BIREP – Representations of finite dimensional algebras at Bielefeld

Noncrossing Parking Functions [slides]

For every Coxeter element in a finite Coxeter group there is a set of group
elements called "noncrossing partitions" that are counted by the Catalan
number of the group. In this talk I will use the noncrossing partitions to
define a set of "noncrossing parking functions" counted by (h+1)^{r}
where h is the Coxeter number and r is the rank of the group. The usual
"nonnesting parking functions" (i.e. regions of the Shi arrangement) are
defined only for crystallographic types but noncrossing parking functions
exist also for noncrystallographic types. I will discuss conjectures relating
noncrossing and nonnesting parking functions. This is joint work with Brendon
Rhoades and Vic Reiner.

Non-crossing partitions and reflection discriminants [slides]

Key numerical invariants (Catalan numbers, Zeta functions) of generalized non-crossing partitions can be expressed in terms of the degrees of the associated reflection groups, suggesting strong links with the geometry of the quotient singularity. This can be explained (partly) via a "canonical decomposition" theorem for reflection singularities, which appears as "pull-backs" of configuration spaces and sets of chains in the non-crossing partition lattice. I will explain this canonical decomposition, how it is used in the proof of the K(π,1) conjecture for complex reflection groups, and its relationship with the cyclic sieving phenomenon.

Milnor fibres and non-crossing partitions [slides]

We use non-crossing partitions to construct simplicial complexes with the homotopy type of Milnor fibres of complexified real reflection arrangements. This is joint work with Mike Falk, Ben Quigley and Colum Watt.

Unordered exceptional sequences and Tamari lattices

Maximal chains in the noncrossing partition lattices can be identified with exceptional sequences in the category of representations of the associated quiver. It seems to be interesting to study the underlying sets of modules (obtained by forgetting the order in an exceptional sequence). In type A, one can describe them using noncrossing trees or quadrangulations. I will explain that one can map them to objects in the derived category of modules over the Tamari lattice in a meaningful way.

Hom-configurations in negative Calabi-Yau triangulated categories

A key way to understand triangulated categories is to look at generators. In the context of positive Calabi-Yau (CY) triangulated categories, such as cluster categories, the most interesting generators seem to be cluster-tilting objects. There are some triangulated categories which are naturally negative CY. In this context, it is unclear which generator we should study. Higher Hom-configurations appear to be a natural candidate and seem to play a similar role as that of cluster-tilting objects. In this talk, we discuss connections between higher Hom-configurations and noncrossing partitions.

Non-crossing partitions and the dual Garside structure of braids [slides]

We shall survey some of the (many) combinatorial problems involving Artin braid groups and the associated Garside structures, which are connected with finite state automata and directly lead to counting questions. We shall mention some results arising in the so-called classical case, where the relevant objects are permutations and the Solomon descent algebras, and, mainly, we shall address the so-called dual case, where the relevant objects are noncrossing partitions. This approach leads to a number of seemingly new questions about the latter. We shall mention some very recent preliminary observations by Ph. Biane in this direction.

Groupoids with weak orders [slides]

We discuss certain groupoids equipped with partial orders satisfying properties which abstract those of weak orders of Coxeter groups. In particular, we describe braid presentations of the underlying groupoids and some of the very strong closure properties of these structures under natural categorical constructions.

Noncrossing partitions and Bruhat order [slides]

The aim of the talk is to give a criterion for Bruhat order on noncrossing partitions in type A for a specific choice of Coxeter element. Firstly we will give a motivation for the study of such a non-natural order coming from bases of Temperley-Lieb algebras. We will then give the criterion which uses amusing sets of vectors having Catalan enumeration. With this criterion one can show that the set of noncrossing partitions with Bruhat order forms a lattice that is isomorphic to the lattice of order ideals in the root poset. It turns out that if one changes the Coxeter element, the lattice property fails. We explain which order to consider in order to get the same lattice structure for any Coxeter element. Using the criterion we can derive properties of the mentioned bases of Temperley-Lieb algebras.

