On the representation theory of partition (easy) quantum groups

Abstract

Abstract Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups ), the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>S</m:mi> <m:mi>N</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> ${S_{N}^{+}}$ , the free orthogonal quantum group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>O</m:mi> <m:mi>N</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> ${O_{N}^{+}}$ as well as the hyperoctahedral quantum group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>H</m:mi> <m:mi>N</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> ${H_{N}^{+}}$ . We then extend our work to unitary easy quantum groups and link it with a “freeness conjecture” of T. Banica and R. Vergnioux.