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

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