Poincaré polynomials of representations of finite groups generated by reflections (Q790255)

Let G be a finite linear group generated by reflections in a real Euclidean space V of dimension N. For \(g\in GL(V)\) let \(V^ g\) be the subspace of g-fixed elements of V and let r(g) be the dimension of \(V^ g\). Let T be a representation of G of dimension d and let \(\chi\) be its character. The polynomial \(P_ T(t)=1/d\sum_{g\in G}\chi(g)t^{r(g)}\) is called the Poincaré polynomial of T. The author calculates explicitly the Poincaré polynomials of irreducible representations of all groups of the infinite series of finite irreducible groups generated by reflections. In particular if \(G=S_ n\) is a symmetric group and \(N=n-1\) then the Poincaré polynomial of the representation T of G afforded by a Young diagram \(\lambda =(\lambda_ 1,...,\lambda_ p)\) is of the form \(P_ T(t)=\sum^{p}_{i=1}\sum^{\lambda_ i}_{j=1}(t-i+1).\)