A formal notion of a Typ T of a self-dual linear code over a finite left R-module V is introduced which allows to give explicit generators of a finite complex matrix group, the associated Clifford-Weil group C(T) ≤ GL|V| (C), such that the complete weight enumerators of self-dual isotropic codes of Type T span the ring of invariants of [Cscr ](T). This generalizes Gleason’s 1970 theorem to a very wide class of rings and also includes multiple weight enumerators (see Section 2.7), as these are the complete weight enumerators cwem (C) = cwe(Rm ⊗ C) of Rm × m -linear self-dual codes Rm ⊗ C ≤ (Vm)N of Type Tm with associated Clifford-Weil group [Cscr ]m(T) = [Cscr ](Tm). The finite Siegel Φ-operator mapping cwem(C) to cwem−1(C) hence defines a ring epimorphism Φm : Inv([Cscr ]m(T)) ⇒ Inv([Cscr ]m−1(T)) between invariant rings of complex matrix groups of different degrees. If R = V is a finite field, then the structure of [Cscr ]m(T) allows to define a commutative algebra of [Cscr ]m (T) double cosets, called a Hecke algebra in analogy to the one in the theory of lattices and modular forms. This algebra consists of self-adjoint linear operators on Inv([Cscr ]m (T)) commuting with Φm. The Hecke-eigenspaces yield explicit linear relations among the cwem of self-dual codes C ≤ VN.
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