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A complete and rigorous determination of the possible ground states for D-wave pairing Bose condensates is presented, using a geometrical invariant theory approach to the problem. The order parameter is argued to be a vector, transforming according to a ten-dimensional real representation of the group G = O(3)xU(1) x [T]. We determine the equalities and inequalities defining the orbit space of this linear group and its symmetry strata, which are in a one-to-one correspondence with the possible distinct phases of the system. We find 15 allowed phases (besides the unbroken one), with different symmetries, that we thoroughly determine. The group-subgroup relations between bordering phases are pointed out. The perturbative sixth degree corrections to the minimum of a fourth degree polynomial G-invariant free energy, calculated by Mermin, are also determined. (C) 2001 American Institute of Physics

Geometric invariant theory approach to the determination of ground states of D-wave condensates in isotropic space