Related papers: On two-coloured noncrossing quantum groups
The total number of noncrossing partitions of type $\Psi$ is the $n$th Catalan number $\frac{1}{n+1}\binom{2n}{n}$ when $\Psi=A_{n-1}$, and the binomial $\binom{2n}{n}$ when $\Psi=B_n$, and these numbers coincide with the correspondent…
Quantum channels, a subset of quantum maps, describe the unitary and non-unitary evolution of quantum systems. We study a generalization of the concept of Pauli maps to the case of multipartite high dimensional quantum systems through the…
We determine the quantum automorphism groups of finite spaces and find they are all compact quantum groups in the sense of Woronowicz. This solves a problem of Connes for finite spaces.
We find that a bipartite quantum state is entangled if and only if it is quantum coherent with respect to complete bases of states in the corresponding system that are distinguishable under local quantum operations and classical…
We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups…
We extend the usual notion of fully commutative elements from the Coxeter groups to the complex reflection groups. Then we decompose the sets of fully commutative elements into natural subsets according to their combinatorial properties,…
A matrix A is image partition regular over Q provided that whenever Q - {0} is finitely coloured, there is a vector x with entries in Q - {0} such that the entries of Ax are monochromatic. It is kernel partition regular over Q provided that…
We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of…
We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification…
In 2009, Blagojevic, Matschke & Ziegler established the first tight colored Tverberg theorem, but no lower bounds for the number of colored Tverberg partitions. We develop a colored version of our previous results (2008), and we extend our…
We determine the structure of automorphism group or each nonsplit metacyclic 2-group. This completes the work on automorphism groups of metacyclic $p$-groups.
We consider indecomposable representations of the Klein four group over a field of characteristic $2$ and of a cyclic group of order $pm$ with $p,m$ coprime over a field of characteristic $p$. For each representation we explicitly describe…
We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum…
We give a general definition of classical and quantum groups whose representation theory is "determined by partitions" and study their structure. This encompasses many examples of classical groups for which Schur-Weyl duality is described…
Given two combinatorial notions $\mathsf{P}_0$ and $\mathsf{P}_1$, can we encode $\mathsf{P}_0$ via $\mathsf{P}_1$. In this talk we address the question where $\mathsf{P}_0$ is 3-coloring of integers and $\mathsf{P}_1$ is product of…
In this paper we will demonstrate that any compact quantum group can be used as symmetry groups for quantum channels, which leads us to the concept of covariant channels. We, then, unearth the structure of the convex set of covariant…
We study the structure of two cointeracting bialgebras on noncrossing partitions appearing in the theory of free probability. The first coproduct is given by separation of the blocks of the partitions into two parts, with respect to the…
Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in…
We study actions of compact quantum groups on type I factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz' results concerning Peter-Weyl theory for…
The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. This extends previous work that appeared in math.QA/0308228. Several important classes of examples of tensor…