A Noncrossing Basis for Noncommutative Invariants of SL(2,C)

Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL(2,\IC) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory free stochastic measures this provides a combinatorial proof of the Molien-Weyl formula in this setting.Comment: AMS LaTeX, 13 pages, using Asymptote picture

Fusion rules for quantum reflection groups

We find the fusion rules for the quantum analogues of the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. The irreducible representations can be indexed by the elements of the free monoid $\mathbb N^{*s}$, and their tensor products are given by formulae which remind the Clebsch-Gordan rules (which appear at $s=1$).Comment: 33 page

Interactions between Algebraic Geometry and Noncommutative Algebra

The workshop discussed the interactions between algebraic geometry and various areas of noncommutative algbebra including finite dimensional algebras, representation theory of algebras and noncommutative algebraic geometry. More than 45 mathematicians participated with a notable number of young mathematicians present

Finite groups acting linearly: Hochschild cohomology and the cup product

When a finite group acts linearly on a complex vector space, the natural semi-direct product of the group and the polynomial ring over the space forms a skew group algebra. This algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics. Its Hochschild cohomology predicts various Hecke algebras and deformations of the orbifold. In this article, we investigate the ring structure of the Hochschild cohomology of the skew group algebra. We show that the cup product coincides with a natural smash product, transferring the cohomology of a group action into a group action on cohomology. We express the algebraic structure of Hochschild cohomology in terms of a partial order on the group (modulo the kernel of the action). This partial order arises after assigning to each group element the codimension of its fixed point space. We describe the algebraic structure for Coxeter groups, where this partial order is given by the reflection length function; a similar combinatorial description holds for an infinite family of complex reflection groups.Comment: 30 page

Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions

The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These objects are no longer in bijection with decreasing (m+1)-ary trees, and a finer congruence, called metasylvester, allows us to build Hopf algebras based on these decreasing trees. At the opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions and quasi-symmetric functions in a natural way. Finally, the algebras of packed words and parking functions also admit such m-analogues, and we present their subalgebras and quotients induced by the various congruences.Comment: 51 page

Nonlocal, noncommutative diagrammatics and the linked cluster Theorems

Recent developments in quantum chemistry, perturbative quantum field theory, statistical physics or stochastic differential equations require the introduction of new families of Feynman-type diagrams. These new families arise in various ways. In some generalizations of the classical diagrams, the notion of Feynman propagator is extended to generalized propagators connecting more than two vertices of the graphs. In some others (introduced in the present article), the diagrams, associated to noncommuting product of operators inherit from the noncommutativity of the products extra graphical properties. The purpose of the present article is to introduce a general way of dealing with such diagrams. We prove in particular a "universal" linked cluster theorem and introduce, in the process, a Feynman-type "diagrammatics" that allows to handle simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams arising from the study of interacting systems (such as the ones where the ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or electric field, by impurities...) or Wightman fields (that is, expectation values of products of interacting fields). Our diagrammatics seems to be the first attempt to encode in a unified algebraic framework such a wide variety of situations. In the process, we promote two ideas. First, Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras. Second, linked cluster-type theorems rely ultimately on M\"obius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingl