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Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: spin-independent interactions and spin-dependent external fields

Analytical expressions are derived for sums of matrix elements and their squared moduli over many-body states with given total spin --- the states built from spin and spatial wavefunctions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. For spin-dependent one-body interactions with external fields and spin-independent two-body ones between the particles, the sum dependence on the many-body states is given by universal factors, which are independent of the interaction details and Hamiltonians of non-interacting particles. The sum rules are applied to perturbative analysis of energy spectra.