Hermite reciprocity for the regular representations of cyclic groups (Q1279786)

Let \(G=C_n\) be the cyclic group of order \(n\), \(V=\mathbb{C} G\) the regular module, \(A:=\text{Sym}(V^*)\) with natural \(G\)-action and \(A^G= \sum^\infty_{i=0} A_i^G\) the corresponding graded ring of polynomial invariants. Using a classical theorem of Molien the authors prove the following theorem: The dimension \(a(m,n)\) of the vector space \(A^G_m\) of degree \(m\) homogeneous invariants is given by \(a(m,n)={1\over n+m} \sum_{d| (n,m)}\varphi(d) {n/d+m/d \choose n/d}\); in particular \(a(n,m)= a(m,n)\). The last equality is, what the authors call `Hermite reciprocity'. After suitable diagonalization, the ring \(A^G\) as a vector space has a monomial basis \(A(n)\) which carries a structure of commutative monoid. The authors prove that for \(n\neq 2\), the automorphism group of the monoid \(A(n)\) is isomorphic to \((\mathbb{Z}/n\mathbb{Z})^*\), the automorphism group of \(G\).