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In the Coxeter group W(R) generated by the root system R, let T(R) be thenumber of conjugacy classes having no eigenvalue 1 and let S(R) be the numberof conjugacy classes having no eigenvalue -1. The algebra H{R) of observablesof the rational Calogero model based on the root system R possesses T(R)independent traces, the same algebra considered as an associative superalgebrawith respect to a certain natural parity possesses S(R) even independentsupertraces and no odd trace or supertrace. The numbers T(R) and S(R) aredetermined for all irreducible root systems (hence for all root systems). It isshown that T(R) =< S(R), and T(R) = S(R) if and only if superalgebra H(R)contains a Klein operator (or, equivalently, W(R) containes -1).

journal of nonlinear mathematical physics

Klein operator and the Numbers of independent Traces and Supertraces on  the Superalgebra of Observables of Rational Calogero Model based on the Root  System