Prove that any morphism of Woronowicz algebras

increases the moments of the main character, and that such a morphism is an isomorphism precisely when all these moments, and so the character laws, are the same.

Consider the symmetric group [math]S_N[/math], regarded as symmetry group of the [math]N[/math] coordinate axes of [math]\mathbb R^N[/math], and so as group of orthogonal matrices:

Compute the main character for this group, then the law of this main character, and work out the [math]N\to\infty[/math] asymptotics.

Work out the formulae of the Gram and Weingarten matrices for all the easy quantum groups introduced so far, up to the size [math]5\times5[/math].

Work out explicitely the asymptotic laws of the main characters for the half-classical quantum groups [math]O_N^*,U_N^*[/math].

Work out explicitely the asymptotic laws of the main characters for the bistochastic quantum groups [math]B_N,B_N^+,C_N,C_N^+[/math].