Related papers: On two-coloured noncrossing quantum groups
Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These…
Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a…
We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one…
We consider a twisted version of quantum groups corepresentations. This generalization amounts to include in the theory the case where quantum space coordinates and its endomorphism matrix entries belong to a non-commutative quadratic…
We consider noncrossing partitions of [n] under the action of (i) the reflection group (of order 2), (ii) the rotation group (cyclic of order n) and (iii) the rotation/reflection group (dihedral of order 2n). First, we exhibit a bijection…
Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group $O_n^+$, the so called easy quantum groups, introduced by Banica and Speicher in 2009. This…
We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products. This illustrates the efficiency of the methods developed in our previous joint…
In this article, we give a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes).…
We describe the notion of a quantum family of maps of a quantum space and that of a quantum commutant of such a family. Quantum commutants are quantum semigroups defined by a certain universal property. We give a few examples of these…
We classify the compact quantum groups $A_u(Q)$ (resp. $B_u(Q)$) up to isomorphism when $Q>0$ (resp. when $Q \bar{Q} \in {\mathbb R} I_n$). We show that the general $A_u(Q)$'s and $B_u(Q)$'s for arbitrary $Q$ have explicit decompositions…
Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by…
We prove that the generalised non-crossing partitions associated to well-generated complex reflection groups of exceptional type obey two different cyclic sieving phenomena, as conjectured by Armstrong, respectively by Bessis and Reiner.…
For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a…
We study glued tensor and free products of compact matrix quantum groups with cyclic groups -- so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In…
We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions -- which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so…
We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of…
Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the orthogonal group as well as Wang's quantum…
Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and quantum Poincar\'e groups are classified. The results extend (under certain assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on those objects…
We present an elementary type preserving bijection between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis.
We introduce Coxeter-sortable elements of a Coxeter group W. For finite W, we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in…