Abstracts

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Mini-courses

Michele D'Adderio (Università di Pisa)

Title: q,t-Combinatorics and Sandpiles

Abstract: In 1987 Bak, Tang and Wiesenfeld introduced their famous sandpile model as a first system showing self-organized criticality. In 1988 Macdonald introduced his famous symmetric polynomials. Each of these two discoveries produced a huge amount of research that is still developing intensely today. But until recently, these two lines of research went on without any relevant interaction. In this minicourse we show how the combinatorics generated by these two important mathematical objects come together in a surprising way, proving that a synergy between these two topics is inevitable.

Nicolas Libedinsky (Universidad de Chile)
Title: Alcovic geometry and Euclidean geometry

 Abstract: A fascinating and quite unexplored relationship is beginning to emerge between alcovic geometry and Euclidean geometry. This connection has surprising depth, as seen in problems such as the enumeration and classification of Bruhat intervals in affine Weyl groups. 

Invited talks

Karim Adiprasito (Sorbonne Université and Université Paris Cité)

Title: Commutative algebra for lattice polytopes

Abstract: I discuss ongoing work with Stavros Papadakis and Vasiliki Petrotou that explores the commutative algebra of lattice polytopes in depth. We extend work of Hochster, and describe their volume polynomials in detail, and then prove Lefschetz type properties for the associated algebras in two ways: We establish generic Lefschetz properties using an identity of the Parseval type for the volume polynomial. And we establish a classical Lefschetz type property that includes the Hodge-Riemann relations.

Angela Carnevale (University of Galway)
Title: Coloured shuffle compatibility and Hadamard products

Abstract: In this talk I will present recent work on coloured shuffle compatibility of permutation statistics and its applications to zeta functions in algebra. I will discuss how we extended recent work of Gessel and Zhuang, introducing shuffle algebras associated with coloured permutation statistics. Our shuffle algebras provide a natural framework for studying Hadamard products of certain rational generating functions. As an application, we will see how to explicitly compute such products in the context of so-called class- and orbit-counting zeta functions of direct products of suitable groups. This is joint work with V. D. Moustakas and T. Rossmann.

James Cruickshank (University of Galway)
Title: Simplicial Complexes and Rigidity

Abstract: Rigidity theory is a subdiscipline of discrete geometry that explores combinatorial and geometric aspects of structures that have some associated notion of flexibility/rigidity. The prototypical example of such a structure is a bar-joint framework, but many variations and generalisations have been studied in the literature. Classically, rigidity theory was most often studied in connection with objects that could be combinatorially modelled by a graph. However, in recent years there have been several interesting developments relating algebraic combinatorial aspects of simplicial complexes to the rigidity theory of graphs and hypergraphs. In this talk I will survey some of these developments, with a focus on recent joint work from a series of papers with Bill Jackson and Shin-ichi Tanigawa. I will not assume any previous familiarity with rigidity theory.

Francesco Esposito (Università di Padova)
Title: Flip Combinatorial Invariance and Weyl groups
Abstract: In this talk, we present an approach via flipclasses to the Combinatorial Invariance Conjecture for Kazhdan--Lusztig polynomials of all Coxeter groups. We prove the combinatorial invariance of Kazhdan--Lusztig \widetilde{R}-polynomials of Weyl groups modulo q^7 and of Kazhdan--Lusztig \widetilde{R}-polynomials of type A Weyl groups modulo q^8. As a consequence, the Combinatorial Invariance Conjecture holds for all intervals up to length 8 in Weyl groups and up to length 10 in type A Weyl groups. This is joint work with M. Marietti and S. Stella.

Christophe Hohlweg (Université du Québec)
Title: On generating functions and automata associated to reflections in Coxeter systems

Abstract: In this talk, I will discuss two combinatorial problems regarding  the set of reflections of a Coxeter system: (1) Is  the language of palindromic reduced words for the reflection regular?  (2) Find nice formulas for the Poincaré series of the length of reflections. These two problems were inspired by the conjecture of Stembridge stating that the Poincaré series of the length of reflections is rational and by the solution provided by de Man.  

