• May 29-June 2, 2017 (Spring school)
• June 5-June 9, 2017 (Conference)

Contact:	coxeter-conv@lacim.ca
Organization:
	Matthew Dyer Christophe Hohlweg Vincent Pilaud Hugh Thomas

This meeting will lie at the interface between algebraic and geometric combinatorial aspects of Coxeter groups and reflection groups. Reflection groups appear in very many domains of mathematics, for instance,

• as symmetry groups of regular polytopes
• as quotient of Artin-Tits (braid) groups and
• via root systems, as Weyl groups of semi-simple Lie algebras, Lie groups, algebraic groups, Kac-Moody algebras or cluster algebras.

Properties of these groups, finite or infinite, are often key to the understanding of related structures. This conference is concerned with bringing leading experts (and their students and postdocs) to discuss the state of the art in strongly interconnected and lively areas such as the following:

• Algebraic Combinatorics of Reflection Groups. This includes questions related to the combinatorics of words in Coxeter groups (structural properties of reduced expressions), the structures of complex reflexion groups and their relations to: weak and Bruhat order and representation of Coxeter groups, Artin-Tits (braid) groups via Garside families, Kazhdan-Luzstig theory.
• Coxeter-Catalan combinatorics. This lively subject arose from two spectacular developments in the beginning of 2000: Fomin and Zelevinsky's cluster algebras on one side and the generalization of non-crossing partitions to complex reflection groups due to Bessis and, independently, Brady and Watt. Among the topics covered by Coxeter-Catalan combinatorics we find generalized associahedra and other Coxeter polytopes, links to representation of quivers, Reading's Cambrian lattices and fans, Shi arrangements etc.

• Geometry and combinatorics of root systems. Even aside from their importance in Lie theory and representation theory, root systems are at the heart of fundamental work on Coxeter groups, such as the proof of their automatic structure (Brink and Holwett), or the conjugacy problem for Coxeter groups (Kraammer). In recent years, groundbreaking studies about combinatorics and geometry of infinite root systems of Coxeter groups have been conducted in an attempt to generalize the notion of reduced words via infinite words and biclosed sets to a more general framework suitable for application to connected structures such as Kazhdan-Lusztig polynomials, loop groups or generalized associahedra in infinite type (c.f. work of Dyer, Lam and Pylyavskyy, Reading and Speyer).

The conference timeline is the following:

• May 29-June 2 (Spring school):
  Monday May 29 will be devoted to introductory lectures, for interested participants, on common background assumed for the school.
  The main part of the school (May 30-June 2) will consist of four mini-courses:
  • Hecke algebras and Kazhdan-Lusztig theory [Cédric Bonnafé]

  • Geometric group theory of Coxeter groups [Piotr Przytycki]

  • Fans, lattices and cluster combinatorics [Nathan Reading*]

  • Reflection groups and noncrossing partitions [Vic Reiner]

• June 5-June 9 (Conference): There will be twenty talks of one hour (including time for questions), four per day, in areas possibly including but not limited to
  • Fans, lattices and cluster combinatorics
  • Reflection groups and noncrossing partitions
  • Geometric group theory of Coxeter groups
  • Hecke Algebras and Kazdhan-Lusztig theory
  • Polytopes and fans arising from Coxeter groups
  • Braid groups and reflection groups
  
  • Combinatorial aspects of reflection groups
  • Infinite Coxeter groups and root systems
  • Reflection groups and enumerative combinatorics.
  • Representation theory and reflection groups.
  • Finite dimensional algebras and quiver representations
  • Flag varieties, Schubert calculus and Bruhat order
  • Algebraic groups, Kac-Moody groups, reductive monoids etc
  • Lie Theory
  • Type A combinatorics with a view toward other reflection groups.
  Speakers include
  • Cédric Bonnafé
  • Piotr Przytycki
  • Nathan Reading*
  • Vic Reiner

(*: To be confirmed.)

For further information, please contact coxeter-conv@lacim.ca.

The mini-courses in the Spring School are as follows:

• Introduction to Coxeter groups and root systems [Speaker(s) TBA]

  These lectures on Monday will cover some of the assumed background for the main part of the Spring School, on Tuesday-Friday. A convenient reference for much of this background is Chapters 1,2,5 and 6 of [1].

  
  Selected Bibliography:
  [1] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press.

• Geometric group theory of Coxeter groups [Piotr Przytycki]

• Reflection groups and noncrossing partitions [Vic Reiner]

• Fans, lattices and cluster combinatorics [Nathan Reading*]

• Hecke algebras and Kazhdan-Lusztig theory [Cédric Bonnafé]

Here are Abstracts for the minicourses and for conference talks.