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

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