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

TL;DR

This paper linearizes the BTK algorithm to produce explicit symmetric Jordan bases for the binary Hamming scheme, providing new proofs and characterizations related to the Terwilliger algebra and Gelfand-Tsetlin bases.

Contribution

It introduces a linearized version of the BTK algorithm that constructs explicit symmetric Jordan bases and links them to Gelfand-Tsetlin bases, offering new insights into the Terwilliger algebra.

Findings

Explicit symmetric Jordan bases for Boolean algebras

Orthogonality of the basis with explicit singular value formulas

Constructive proof of block diagonalization of the Terwilliger algebra

Abstract

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 (upto scalars) symmetric Gelfand-Tsetlin basis.