Regarding Problem 1, I will  discus the notion of reflection-prefixes, a class of elements in W arising naturally from palindromic reduced words of reflections, and discuss their properties in relation to the  root poset, the dominance order on roots  and dihedral reflection subgroups. Then show that language of reduced words for reflection-prefixes is regular and recognized by the family of automata arising from m-Shi arrangements,  As for Problem 2, I will present,  in the case of  affine Coxeter groups, simple formulas  for the Poincaré series of the length of reflections in term of symmetries of the Hasse diagram of the root poset.

(This talk is based on a joint work with Riccardo Biagioli and Elisa Sasso)

Emily Norton (University of Kent)
Title: Enumerating defect zero blocks

Abstract: A staircase partition cannot be tiled in such a way that upon removing a domino-shaped tile from the staircase, you still have a partition. We say that the staircase partition is a 2-core partition. The notion of an e-core partition is similar, but with e-ribbons in place of dominoes. The e-core partitions describe blocks in the representation theory of symmetric groups in positive characteristic, but also rational Cherednik algebras and Hecke algebras at roots of unity. In the modular representation theory of the finite general linear group, the e-core partitions describe the unipotent blocks.  In 1996, Granville and Ono proved that there exists an e-core partition of every size n if e is at least 4 (when e is 2 or 3, there are infinitely many values of n without an e-core partition of size n). We may restate Granville and Ono’s result as saying that in quantum characteristic at least 4, there exists a defect 0 unipotent block of GL(n,q) for every natural number n. We may then ask if there is an analogue of this theorem for other finite classical groups, for cyclotomic Hecke algebras at appropriate parameters, etc. This a project with Thomas Gerber.

Giovanni Paolini (Università di Bologna)
Title: Factoring isometries into reflections
Abstract: Any element of the orthogonal group O(n) can be expressed as a product of at most n reflections. Less well known is the structure of the poset formed by the minimal such factorizations, or its analogue for isometries of arbitrary quadratic spaces. In this talk, I will survey these questions at the crossroads of linear algebra and combinatorics, and illustrate how the resulting insights help us study Coxeter and Artin groups.

Leonardo Patimo (Università di Pisa)

Title: Atomic decompositions and charge statistics in representation theory
Abstract: The Kazhdan-Lusztig bases of the Hecke algebra have remarkable positivity properties, some of which remain to  be fully explored. In the case of type A spherical Hecke algebras, the KL basis is positive with respect to the atomic basis, a basis in which each weight smaller than a given dominant weight appears with multiplicity one.
A counterpart in representation theory of this positivity is the atomic decomposition of crystal graphs. Such a decomposition is obtained by considering the closure of Weyl group orbits under an extremal crystal operator. The grading in KL polynomials corresponds to the charge statistics defined by Lascoux and Schuetzenberger.
In the talk, we will present refinements of these decompositions, both in the Hecke algebra and in crystal graphs, which conjecturally agree on regular weights. We will also discuss what happens beyond type A and how this approach can help to find charge statistics in general.

Salvatore Stella (Università dell'Aquila)
Title: Cluster scattering diagrams of affine type

Abstract: Cluster scattering diagrams, since their introduction, played a central role in shaping the structure theory of cluster algebras. They consist of a combinatorial datum, a fan, together with the assignment of a formal power series for each of its codimension-1 cones. While their recursive definition is in theory explicit, constructing cluster scattering diagrams is usually a difficult task. 

In this talk, based on joint work with N. Reading, I will address this problem in the special case of acyclic cluster algebras of affine type where one can leverage previous constructions using the machinery of root systems, Coxeter groups, and lattice theory to relate cluster scattering diagrams to two other fans: the mutation fan and the fan of almost-positive roots.