2002
DOI: 10.1006/eujc.2000.0514
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The Terwilliger Algebra of the Hypercube

Abstract: We give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube Q D of dimension D. Let X denote the vertex set of Q D . Fix a vertex x ∈ X , and let T = T (x) denote the associated Terwilliger algebra. We show that T is the subalgebra of Mat X (C) generated by the adjacency matrix A and a diagonal matrix A * = A * (x), where A * has yy entry D − 2∂(x, y) for all y ∈ X , and where ∂ denotes the path-length distance function. We show that A and A * satisfyUsing the abov… Show more

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Cited by 102 publications
(121 citation statements)
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“…We then display the transition matrices between each pair of these bases. We remark that our paper extends the work of Go [10].…”
Section: Introductionsupporting
confidence: 76%
See 1 more Smart Citation
“…We then display the transition matrices between each pair of these bases. We remark that our paper extends the work of Go [10].…”
Section: Introductionsupporting
confidence: 76%
“…Consequently, results concerning Leonard triples also have interpretations as results concerning such polynomials. Leonard pairs play a role in representation theory [12,14,20,21,28] and combinatorics [4,7,8,10,12,[17][18][19]23]. Consequently, also Leonard triples play a role in representation theory and combinatorics.…”
Section: Introductionmentioning
confidence: 99%
“…Basic properties of the Terwilliger algebra of the Hamming cube were found by Go [6]. In particular, Go identified the irreducible T -modules of the algebra, which implies the block sizes and block multiplicities of A n .…”
Section: Some Backgroundmentioning
confidence: 98%
“…Setting the expression (20) equal to zero for i = 0, 1 and using q = 1, q = −1, we routinely find b = 0, c = 0. But this is inconsistent with (19) and the requirement σ 0 = σ 1 . We have now proved the claim.…”
Section: Let D Denote a Nonnegative Integer And Pickmentioning
confidence: 99%