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The de Bruijn-Tengbergen-Kruyswijk (BTK) construction is a simple algorithmthat produces an explicit symmetric chain decomposition of a product of chains.We linearize the BTK algorithm and show that it produces an explicit symmetricJordan basis (SJB). In the special case of a Boolean algebra the resulting SJBis orthogonal with respect to the standard inner product and, moreover, we canwrite down an explicit formula for the ratio of the lengths of the successivevectors in these chains (i.e., the singular values). This yields a new,constructive proof of the explicit block diagonalization of the Terwilligeralgebra of the binary Hamming scheme. We also give a representation theoreticcharacterization of this basis that explains its orthogonality, namely, that itis the canonically defined (upto scalars) symmetric Gelfand-Tsetlin basis.

Symmetric chains, Gelfand-Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme