An action of the tetrahedron algebra on the standard module of the shrikhande graph
Date of Publication
12-2010
Document Type
Master's Thesis
Degree Name
Master of Science in Mathematics
Subject Categories
Algebra
College
College of Science
Department/Unit
Mathematics and Statistics
Thesis Adviser
Arlene A. Pascasio
Defense Panel Chair
Ederlina G. Nocon
Defense Panel Member
Edmundo R. Perez
Diana R. Cerzo
Abstract/Summary
In 2007, Hartwig and Terwilliger obtained a presentation of the three-point sl2 loop algebra via generators and relations. In order to do this, they defined a complex Lie algebra , called the tetrahedron algebra, using generators {xij | i, j ∈ {1, 2, 3, 4}, i 6= j} and relations: (i) xij + xji = 0, (ii) [xhi, xij ] = 2xhi + 2xij for mutually distinct h, i, j and (iii) [xhi, [xhi, [xhi, xjk]]] = 4[xhi, xjk] for mutually distinct h, i, j, k.
The Shrikhande graph S was first introduced by S. S. Shrikhande in 1959. It is a distance-regular graph that is not distance-transitive and its intersection numbers coincide with that of the Hamming graph H(2, 4). Let X be the vertex set of S. Let A1 denote the adjacency matrix of S. Fix x ∈ X and let A∗ 1 = A∗ 1 (x) denote the dual adjacency matrix of S. Let T = T(x) denote the subalgebra of M atX(C) generated by A1 and A∗ 1 . In this paper, we exhibit an action of on the standard module of S. To do this, we use the complete set of pairwise non-isomorphic irreducible T−modules Ui’s of S and the standard basis Bi of each Ui which were obtained by Tanabe in 1997. We define matrices A, A ∗ ,B,B ∗ , K, K ∗ , Φ and Ψ in M atX(C) by giving the matrix representations of the restriction maps on Ui with respect to the basis Bi . Finally, we take A ∗ + Ψ + Φ, B ∗ − Φ, A − Ψ + Φ, B − Φ, K − Ψ and K ∗ − Ψ, and show that these matrices satisfy the relations of .
Abstract Format
html
Language
English
Format
Electronic
Electronic File Format
MS WORD
Accession Number
CDTG004870
Shelf Location
Archives, The Learning Commons, 12F Henry Sy, Sr. Hall
Physical Description
1 computer optical disc ; 4 3/4 in.
Keywords
Algebra; Graph theory
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Recommended Citation
Morales, J. S. (2010). An action of the tetrahedron algebra on the standard module of the shrikhande graph. Retrieved from https://animorepository.dlsu.edu.ph/etd_masteral/7193
Embargo Period
3-1-2024
