Trace of powers of representations of finite quantum groups

Abstract

In this paper we study (asymptotic) properties of the * -distribution of irreducible characters of finite quantum groups. We proceed in two steps, first examining the representation theory to determine irreducible representations and their powers, then we study the * -distribution of their trace with respect to the Haar measure. For the Sekine family we look at the asymptotic distribution (as the dimension of the algebra goes to infinity).

Random Walks on Finite Quantum Groups

“…This is the smallest Hopf-von Neumann algebra which is neither commutative nor cocommutative. We use the notations given in [1]. In particular, the multimatrix algebra C(KP) = C ⊕ C ⊕ C ⊕ C ⊕ M 2 (C) is endowed with the canonical basis {e 1 , e 2 , e 3 , e 4 , E 11 , E 12 , E 21 , E 22 }, where E i j is the image of a matrix unit.…”

“…We use the definition and the representation theory of the Sekine family of finite quantum groups presented in [1]. Let us recall that for each n greater than or equal to 2,…”

“…This is the smallest Hopf-von Neumann algebra which is neither commutative nor cocommutative. We use the notations given in [1]. In particular, the multimatrix algebra…”

“…Then, by looking at the different cases for g to satisfy (1) and not (4), we get: Proposition 2.1. Assume that the random walk associated to g does not converge to the Haar state φ 8 .…”

“…g 1 which define a walk on the Cayley graph of G. A frequently asked question is when the distribution π ⋆k of the current position h k is close to the uniform distribution on G. Diaconis and Shahshahani answer this question in [4] for the case of permutation groups, by using representation theory. They also observe that after less than t n random transpositions a deck of n cards is not well mixed, but after more than t n random transpositions it is well mixed, where t n = 1 2 n log(n). This is called a cut-off.…”