Computing the Haar state on $\mathbb{O}(S{L}_q(3))$

This paper shows that to compute the Haar state on $\mathbb{O}(S{L}_q(n))$, 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 $\mathbb{O}(S{L}_q(3))$ with reasonable computational cost. The numerical results on $\mathbb{O}(S{L}_q(3))$ will be used in the future study of the $q$-deformed Weingarten function.