The Haar State values of monomials and a method to pick orthonormal bases on $O(U_q(3))$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912546500378624 |
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| author | Lu, Ting |
| author_facet | Lu, Ting |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Haar State values of monomials and a method to pick orthonormal bases on $O(U_q(3))$ Lu, Ting Quantum Algebra Mathematical Physics 20G42, 33D80 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. |
| title | The Haar State values of monomials and a method to pick orthonormal bases on $O(U_q(3))$ |
| topic | Quantum Algebra Mathematical Physics 20G42, 33D80 |
| url | https://arxiv.org/abs/2508.15346 |