The Q-polynomial idempotents of a distance-regular graph

Jurišić, Aleksandar; Terwilliger, Paul; itnik, Arjana

doi:10.1016/j.jctb.2010.07.002

Cited by 7 publications

(10 citation statements)

References 9 publications

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“…In (38), evaluate θ * i+1 − θ * i using (14) and evaluate φ i+1 using (16). Simplify the result to get (35).…”

Section: (38)mentioning

confidence: 99%

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Q -polynomial distance-regular graphs and a double affine Hecke algebra of rank one

Lee

2013

Linear Algebra and its Applications

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We study a relationship between Q-polynomial distance-regular graphs and the double affine Hecke algebra of type (C ∨ 1 , C 1 ). Let Γ denote a Q-polynomial distance-regular graph with vertex set X. We assume that Γ has q-Racah type and contains a Delsarte clique C. Fix a vertex x ∈ C. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a C-vector space W.The universal double affine Hecke algebra of type (C ∨ 1 , C 1 ) is the C-algebraĤ q defined by generators {t ±1 n } 3 n=0 and relations (i) t n t −1 n = t −1 n t n = 1; (ii) t n +t −1 n is central; (iii) t 0 t 1 t 2 t 3 = q −1/2 . In this paper, we display anĤ q -module structure for W. For this module and up to affine transformation,• t 0 t 1 + (t 0 t 1 ) −1 acts as the adjacency matrix of Γ;• t 3 t 0 + (t 3 t 0 ) −1 acts as the dual adjacency matrix of Γ with respect to C;• t 1 t 2 + (t 1 t 2 ) −1 acts as the dual adjacency matrix of Γ with respect to x.To obtain our results we use the theory of Leonard systems.

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“…In (38), evaluate θ * i+1 − θ * i using (14) and evaluate φ i+1 using (16). Simplify the result to get (35).…”

Section: (38)mentioning

confidence: 99%

“…Proof. To get (53) evaluate (52) using (14) and Lemma 4.5 together with (39), and simplify. We now show h * = 0.…”

Section: Define the Diagonal Matrixmentioning

confidence: 99%

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Q -polynomial distance-regular graphs and a double affine Hecke algebra of rank one

Lee

2013

Linear Algebra and its Applications

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“…In [5], M. S. Lang introduced the notion of a tail for bipartite distance-regular graphs. In [4], the tail notion was applied to general distance-regular graphs. In [4, Theorem 1.1], these tails were used to characterize Q-polynomial distance-regular graphs.…”

Section: Introductionmentioning

confidence: 99%

A characterization of bipartite Leonard pairs using the notion of a tail

Hanson

2014

Linear Algebra and its Applications

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Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and A * : V → V that satisfy (i) and (ii) below.(i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A * is diagonal.(ii) There exists a basis for V with respect to which the matrix representing A * is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V . Very roughly speaking, a Leonard pair is a linear algebraic abstraction of a Q-polynomial distance-regular graph. There is a well-known class of distance-regular graphs said to be bipartite and there is a related notion of a bipartite Leonard pair. Recently, M. S. Lang introduced the notion of a tail for bipartite distance-regular graphs, and there is an abstract version of this tail notion. Lang characterized the bipartite Q-polynomial distance-regular graphs using tails. In this paper, we obtain a similar characterization of the bipartite Leonard pairs using tails. Whereas Lang's arguments relied on the combinatorics of a distance-regular graph, our results are purely algebraic in nature.

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“…In his thesis [12] Delsarte introduced the Q-polynomial property for a distance-regular graph Γ (see Section 2 for formal definitions). Since then the Q-polynomial property has been investigated by many authors, such as Bannai and Ito [1], Brouwer, Cohen and Neumaier [3], Caughman [4,5,6,7,8,9], Curtin [10,11], Jurišić, Terwilliger, and Žitnik [14], Lang [15,16], Lang and Terwilliger [17], Miklavič [18,19,20,21], Pascasio [22,23], Tanaka [24,25], Terwilliger [26,27,30,32], and Weng [33,34].…”

Section: Introductionmentioning

confidence: 99%

Bipartite Q-polynomial distance-regular graphs and uniform posets

Miklavič

Terwilliger

2012

J Algebr Comb

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Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D ≥ 3. Fix x ∈ X and let L (resp. R) denote the corresponding lowering (resp. raising) matrix. We show that each Q-polynomial structure for Γ yields a certain linear dependency among RL 2 , LRL, L 2 R, L. Define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ∂(x, y) + ∂(y, z) = ∂(x, z), where ∂ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.

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The Q-polynomial idempotents of a distance-regular graph

Cited by 7 publications

References 9 publications

Q -polynomial distance-regular graphs and a double affine Hecke algebra of rank one

Q -polynomial distance-regular graphs and a double affine Hecke algebra of rank one

A characterization of bipartite Leonard pairs using the notion of a tail

Bipartite Q-polynomial distance-regular graphs and uniform posets

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