The Haar State values of monomials and a method to pick orthonormal bases on $O(U_q(3))$

In this paper, we investigate the evaluation problem of the Haar state on the quantum group $O(U_q(n))$ ($n\ge 3$) which is a $q$-deformation of the Haar measure on the Lie group $U(n)$. The relation between the Haar state values of monomials on $O(U_q(n))$ is studied. On $O(U_q(3))$, the Haar state values of monomials are explicitly computed and these values are expressed as a finite summation of rational polynomials in $q$. As an application, we compute the Gram matrices of the irreducible co-representations of $O(U_q(3))$ which is essential to the method of constructing orthonormal bases on $O(U_q(3))$ proposed by Noumi, Yamada, and Mimachi. New connections between the Haar state values of monomials and basic hypergeometric multi-summations are found during our computation.