Regular representation of the group \(\mathrm{GL}(N, \mathbb{R})\): factorization, Casimir operators and Toda chain (Q2160258)

The Regular representation appears in the reduction of a lot of irreducible representations of finite and continuous groups. In this context, this note is devoted to the factorization formula for the matrix with the generators of the group \(\mathrm{GL}(N, \mathbb R)\) in the regular representation. Moreover the factorization formula helps to evaluate these generators and Casimir operators in the case of arbitrary \(N\) (Gauss parametrization). The approach is based on the following observation: the matrix whose elements are the generators in the regular representation admits a factorization into the simpler parts, which can be easily written in an explicit form for an arbitrary \(N\) and clarifies the connection between \(\mathrm{GL}(N, \mathbb R)\) and the quantum Toda chain.