Quantum permanents and Hafnians via Pfaffians

TL;DR

This paper introduces quantum determinants, Pfaffians, permanents, and Hafnians within the framework of two-parameter quantum groups, establishing fundamental identities and special cases with integral Laurent polynomial coefficients.

Contribution

It develops a general theory of quantum Pfaffians and Hafnians, proving key identities and identifying special quantum algebras where these invariants have integral Laurent polynomial coefficients.

Findings

Quantum Pfaffians and Hafnians are defined on two-parameter quantum groups.

Fundamental identities among quantum Pf, Hf, and det are established.

Quantum Hafnian equals the quantum permanent and Pfaffian on special quantum algebras.

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.