$\Gamma $-conjugate weight enumerators and invariant theory

Abstract

Let $K$ be a field, $\Gamma $ a finite group of field automorphisms of $K$, $F$ the $\Gamma $-fixed field in $K$ and $G\leq $GL$_v(K)$ a finite matrix group. Then the action of $\Gamma $ defines a grading on the symmetric algebra of the $F$-space $K^v$ which we use to introduce the notion of homogeneous $\Gamma $-conjugate invariants of $G$. We apply this new grading in invariant theory to broaden the connection between codes and invariant theory by introducing $\Gamma $-conjugate complete weight enumerators of codes. The main result of this paper applies the theory from Nebe, Rains, Sloane to show that under certain extra conditions these new weight enumerators generate the ring of $\Gamma $-conjugate invariants of the associated Clifford-Weil groups. As an immediate consequence we obtain a result by Bannai etal that the complex conjugate weight enumerators generate the ring of complex conjugate invariants of the complex Clifford group. Also the Schur-Weyl duality conjectured and partly shown by Gross etal can be derived from our main result.