Coinvariant Theory of a Coxeter Group

Let G be a finite group represented on a real vector space V. We can make G act on the polynomial algebra S(V) on V by g · f(x) = f(g−1x). Classical invariant theory studies the invariant subalgebra $$ S{\left( V \right)^G}\;{\rm{ = }}\mathop \oplus \limits_{j = 0}^\infty \;{S_j}{\left( V \right)^G}. $$ Alternatively, one has the graded, homogeneous ideal I _G , generated by the positive components of S(V)G, and we can form the quotient algebra S _G = S(V)/I _G . For convenience, we call this the coinvariant algebra of G and its elements coinvariants (though this terminology has been used for other purposes).