Symmetric chains, Gelfand-Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme
Murali K. Srinivasan
DOI: 10.1007/s10801-010-0272-2
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.
Pages: 301–322
Keywords: keywords symmetric chain decomposition; Gelfand-tsetlin bases; symmetric group; Terwilliger algebra; explicit block diagonalization
Full Text: PDF
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2. Canfield, E.R.: A Sperner property preserved by product. Linear Multilinear Algebra 9, 151-157 (1980)
3. Engel, K.: Sperner Theory. Cambridge University Press, Cambridge (1997) 4. de Bruijn, N.G., Tengbergen, C.A.v.E., Kruyswijk, D.: On the set of divisors of a number. Nieuw Arch. Wiskd. 23, 191-193 (1951)
5. Dunkl, C.F.: A Krawtchouk polynomial addition theorem and wreath product of symmetric groups. Indiana Univ. Math. J. 26, 335-358 (1976)
6. Dunkl, C.F.: Spherical functions on compact groups and applications to special functions. Symp. Math. 22, 145-161 (1977)
7. Gijswijt, D., Schrijver, A., Tanaka, H.: New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Comb. Theory, Ser. A 113, 1719-1731 (2006)
8. Go, J.T.: The Terwilliger algebra of the hypercube. Eur. J. Comb. 23, 399-429 (2002)
9. Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Graduate Texts in Mathematics. Springer, Berlin (2009)
10. James, G., Liebeck, M.: Representations and Characters of Groups. Cambridge University Press, Cambridge (2001)
11. Proctor, R.A.: Representations of sl(2, C) on posets and the Sperner property. SIAM J. Algebr. Discrete Methods 3, 275-280 (1982)
12. Proctor, R.A., Saks, M.E., Sturtevant, P.G.: Product partial orders with the Sperner property. Discrete Math. 30, 173-180 (1980)
13. Schrijver, A.: New code upper bounds from the Terwilliger algebra and semidefinite programming.
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