The Weyl group of the fine grading of $sl(n,\mathbb{C})$ associated with tensor product of generalized Pauli matrices

TL;DR

This paper characterizes the Weyl group of the fine grading of sl(n,C) induced by tensor products of generalized Pauli matrices, showing it is the isometry group of a symplectic abelian group and detailing its structure.

Contribution

It proves that the Weyl group corresponds to the isometry group of a symplectic abelian group, providing a detailed structural understanding of the symmetry in this grading.

Findings

Weyl group equals the isometry group of the symplectic abelian group

Any finite maximal diagonalizable subgroup is a symplectic abelian group

Isometry groups are generated by transvections

Abstract

We consider the fine grading of $sl(n,\mb C)$ induced by tensor product of generalized Pauli matrices in the paper. Based on the classification of maximal diagonalizable subgroups of $PGL(n,\mb C)$ by Havlicek, Patera and Pelantova, we prove that any finite maximal diagonalizable subgroup $K$ of $PGL(n,\mb C)$ is a symplectic abelian group and its Weyl group, which describes the symmetry of the fine grading induced by the action of $K$, is just the isometry group of the symplectic abelian group $K$. For a finite symplectic abelian group, it is also proved that its isometry group is always generated by the transvections contained in it.