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

The note is devoted to a factorization formula for the matrix constructed from the generators of the group $\mathrm{GL}(N,\mathbb{R})$ in its regular representation. The factorization formula makes it possible to calculate these generators together with Casimir operators in the case of an arbitrary $N$, and it also clarifies a link between the group $\mathrm{GL}(N,\mathbb{R})$ and the quantum Toda chain.