q-Krawtchouk polynomials as spherical functions on the Hecke algebra of type B

The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group of type B, contains the generic Hecke algebra for the symmetric group, i.e. the Weyl group of type A, as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of q-Krawtchouk polynomials. The result covers a number of previously established interpretations of (q-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum SU(2) group. Jimbo's analogue of the Frobenius-Schur-Weyl duality is a key ingredient in the proof.Comment: AMS-TeX v. 2.1, 30 page

Hyperoctahedral Chen calculus for effective Hamiltonians

The algebraic structure of iterated integrals has been encoded by Chen. Formally, it identifies with the shuffle and Lie calculus of Lyndon, Ree and Sch\"utzenberger. It is mostly incorporated in the modern theory of free Lie algebras. Here, we tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics, quantum field theory or engineering. We show, among others, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion.Comment: Minor corrections. Some misleading notations and typos in the first version have been fixe

Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators $\{x_r\}_{r \in \Lambda}$ in a tracial C$^\ast$-probability space $(\mathcal A, \phi)$, whose joint $\ast$-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups $\{H_n^+\}_{n \in \N}$. We prove a strong form of Haagerup's inequality for the non-self-adjoint operator algebra $\mathcal B$ generated by $\{x_r\}_{r \in \Lambda}$, which generalizes the strong Haagerup inequalities for $\ast$-free R-diagonal families obtained by Kemp-Speicher \cite{KeSp}. As an application of our result, we show that $\mathcal B$ always has the metric approximation property (MAP). We also apply our techniques to study the reduced C$^\ast$-algebra of the free unitary quantum group $U_n^+$. We show that the non-self-adjoint subalgebra $\mathcal B_n$ generated by the matrix elements of the fundamental corepresentation of $U_n^+$ has the MAP. Additionally, we prove a strong Haagerup inequality for $\mathcal B_n$, which improves on the estimates given by Vergnioux's property RD \cite{Ve}

Stochastic aspects of easy quantum groups

We consider several orthogonal quantum groups satisfying the easiness assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(u^k) with respect to the Haar measure, u being the fundamental representation. For the classical groups O_n, S_n we recover in this way some well-known results of Diaconis and Shahshahani.Comment: 28 page

Liberation of orthogonal Lie groups

We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: $O_n,S_n,H_n,B_n,S_n',B_n'$. We investigate the representation theory aspects of the correspondence, with the result that for $O_n,S_n,H_n,B_n$, this is compatible with the Bercovici-Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.Comment: 42 page