A Brief Note On The Definition of Signature

Abstract

It is known that signature of a Weyl group element is defined in terms of the number of its simple Weyl reflections. Actual calculations hence are not always possible especially for Weyl groups with higher order like $E_8$ Weyl group. By extending the concept from signature of a Weyl reflection to signature of a weight, we show that signature of a weight is defined without referring to Weyl reflections, Though both have the same result, the signature of a weight can be calculated for any Lie algebra.