Der Grad erzeugender Funktionen von Invariantenringen. (The degree of generating functions of rings of invariants)
Let G be a semisimple group defined over an algebraically closed field of characteristic zero and let V be a faithful representation of G. Denote by \(a_ d\) the dimension of the space of G-invariant polynomials on V which are homogeneous of degree d. It is known that the generating function \(\chi (t):=\sum^{\infty}_{d=0}a_ dt\) d is the power series expansion of a rational function f/g, f,g\(\in {\mathbb{Z}}[t]\), and \(\chi\) (t) fulfills the functional equation \(\chi (t^{-1})=\pm t\) \(q\chi\) (t) with \(q=\deg g-\deg f\). The first named author has shown [in Manuscr. Math. 56, 419-427 (1986; Zbl 0585.14033)] that \(q\leq \dim V\) and equality holds iff the set \(Z(V):=\{v\in V|\) dim \(G_ V>0)\}\) has codimension at least two. In this paper we classify all representations of simple groups and all irreducible representations of semisimple groups which do not have this property. For these representations we determine the corresponding q.
scientific article; zbMATH DE number 4033996
scientific article