Let K be a field, \(\Gamma \) a finite group of field automorphisms of K, F the \(\Gamma \)-fixed field in K, and \(G\le {{\,\textrm{GL}\,}}_v(K)\) 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 et al. 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 et al. can be derived from our main result.

设K为一个域，\(\Gamma \) K的有限域自同构群，F为K中的\(\Gamma \)固定域，且\(G\le {{\,\textrm{GL} \,}}_v(K)\)有限矩阵群。然后\(\Gamma \)的作用定义了F空间\(K^v\)对称代数的分级，我们用它来引入G的齐次\(\Gamma \)共轭不变量的概念。我们在不变理论中应用这种新的分级，通过引入\(\Gamma \) - 代码的共轭完全权重枚举器来扩大代码和不变理论之间的联系。本文的主要结果应用 Nebe、Rains、Sloane 的理论来表明，在某些额外条件下，这些新的权重枚举器生成相关 Clifford-Weil 群的\(\Gamma \)共轭不变量环。直接的结果是，我们得到了 Bannai 等人的结果。复共轭权重枚举器生成复 Clifford 群的复共轭不变量环。Gross 等人也推测并部分展示了 Schur-Weyl 对偶性。可以从我们的主要结果中得出。