The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Terwilliger, Paul

doi:10.1007/s00373-004-0594-8

Cited by 23 publications

(29 citation statements)

References 37 publications

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“…Using (78) it is routine to verify that (81), (82) satisfy (20), (21), respectively. Therefore the Leonard system has dual q-Krawtchouk type.…”

Section: Irreducible T -Modules and Leonard Systems Of Dual Q-krawtchmentioning

confidence: 99%

Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Worawannotai

2013

Linear Algebra and its Applications

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In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra U q (sl 2 );(iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two U q (sl 2 )-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let denote a dual polar graph. Fix a vertexx of and let T = T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of and a certain diagonal matrix A * = A * (x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A * to W induce a Leonard system of dual qKrawtchouk type. We now describe how (i) and (ii) are related. We obtain two U q (sl 2 )-module structures on the standard module of . We describe how these two U q (sl 2 )-module structures are related. Each of these U q (sl 2 )-module structures induces a C-algebra homomorphism U q (sl 2 ) → T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of we obtain a generating set L, F, R, K of T along with some attractive relations satisfied by these generators.

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“…Using (78) it is routine to verify that (81), (82) satisfy (20), (21), respectively. Therefore the Leonard system has dual q-Krawtchouk type.…”

Section: Irreducible T -Modules and Leonard Systems Of Dual Q-krawtchmentioning

confidence: 99%

Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Worawannotai

2013

Linear Algebra and its Applications

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“…For 1 i n we use the decomposition {U i (s)} 2d i s=0 of V from Lemma 6.1 and the decomposition {V i (s)} 2d i s=0 of V from Lemma 7.1 to construct a third "split" decomposition {W i (s)} 2d i s=0 of V . This construction is motivated by similar split decompositions in [1], [8], and [13]. …”

Section: Split Decompositionsmentioning

confidence: 99%

Irreducible modules for the quantum affine algebra Uq(g) and its Borel subalgebra Uq
Bowman¹
2007
Journal of Algebra
11
0
1
0
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Let g be an affine Kac-Moody Lie algebra, and let U q (g) be its quantized universal enveloping algebra. Let the Borel subalgebra U q (g) 0 of U q (g) be the nonnegative part of U q (g) with respect to the standard triangular decomposition. Suppose ε ∈ {−1, 1} n , where n is the number of simple roots of g. We construct a bijection between finite-dimensional irreducible U q (g) 0 -modules of type ε and finite-dimensional irreducible U q (g)-modules of type ε. In particular:(i) Let V be a finite-dimensional irreducible U q (g) 0 -module of type ε. Then the action of U q (g) 0 on V extends uniquely to an action of U q (g) on V . The resulting U q (g)-module structure on V is irreducible and of type ε. (ii) Let V be a finite-dimensional irreducible U q (g)-module of type ε. When the U q (g)-action is restricted to U q (g) 0 , the resulting U q (g) 0 -module structure on V is irreducible and of type ε.

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“…We now consider for each eigenvalue of H(D, r), what is the corresponding eigenspace and its dimension? To do this, it is convenient to bring in the split decomposition of V (see [10]). We now recall this decomposition.…”

Section: Theorem 23 For the Graph H(d R) The Matrixmentioning

confidence: 99%

A Group Commutator Involving the Last Distance Matrix and Dual Distance Matrix of a Q-Polynomial Distance-Regular Graph: The Hamming Graph Case

Mamart

2018

Graphs and Combinatorics

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Let Γ denote the Hamming graph H(D, r) with r ≥ 3. Consider the distance matrices {A i } D i=0 of Γ. Fix a vertex x of Γ, and consider the dual distance matrices {A * i } D i=0 of Γ with respect to x. We investigate the group commutatorWe show that this matrix is diagonalizable. We compute its eigenvalues and their eigenspaces. Let T denote the subconstituent algebra of Γ with respect to x. We describe the action of A −1 D A * −1 D A D A * D on each irreducible T -module.

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The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Cited by 23 publications

References 37 publications

Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Irreducible modules for the quantum affine algebra Uq(g) and its Borel subalgebra Uq
Bowman¹
2007
Journal of Algebra
11
0
1
0
View full text Add to dashboard Cite

A Group Commutator Involving the Last Distance Matrix and Dual Distance Matrix of a Q-Polynomial Distance-Regular Graph: The Hamming Graph Case

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The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Cited by 23 publications

References 37 publications

Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Irreducible modules for the quantum affine algebra Uq(g) and its Borel subalgebra UqBowman1 2007Journal of Algebra11010View full textAdd to dashboardCite

A Group Commutator Involving the Last Distance Matrix and Dual Distance Matrix of a Q-Polynomial Distance-Regular Graph: The Hamming Graph Case

Contact Info

Product

Resources

About

Irreducible modules for the quantum affine algebra Uq(g) and its Borel subalgebra Uq
Bowman¹
2007
Journal of Algebra
11
0
1
0
View full text Add to dashboard Cite