The Casimir invariants and Gel'fand basis of the graded unitary group SU(m/n)

Abstract

In the space of f-particle product states, the Casimir invariants of the graded unitary group SU(m/n) are functions of the class operators of the graded coordinate permutation group S(f), which are equal to the class operators of the graded state permutation group S(f). Formulae for the quadratic and cubic Casimir invariants are given. It is shown that in general the eigenvalue formulae for the U(m/n) Casimir invariants as functions of the partition can be obtained from those for U(m+n) by simply changing the argument of the functions from (m+n) to (m-n). The quasi-standard basis of S(f) is identified with the Gel'fand basis of SU(m/n), i.e. the irreducible basis classified according to the group chain SU(m/n) contains/implies SU(m/n-1) contains/implies ...SU(m) contains/implies ...SU(2) contains/implies U(1). The special Gel'fand basis of SU(m/n) (for which the weight is restricted to (1,1,...,1)) can be easily obtained from the Gel'fand basis of SU(m+n) with due allowances for modifying sign factors arising from the grading. The construction of a general SU(m/n) Gel'fand basis is also discussed.