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In an orthonormal basis of localized atomic orbitals, a ring of three equivalent sites with real hopping $t$ has Hamiltonian $H=(E_0-t)I+tJ$, where $J$ is the three-by-three all-ones matrix. The normalized symmetric state $(|1⟩+|2⟩+|3⟩)/\sqrt{3}$ has energy $E_0+2t$. Its two-dimensional orthogonal complement has energy $E_0-t$. Thus the symmetric state is the unique spatial ground state for $t<0$, with an energy difference $3|t|$ to the excited eigenstates. Equivalently, the three discrete Bloch states have energies $E_0+2t\cos (2\pi l/3)$, $l=0,\pm 1$. Nonorthogonal atomic orbitals instead require a generalized eigenvalue problem.

The last-column map has fibre the subgroup fixing the last standard unit vector. Unitary completion gives surjectivity, and compactness gives the quotient homeomorphism. Projecting the other standard vectors onto the orthogonal complement of a nearby unit vector and applying Gram-Schmidt gives a local section, so the projection is a fiber bundle. The bundle $U(n)\to U(n+1)\to S^{2n+1}$ then gives ${\pi }_{i}U(n)\cong {\pi }_{i}U(n+1)$ for $i\le 2n-1$ because both adjacent sphere homotopy groups vanish.