The subconstituent algebra of a strongly regular graph
Journal of Algebraic Combinatorics, Tome 22 (2005) no. 1, pp. 5-38.
Voir la notice de l'article dans Electronic Library of Mathematics
Summary: Let $\Gamma $be a distance-regular graph of diameter $D$. Let $X$ denote the vertex set of $\Gamma $and let $Y$ be a nonempty subset of $X$. We define an algebra $\tau = \tau ( Y)$. This algebra is finite dimensional and semisimple. If $Y$ consists of a single vertex then $\tau $is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible $\tau $-modules. We define endpoints and thin condition on irreducible $\tau $-modules as a generalization of the case when $Y$ consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of $Y$, it is thin if and only if $Y$ is a completely regular code of $\Gamma $. By considering a suitable subset $Y$, every irreducible $\tau ( x)$-module of endpoint $i$ can be regarded as an irreducible $\tau ( Y)$-module of endpoint 0.
Mots-clés :
keywords distance-regular graph, association scheme, subconstituent algebra, Terwilliger algebra, tight graph, completely regular code
@article{JAC_2005__22_1_a4,
author = {Suzuki, Hiroshi},
title = {The subconstituent algebra of a strongly regular graph},
journal = {Journal of Algebraic Combinatorics},
pages = {5--38},
publisher = {mathdoc},
volume = {22},
number = {1},
year = {2005},
language = {en},
url = {https://geodesic-test.mathdoc.fr/item/JAC_2005__22_1_a4/}
}
Suzuki, Hiroshi. The subconstituent algebra of a strongly regular graph. Journal of Algebraic Combinatorics, Tome 22 (2005) no. 1, pp. 5-38. https://geodesic-test.mathdoc.fr/item/JAC_2005__22_1_a4/