Permutation Polynomials on Symmetric Matrices
This paper contains a general characterization for the permutation polynomials of the symmetric matrices over any field. Specific characterizations are for symmetric matrices over algebraically closed fields, principal axis fields, and finite fields. In the latter case enumeration formulas are established as well. More general results concerning permutation polynomials on sets other than the symmetric matrices are sometimes stated.
Teitloff, T. C. (1999). Permutation polynomials on symmetric matrices. Linear Algebra and its Applications, 296(1-3), 233-243. doi:DOI: 10.1016/S0024-3795(99)00132-9