Semiinvariants of Finite Reflection Groups

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Let G be a finite group of complex n by n unitary matrices generated by reflections acting on Cⁿ. Let R be the ring of invariant polynomials, and χbe a multiplicative character of G. Let Ωχbe the R-module of χ-invariant differential forms. We define a multiplication in Ωχand show that under this multiplication Ωχhas an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between χ-invariant forms and logarithmic forms.