Quantum-mechanical calculations in the algebraic group theory (Q749670)

[This review concerns also the following two items Zbl 0713.20042 and Zbl 0713.20043.] Methods of quantum mechanics can be used not only to derive some formulas in algebraic group theory, but also to give to these formulas a physical interpretation. Given a simple Lie algebra \({\mathfrak g}\) and a finite field k, a Chevalley group is a finite group generated by \(\exp (tX_{\alpha})\), \(t\in k\), \(X_{\alpha}\in {\mathfrak g}\) (in a Chevalley basis). Here the orders of all Chevalley groups of Lie type over finite fields are shown to be simply related to the partition function Z of n harmonic oscillators over states of a particular symmetry type. There is also an interesting conformal limit when \(n\to \infty\).