Abstracts

Mini-courses

Fabrizio Caselli: "Flags, reflections, Coxeter matroids and Bruhat polytopes"

The Kostant-Toda lattice is an integrable hamiltonian system whose associated moment map images produce nice combinatorial polytopes, the Bruhat polytopes,  which are strictly related to the combinatorics of the flag variety and of the Bruhat ordering of the symmetric group. The study of these polytopes and their generalization for all finite Coxeter groups lead to the study of new purely algebraic properties of Bruhat intervals such as the so called generalized lifting property and the fact that every Bruhat interval in a finite Coxeter group is a Coxeter matroid, i.e. every translation of it has a unique maximal element. In this minicourse I will also sketch the proofs of these results which make use of the theory of reflection subgroups and old and new properties of reflection orderings, and I will show the main applications in the study of Bruhat polytopes.

Axel Hultman: "Inversion arrangements"

Every element w of a finite, real reflection group comes with the collection of reflecting hyperplanes corresponding to the inversions of w. Many properties of such inversion arrangements turn out to be closely related to other algebraic, geometric, and combinatorial objects indexed by w, such as Schubert varieties. Some of these connections are well understood, some are still mysterious. I shall survey some constructions, results, and open questions in this field.

Invited talks

Nancy Abdallah: "Combinatorial Invariance of KLV Polynomials for Fixed Points Free Involutions"

Let S_{2n} be the symmetric group of permutations of {1,..., 2n}, and F_{2n} be the set of fixed points free involutions of S_{2n}. To every interval [u,v] in the poset F_{2n} ordered by the Bruhat order, we associate a KLV-polynomial P_{u,v}. Using a combinatorial concept called Special Partial Matching or SPM, we prove that these polynomials are combinatorially invariant for upper intervals, i.e. for intervals [u,w_0] where w_0 is the maximal element of F_{2n}. This gives a generalization of the combinatorial invariance of the classical Kazhdan-Lusztig polynomials in the symmetric case. This is a joint work with Axel Hultman.

Riccardo Biagioli: "321-avoiding affine permutations and periodic parallelogram polyominoes"

Among permutations, those that avoid the pattern 321 are of greatinterest in combinatorics and algebra. They are counted Catalannumbers. From an algebraic point of view, a permutation is321-avoiding if, and only if its corresponding element in the Coxetergroup of type A is fully commutative.In this talk we consider their affine analogues, so 321-avoidingaffine permutations. We show some combinatorial characterizations ofthem, and we prove a formula for their enumeration with respect to theinversion number.First, we introduce periodic parallelogram polyominoes, which are newcombinatorial objects of independent interest. We enumerate them byextending the approach of Bousquet-Mélou and Viennot used forclassical parallelogram polyominoes. We then establish a connection between these new objects and 321-avoiding affine permutations.This talk is based on a joint work with Frédéric Jouhet and Philippe Nadeau.

Francesco Brenti: "Permutations, tensor products, and Cuntz algebra automorphisms"

We introduce and study a new class ofpermutations which arises from the automorphisms of the Cuntz algebra. I will define this class, explain its relation to the Cuntz algebra, present resultsabout symmetries, constructions,characterizations, and enumeration of these permutations, and discusssome open problems and conjectures. This is joint work with Roberto Conti.

Giovanna Carnovale: "Quotient stratifications for affine hyperplane arrangements"

An affine hyperplane arrangement  H in a Euclidean space E induces a natural stratifications on E and on its complexification by taking intersection of hyperplanes. For W a group of orthogonal transformations of E preserving H, we also have a stratification on  E/W for W and its complexification. When W is a Weyl group or a subgroup of an extended affine Weyl group, the strata in the latter are quotients of closures of Jordan classes in a reductive Lie algebra or  in a reductive algebraic group and they have been studied for representation theoretic purposes. We shall discuss how to detect geometric properties such as normality, unibranchedness and  smoothness of this strata. An easy algorithm for normality is given for the case of quotients of closures of Jordan classes in a simple algebraic group.

Paola Cellini: "General triangulations of root polytopes"

The root polytope associated to a finite root system r is the convex hull of all the roots in r. I will describe a uniform construction, for all irreducible crystallographic root systems, of a triangulation of the root polytope.

Maria Chlouveraki: "The ΒΜΜ symmetrising trace conjecture for Hecke algebras"

Exactly twenty years ago, Broué, Malle and Rouquier published a paper in which they associated to every complex reflection group two objects which were classically associated to real reflection groups: a braid group and a Hecke algebra. Their work was further motivated by the theory, developed together with Michel, of “Spetses”, which are objects that generalise finite reductive groups in the sense that their associated Weyl groups are complex reflection groups. The four of them advocated that several nice properties of braid groups and Hecke algebras generalise from the real to the complex case, culminating in two main conjectures as far as the Hecke algebras are concerned: the “freeness conjecture” [BMR] and the “symmetrising trace conjecture” [BMM]. The two conjectures are the cornerstones in the study of several subjects that have flourished in the past twenty years, but had remained open until recently for the exceptional complex reflection groups. In the past five years, the proof of the “freeness conjecture” was completed for all exceptional complex reflection groups. In this talk, we will discuss our proof of the “symmetrising trace conjecture” for the first five exceptional groups. This is joint work with Christina Boura, Eirini Chavli and Konstantinos Karvounis.

Stéphane Gaussent: "Satake isomorphism and Macdonald formula in Kac-Moody setting"

Starting with a reductive group G over a local field, one can define the spherical Hecke algebra as bi-invariant functions on the group with compact support. This algebra can be also seen as the center of the affine Iwahori-Hecke algebra. The Macdonald formula we are discussing here is the one giving the image, via the Satake isomorphism, of a characteristic function in the spherical Hecke algebra into the algebra of invariant polynomials under the Weyl group of G. If G is the general linear group, the latter is nothing else than the symmetric algebra. The purpose of the talk is to discuss that Macdonald formula in the context of Kac-Moody groups over local fields. Some combinatorics on paths will be involved in the proof.

Thomas Gerber: "Generalised Mullineux involution and applications"

Let S_n be the symmetric group on a set with n elements. Tensoring an irreducible representation of S_n with the sign representation yields another irreducible representation of S_n. Over C, we can label the irreducible representations of S_n by the partitions of n, and the above procedure is simply given by to taking the transpose of the starting partition.Over F_p (with p prime), we can label the irreducible representations of S_n by the p-regular partitions of n, and the above procedure is the so-called Mullineux involution. In 1995, Kleshchev gave a combinatorial formula for computing the Mullineux involution, which can be interpreted in terms of crystals.In this talk, I will explain how to define a version of the Mullineux involution on multipartitions (= tuples of partitions) using crystals, and I will give representation-theoretic interpretations of this construction. This is joint work with Nicolas Jacon and Emily Norton.

Paolo Papi: "B-orbits and abelian ideals"

Let G be a quasi simple algebraic group and B a Borel subgroup of G. I will review work of Panyushev on the classification of B-orbits in an abelian ideal of the Lie algebra of B and I'll give a description of the closure relations among these orbits. Joint work with Jacopo Gandini, Andrea Maffei and Pierluigi Moseneder Frajria.