Positive m-divisible non-crossing partitions and their cyclic sieving [slides]

Buan, Reiten and Thomas (implicitly) defined positive m-divisible non-crossing partitions, by setting up a bijection between the facets of the generalized cluster complex of Fomin and Reading and m-divisible non-crossing partitions, positive clusters corresponding to positive m-divisible non-crossing partitions. We embark on a finer enumerative study of these combinatorial objects associated to finite reflection groups. In particular, we define a cyclic action on them, which - together with the "obvious" q-analogue of positive Fuß–Catalan numbers - seems to satisfy the cyclic sieving phenomenon of Reiner, Stanton and White. We approach this conjecture case-by-case, and are able to prove it for several types (which may have become more by the actual dates of the workshop). This is joint work with Christian Stump.

The second fundamental theorem of invariant theory – old, new and super

The second fundamental theorem gives all relations among the invariants of a
classical group acting on tensor space. I shall show how an old idea of Atiyah
in algebraic geometry may be used to reduce the second fundamental theorem for
a classical group to the case of GL_{n}, which is understood. The most
efficient description of this reduction is in terms of the Brauer category.
This provides a new theorem for the orthosymplectic supergroups, and new
proofs of the classical second fundamental theorems for the classical groups,
independent of the classical work of Hermann Weyl. Quantum generalisations may
also be discussed in this context. This is joint work with Ruibin Zhang.

Hurwitz action on presentations of exceptional reflections groups [slides]

I will describe part of a work with Gunter Malle several years ago. The
product in some order of the minimal generators of a well-generated complex
reflection group is a Coxeter element. The Hurwitz orbit of such a
decomposition of a Coxeter element provides the generators of the dual braid
monoid. In the case of the exceptional groups G_{24} to G_{34}
it also provides other sets of minimal generators leading to interesting
presentations of the braid group.

On orbits of antichains associated with weight multiplicity free representations

For any finite poset P, there is a natural operator X acting on the set of antichains of P. The goal of my talk is to discuss properties of X if P is the (po)set of weights of a weight multiplicity free representation of a semisimple Lie algebra. Our results suggest that X has some good properties whenever the representation in question is associated with a Z-grading of larger semisimple Lie algebra.

Noncrossing diagrams and canonical join representations

Classical noncrossing partitions are often realized by diagrams consisting of (1) some points on a line, (2) some arcs connecting the points satisfying certain rules, and (3) some compatibility rules for arcs to appear in the same diagram, including of course the requirement that arcs not cross. Interesting things happen when we modify the rules. Various modifications yield noncrossing diagrams enumerated for example by the Baxter numbers, the secant numbers, etc. A very general class of noncrossing diagrams encodes lattice-theoretic information about the weak order on permutations, namely canonical join-representations of permutations. Other classes of noncrossing diagrams correspond to lattice-theoretic quotients of the weak order. As an added bonus, the arcs occurring in diagrams encode, in a natural way, complete information about lattice congruences on the weak order.

Crossing and nesting

Dealing with simply laced Dynkin diagrams, Ingalls and Thomas (Compos. Math.
145, 2009) gave an interpretation of the set of non-crossing partitions in
terms of the representation category of a Dynkin quiver: they exhibited, for
example, a bijection between the non-crossing partitions and the wide
subcategories or also the torsion classes. These results can be reformulated
in terms of antichains in additive categories and extended to the non-simply
laced cases B_{n}, C_{n}, F_{4}, G_{2} and the
corresponding hereditary abelian categories. We will show in which way the
representation theory approach sheds light on the relationship between
crossing and nesting; this relationship is well-known in the case
A_{n}, but seemed to be quite mysterious in the remaining cases.

Non-crossing partitions, free probability, and quantum symmetries

Non-crossing partitions show up in free probability theory in at least two different ways. On one hand, they are at the basis of the combinatorial approach to freeness, where the connection between moments and free cumulants is given by a summation over non-crossing partitions. On the other hand, they encode also the representation theory, in the form of intertwiner spaces, of fundamental non-commutative symmetries (like quantum permutation group or quantum orthogonal group). de Finetti like theorems relate these two occurrences of non-crossing partitions.

Aisles in derived categories of finite type hereditary algebras

Let Q be a Dynkin quiver. Colin Ingalls and I showed that torsion classes in kQ-mod correspond to the inversion sets of c-sortable elements (in the sense of Reading) of the Weyl group associated to Q. I will report on joint work with Christian Stump and Nathan Williams, in which we consider the extension of this problem to aisles in the bounded derived category of H-mod, for H a hereditary finite representation type algebra. We introduce a notion of c-sortability for Artin groups, and we show that the generating, separated aisles correspond to inversion sets of c-sortable elements in the Artin group corresponding to W.