Trace form preservers Original Research Article

If image, the set of n × n matrices over a field, define Td(X) = tr(Xd). A “preserver” of Td is a linear map image with: tr(f(X)d) = tr(Xd) for every X. A transformation is “standard” if it lies in the group generated by maps of the types: (1) conjugation by something in GLn(K); (2) transpose; and (3) multiplication by a scaler dth root of unity. If d> 2 and d! ≠ 0 in K, OʹRyan-Shapiro (M. OʹRyan, D.B. Shapiro, Linear Alg. Appl. 246 (1996) 313–333) proved that every preserver of Td is standard. Guralnick (R.M. Guralnick, Linear Alg. Appl. 212/213 (1994) 249–257), using methods of algebraic group theory, extended this to infinite fields of characteristic p> 0. When d = 1 + pm, he found a fourth type of Td-preserver: (4) fλ(X) = X + λ(tr X), where λ ε K satisfies nλd + λd−1 + λ = 0. Call a transformation “λ-standard” if it is the group generated by the standard maps together with these fλʹs, if they exist.