollivier, together with some new techniques on higher rank buildings to give strong evidence towards the same equivalence holding for SL_3(K).
  • Eoghan McDowell: “(Spin) representations of the symmetric group with the same p-modular reductions”
    Abstract:     When do two ordinary irreducible representations of a group have the same p-modular reduction? Equivalently, when do two ordinary irreducible characters have the same values on p-regular classes? I will present an (almost complete) answer to this question for the double cover of the symmetric group. I will discuss one of the techniques used in the proofs, which involves the use of induction and restriction functors to swap a pair of runners in an abacus display for the labelling partition. Part of this work is joint with Matt Fayers.
  • Marco Praderio Bova: “Fusion systems and homotopy theory”
    Abstract: During this talk we will start by briefly review the concepts of fusion systems and localities. We will then use the latter to provide a connection between fusion systems and homotopy theory, we will then use this relation tointroduce what is known as the “sharpness conjecture for fusion sysytems”. After reviewing what is known regarding this conjecture we will conclude by presenting the results developed in collaboration with G. Carrión Castillo and wich will allow us to prove the conjecture for a variety of different exotic fusion systems related to sporadic simple groups.
  • Marie Roth: “Unitriangularity of decomposition matrices: the simple adjoint finite groups of exceptional type”
    Abstract:     In 2020, Brunat–Dudas–Taylor showed that the decomposition matrix of the unipotent ℓ-blocks of a finite reductive group G in good characteristic has unitriangular shape. Their theorem holds under some conditions on the prime ℓ, in particular ℓ being good.
    In this talk, we’ll discuss how to extend this result, firstly to ℓ bad (for any G simple) and then to other blocks, called isolated (for G simple of type G2 and F4). This work was part of my PhD thesis under the supervision of G.Malle and O.Dudas. 
  • Jialin Wang: “On the solvability of the Lie algebra (HH^1(B)) for blocks of finite groups.”
    Abstract:     We give some criteria for the Lie algebra of first degree Hochschild cohomology of the twisted group algebra, i.e. HH^1(k_alpha(P ⋊ E)), to be solvable, where P is a finite abelian p-group, E is an abelian p′-subgroup of Aut(P) and α ∈ Z^2(E; k^×) inflated to P ⋊ E via the canonical surjection P ⋊ E → E. As a special case, this gives the criterion to the solvability of the Lie algebra HH^1(B) where B is a p-block of a finite group algebra with abelian defect P and inertial quotient E.

Local eating options

All lunches (Mon-Thu) and dinners on Tuesday and Wednesday are provided by AGGITATE for all participants. For Monday and Thursday, participants may consider one of the following options:

  • Wivenhoe: walking by the riverside path (45 minutes) one can reach this picturesque village, which has three local pubs (the Greyhound, the Black Buoy and Rose and Crown). Alternatively, one can reach them by bus (51, 74, 85) from Boundary Road on campus (15 minutes). Wivenhoe’s riverside is nice on a sunny day.
  • Colchester city centre: there are a lot of options downtown, with several cafes in the pedestrian area in the centre of Colchester, around Sir Isaac’s Walk, Lion Walk, or on and off the High Street. These include Mirra (middle-eastern), The George (British food), Roxi (Mediterranean), Slug & Lettuce (lively bar also serving food), Moto Pizza (bottomless pizza, booking online advised) or chains like Wagamama. There’s also a couple of options in Queen Street, including Kimchi House (Korean), Miseria e Nobilta’ (Italian, phone-booking advised), Chez Afrique (African) and a number of take-aways.
  • Local campus options: in principle we’ve been told Buffalo Joe’s (hamburgers, square 4) is open till 8pm. Crumbs (pastries, square 3) is open till 5pm. Street Food Pizza (square 4) is open till 6.30pm.

About the AGGITatE series

The first AGGITatE conference was to take place in 2020 but was delayed to 2022 due to the CoViD pandemic and, back then it stood for “Algebraic Groups and Geometric Invariant Theory at Essex” as a way to bring together researchers working in algebraic groups and algebraic geometry. The second AGGITatE took place in 2023, focused in algebraic groups and the Cremona group. The third AGGITatE (2024) was focused on moduli theory of varieties. The fifth edition of AGGITatE (2026) will be part of a one-month INI Satellite Programme.