Algebra Days in Caen 2023: Cactus and Diagram Groups

March 06-08, 2023

This edition of Algebra Days in Caen aims to bring together experts, national and international, around the two interconnected themes: Cactus and Diagram groups. These groups appear at the interface between several mathematical fields: combinatorics, representation theory, geometric group theory and operads.

The conference is planned to be in-person.


Adrien Brochier (Paris)

Anthony Genevois (Montpellier)

Iva Halacheva (Massachusetts)

Anton Khoroshkin (Haifa)

Jacob Mostovoy (Mexico)

Neha Nanda (Caen)

Luis Paris (Dijon)

                                                                                                                                                   Photo: David Mercado

Practical information

If you have any questions, please contact the organisers, Paolo Bellingeri, John Guaschi and Neha Nanda.

Conference venue: Université de Caen Normandie, Campus 2, Sciences 3, LMNO in lecture halls S3-057 (ground floor) and salle des thèses S3-102 (first floor) [Please refer to schedule for precise venue for respective days] .

To get there, you may follow these instructions.

Accommodation: Participants should reserve and pay for their own accommodation during the meeting. Here are some hotels in Caen:

Hôtel du Château

Hôtel des Quatrans

Hôtel Bernières

Hôtel Le Dauphin


Monday, March 06
At S3-057 (ground floor)

14:00-14:30 Welcome and registration
14:30-15:30 Anthony Genevois
15:30-16:00 Coffee break
16:00-17:00 Neha Nanda

Tuesday, March 07
At salle des thèses S3-102 (first floor)

9:30-10:30 Iva Halacheva
10:30-11:00 Coffee break
11:00-12:00 Anthony Genevois
12:00-14:00 Lunch
14:00-15:00 Luis Paris
15:00-15:30 Coffee break
15:30-16:30 Jacob Mostovoy
16:45-17:45 Anton Khoroshkin

Wednesday, March 08
At salle des thèses S3-102 (first floor)

09:30-10:30 Anthony Genevois
10:30-11:00 Coffee break
11:00-12:00 Adrien Brochier

Titles and abstracts

Adrien Brochier

Title : RSK, Gaudin algebras and Calogero-Moser cells
Abstract : The RSK correspondence is a bijection between certain words and certain pairs of tableaux, which encodes various aspects of the representation theories of GL_n, S_n and of their q-deformations. One "materialization" of this correspondance is given by a labelling of a certain distinguished basis (the Gelfand-Zeitlin basis) of the kth tensor power of the defining representation of GL_n. This basis can also be described from the eigendecomposition of that space under the action of a certain commutative algebra. This algebra, in turn, is a certain limiting case of a family of commutative algebras (the Gaudin algebras) parameterized by the real part of the compactified moduli space of genus 0 curves, whose fundamental group is the so-called cactus group. Bonnafé-Rouqier have introduced a certain partition of S_n into "cells" based on the geometry of the so-called Calogero-Moser space, and conjectured that it can be described using RSK. We'll explain a proof of this conjecture, based on fundamental work of Mukhin--Tarasov-Vaerchenko, which allows to translates this question into one about the monodromy of Gaudin algebras, and of Halacheva-Kamnitzer-Rybnikov-Weekes which describes this monodromy action explicitly. This is based on joint work with Iain Gordon and Noah White.

Anthony Genevois

Title : An introduction to diagram groups
Abstract : In this mini-course, the goal will be to explain what diagram groups are, to explain the main results known about them, and to describe several explicit examples. Prerequisites will be minimal.

Iva Halacheva

Title :  Categorical braid group actions, perverse equivalences, and the cactus group
Abstract :Given a semisimple Lie algebra g, the representations of its quantum group have additional symmetries captured by the braid group. From a combinatorial perspective, a large part of the information of these representations is captured by g-crystals. The analogue of the braid group action in that setting turns out to be an action of the cactus group. This action can be recovered in several ways – we show that one can do so from a categorical perspective by studying the braid group action realized through Rickard complexes, and restricting our attention to the braids coming from parabolic longest Weyl group elements, or half-twists. This is joint work with A. Licata, I. Losev, and O. Yacobi.

Anton Khoroshkin

Title : Cacti groups from operadic point of view
Abstract: I will explain different algebraic models of the operad of the real locus of the moduli spaces of stable rational curves with marked points and show how to gain the rational homotopy type of these spaces whose fundamental groups coincides with the pure cacti groups. In order to make an advertisement to the operad theory, the relationships with the pure braid groups will be also formulated on the operadic level. The talk is based on the joint work with T.Willwacher arXiv:1905.04499.

Jacob Mostovoy

Title : Cactus groups, braid groups and the Crisp-Paris Theorem
Abstract: In this talk, I will show how to construct a family of embeddings of a certain finite index subgroup of the pure cactus group into the pure braid group.

Neha Nanda

Title : On twin groups
Abstract: Quite recently, it is proved that the twin groups can be viewed as subgroups of cactus groups in the work of Bellingeri-Chemin-Lebed. These groups relate to the study of certain kind of immersed circles in the 2-sphere. In this expository talk, introduction and properties of twin groups and its pure subgroup will be discussed. Results are based on work in collaboration with Tushar Naik and Mahender Singh.

Luis Paris

Title : Groupes de trickle
Abstract : Cet exposé a pour but de présenter un travail en collaboration avec Paolo Bellingeri et Eddy Godelle.

Notre projet consiste en l'étude combinatoire de certains groupes, que nous appelons groupes de trickle, définis par des relations

de la forme ab=bc, les générateurs pouvant être d'un certain ordre. Nous sommes capable de résoudre le problème du mot

et de définir des formes normales pour ces groupes, mais mon but dans cet exposé sera surtout
de présenter des exemples, dont les groupes de cactus et les groupes de cactus virtuels.



Adrien BROCHIER (Paris)


Adrien CLEMENT (Nantes)

Raquel CRUZ (LMNO Caen)

Jacques DARNÉ (UCLouvain)


Anthony GENEVOIS (Montpellier)

Emmanuel GRAFF (LMNO Caen)


Iva HALACHEVA (Massachusetts)

Anton KHOROSHKIN (Haifa)

Victoria LEBED (LMNO Caen)

Bernard LECLERC (LMNO Caen)

Gilbert LEVITT (LMNO Caen)

Ilaria MAZZOTTI (Università di Bologna)

Jacob MOSTOVOY (Mexico)

Neha NANDA (LMNO Caen)

Philippe SATGE (LMNO Caen)


Luis PARIS (Dijon)