Computing the Haar state on ${\mathbb{O}(SL_q(3))}$

TL;DR

This paper develops an algorithm to compute the Haar state on quantum groups $ ext{O}(SL_q(3))$, focusing on standard monomials, facilitating future research on $q$-deformed Weingarten functions.

Contribution

It introduces a method to compute Haar states on $ ext{O}(SL_q(n))$ by reducing the problem to standard monomials and provides an explicit algorithm for $ ext{O}(SL_q(3))$.

Findings

Algorithm for Haar state computation on $ ext{O}(SL_q(3))$

Efficient computation of Haar states for standard monomials

Foundation for future studies of $q$-deformed Weingarten functions

Abstract

This paper shows that to compute the Haar state on $\mathbb{O}(SL_q(n))$, it suffices to compute the Haar states of a special type of monomials which we define as standard monomials. Then, we provide an algorithm to explicitly compute the Haar states of standard monomials on $\mathbb{O}(SL_q(3))$ with reasonable computational cost. The numerical results on $\mathbb{O}(SL_q(3))$ will be used in the future study of the $q$-deformed Weingarten function.