Geodesic
The irreducible modules of the Terwilliger algebras of Doob schemes
Journal of Algebraic Combinatorics, Tome 6 (1997) no. 2, pp. 173-195.

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Summary: Let $Y$ be any commutative association scheme and we fix any vertex $x$ of $Y$. Terwilleger introduced a non-commutative, associative, and semi-simple C-algebra $T=T(x)$ for $Y$ and $x$ in [4]. We call $T$ the Terwilliger (or subconstituent) algebra of $Y$ with respect to $x$. Let $W$( Ì $C ^{ | X |} ) W( \subset C^{|X|} )$ be an irreducible $T(x)$-module. $W$ is said to be thin if $W$ satisfies a certain simple condition. $Y$ is said to be thin with respect to $x$ if each irreducible $T(x)$ -module is thin. $Y$ is said to be thin if $Y$ is thin with respect to each vertex in $X$.The Doob schemes are direct product of a number of Shrikhande graphs and some complete graphs $K _{4}$ . Terwilliger proved in [4] that Doob scheme is not thin if the diameter is greater than two. I give the irreducible $T(x)$-modules of Doob schemes. 
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     author = {Tanabe, Kenichiro},
     title = {The irreducible modules of the {Terwilliger} algebras of {Doob} schemes},
     journal = {Journal of Algebraic Combinatorics},
     pages = {173--195},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_1997__6_2_a0/}
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Tanabe, Kenichiro. The irreducible modules of the Terwilliger algebras of Doob schemes. Journal of Algebraic Combinatorics, Tome 6 (1997) no. 2, pp. 173-195. http://geodesic.mathdoc.fr/item/JAC_1997__6_2_a0/