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

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 (up to scalars) symmetric Gelfand-Tsetlin basis.

Wreath Product Action on Generalized Boolean Algebras

“…The following formulation is motivated by the one given in [8] (also see [9]). We do not use part (v) (a classical result of Delsarte [4]) in this paper but we have included it for completeness.…”

“…For a subspace analog (or q-analog) of Theorem 2.1, see [1,10]. For explicit constructions of orthogonal SJB's of V (B(n)) and V (B q (n)) (the subspace analog of V (B(n))) see [9,11].…”

Notes on Explicit Block Diagonalization

“…Theorem 2 was proved by Go [10] using the sl(2, C) method. For an explicit construction of an orthogonal SJB J (n), together with a representation theoretic interpretation, see Srinivasan [18]. It would be interesting to give an explicit construction of an orthogonal SJB J (q, n) of V (B(q, n)).…”

“…2, we recall (without proof) a result of Terwilliger [22] on the singular values of the up operator on subspaces. In Srinivasan [18], the q = 1 case of this result, together with binomial inversion, was used to derive Schrijver's [17] explicit block diagonalization of the commutant of the S n action on B(n). In Sect.…”

The Goldman-Rota Identity and the Grassmann Scheme

“…Substituting q = 1 in Theorem 1.2 we recover the explicit orthogonal SJB of V (B(n)) constructed in [17]. This basis was given a representation theoretic characterization in [17], namely, that it is the canonically defined symmetric Gelfand-Tsetlin basis of V (B(n)). Similarly, the basis J q (n) should also be studied from a representation theoretic viewpoint.…”