Sp(N) relies on two identities of Littlewood: where Πf,0 is the irreducible representation of U(2N) with signature f1 ≥ ··· ≥ fN ≥ 0 ≥ ··· ≥ 0. where fi ≥ 0. The branching rule from U(2N) to Sp(N) is given by {| border="1" cellspacing="0" cellpadding="5" | |} where all the signature are non-negative and the coefficient M (g, h; k) is the multiplicity of the irreducible representation k of U(N) in the tensor product g h. It is given combinatorially by the Littlewood–Richardson rule, the number of lattice permutations of the skew diagram k/h of weight g. There is an extension of Littelwood's branching rule to arbitrary signatures due to . The Littlewood–Richardson coefficients M (g, h; f) are extended to allow the signature f to have 2N parts but restricting g to have even column-lengths (g2i – 1 = g2i). In this case the formula reads {| border="1" cellspacing="0" cellpadding="5" | |} where MN (g, h; f) counts the number of lattice permutations of f/h of weight g are counted for which 2j + 1 appears no lower than row N + j of f for 1 ≤ j ≤ |g|/2. Example. The special orthogonal group SO(N) has irreducible ordinary and spin representations labelled by signatures for N = 2n; for N = 2n+1. The fi are taken in Z for ordinary representations and in ½ + Z for spin representations. In fact if an orthogonal matrix U has eigenvalues zi±1 for 1 ≤ i ≤ n, then the character of the corresponding irreducible representation f is given by for N = 2n and by for N = 2n+1. The branching rules from SO(N) to SO(N – 1) state that {| border="1" cellspacing="0" cellpadding="5" | |} for N = 2n + 1 and {| border="1" cellspacing="0" cellpadding="5" | |} for N = 2n, where the differences fi − gi must be integers. Gelfand–Tsetlin basis Since the branching rules from to or to have multiplicity one, the irreducible summands corresponding to smaller and smaller N will eventually terminate in one-dimensional subspaces. In this way Gelfand and Tsetlin were able to obtain a basis of any irreducible representation of or labelled by a chain of interleaved signatures, called a Gelfand–Tsetlin pattern. Explicit formulas for the action of the Lie algebra on the Gelfand–Tsetlin basis are given in . Specifically, for , the Gelfand-Testlin basis of the irreducible representation of with