Quantum Permanents and Hafnians via Pfaffians

Abstract

Abstract. Quantum determinants and Pfaffians or permanents and Hafnians are introduced on the two parameter quantum general linear group. Fundamental identities among quantum Pf, Hf, and det are proved in the general setting. We show that there are two special quantum algebras among the quantum groups, where the quantum Pfaffians have integral Laurent polynomials as coefficients. As a consequence, the quantum Hafnian is computed by a closely related quantum permanent and identical to the quantum Pfaffian on this special quantum algebra.

Capelli identity on multiparameter quantum linear groups

“…Further results have been established for two-parameter and multi-parameter quantum groups [21,2,8,14,11] such as the multiparameter quantum determinant, which converts quantum semigroups into quantum groups. Recently it is known that the quantum Pfaffians can be extended to two-parameter quantum groups as well [13].…”

Multiparameter quantum Pfaffians

“…Further results have been established for two-and multi-parameter quantum groups [11,2,4,8,5] such as the existence of the quantum determinant, which helps to transform quantum semigroups into quantum groups. Recently it is known that the quantum Pfaffians can be extended to two-parameter quantum groups as well [7].…”