Let V=∧NV\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{V}=\wedge^{N} V}$$\end{document} be the N-fermion Hilbert space with M-dimensional single particle space V and 2N ≤ M. We refer to the unitary group G of V as the local unitary (LU) group. We fix an orthonormal (o.n.) basis |v1⟩,...,|vM〉 of V. Then the Slater determinants ei1,⋯,iN:=|vi1∧vi2∧⋯∧viN⟩\documentclass[12pt]{minimal}
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\begin{document}$${e_{i_1,\cdots,i_N}:= |{v_{i_1}\wedge v_{i_2} \wedge\cdots\wedge v_{i_N}}\rangle}$$\end{document} with i1 < ... < iN form an o.n. basis of V\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{V}}$$\end{document} . Let S⊆V\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{S}\subseteq\mathcal{V}}$$\end{document} be the subspace spanned by all ei1,⋯,iN\documentclass[12pt]{minimal}
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\begin{document}$${e_{i_1,\cdots,i_N}}$$\end{document} such that the set {i1,...,iN} contains no pair {2k−1,2k}, k an integer. We say that the |ψ⟩∈S\documentclass[12pt]{minimal}
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\begin{document}$${|{\psi}\rangle \in\mathcal{S}}$$\end{document} are single occupancy states (with respect to the basis |v1⟩,...,|vM⟩). We prove that for N = 3 the subspace S\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{S}}$$\end{document} is universal, i.e., each G-orbit in V\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{V}}$$\end{document} meets S\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{S}}$$\end{document} , and that this is false for N > 3. If M is even, the well known BCS states are not LU-equivalent to any single occupancy state. Our main result is that for N = 3 and M even there is a universal subspace W⊆S\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{W}\subseteq\mathcal{S}}$$\end{document} spanned by M(M−1)(M−5)/6 states ei1,…,iN\documentclass[12pt]{minimal}
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\begin{document}$${e_{i_1,\ldots,i_N}}$$\end{document} . Moreover, the number M(M−1)(M−5)/6 is minimal.
