Representations of quantum permutation algebras

TL;DR

This paper introduces a combinatorial framework for quantum permutation algebras, exploring their structural properties and applications to quantum invariants of complex Hadamard matrices, with computations up to size six.

Contribution

It develops a new combinatorial approach to quantum permutation algebras and investigates their structural and Tannakian properties, advancing understanding of their representations.

Findings

Analyzed quantum permutation algebras via Hopf images of representations.

Explored conditions for commutativity, cocommutativity, and product decompositions.

Performed computations for quantum invariants of complex Hadamard matrices up to n=6.

Abstract

We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type $\pi:A_s(n)\to B(H)$. We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to $n=6$.