A note on De Concini and Procesi's curious identity

TL;DR

This paper provides a combinatorial proof of de Concini and Procesi's volume formula for simplicial cones generated by simple roots, extending it to non-crystallographic root systems.

Contribution

It offers a case-free, combinatorial proof of the volume formula and generalizes it to include non-crystallographic root systems.

Findings

Proof is case-free and combinatorial

Formula extended to non-crystallographic root systems

Simplifies understanding of root system volumes

Abstract

We give a short, case-free and combinatorial proof of de Concini and Procesi's formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.