On the Sum of Class Functions of Weyl Groups

Abstract

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].