INVARIANTS AND COINVARIANTS OF FINITE PSEUDOREFLECTION GROUPS, JACOBIAN DETERMINANTS AND STEENROD OPERATIONS

Let $\rho:G\hookrightarrow\GL(n,\F)$ be a representation of a finite group $G$ over a finite field $\F$ and $f_1,\dots,f_n\in\F[V]^G$ such that the ring of invariants is a polynomial algebra $\F[f_1,\dots,f_n]$. It is known that in this case the algebra of coinvariants $\F[V]_G$ is a Poincaré duality algebra, and if, moreover, the order of $G$ is invertible in $\F$, that a fundamental class is represented by the Jacobian determinant $\mathchoice{\det\biggl[\frac{\partial f_i}{\partial z_j}\biggr]}{\det[\partial f_i/\partial z_j]}{\det[\partial f_i/\partial z_j]} {\det[\partial f_i/\partial z_j]}$, and is therefore a $\det^{-1}$-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.AMS 2000 Mathematics subject classification: Primary 13A50; 55S10