TY - JOUR
T1 - The terwilliger algebra of the twisted grassmann graph
T2 - The thin case
AU - Tanaka, Hajime
AU - Wang, Tao
N1 - Funding Information:
TW gratefully acknowledges financial support from Professor Tatsuro Ito so that TW can concentrate on this research. HT was supported by JSPS KAKENHI Grant Numbers JP17K05156 and JP20K03551.
Publisher Copyright:
© The authors.
PY - 2020
Y1 - 2020
N2 - The Terwilliger algebra T (x) of a finite connected simple graph Γ with respect to a vertex x is the complex semisimple matrix algebra generated by the adjacency matrix A of Γ and the diagonal matrices Ei∗(x) = diag(vi) (i = 0, 1, 2, … ), where vi denotes the characteristic vector of the set of vertices at distance i from x. The twisted Grassmann graph Jq (2D +1, D) discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that T (x) is thin whenever x is chosen from it, i.e., every irreducible T (x)-module W satisfies dim Ei∗(x)W ≤ 1 for all i. In this paper, we determine all the irreducible T (x)-modules of Jq (2D + 1, D) for this “thin” case.
AB - The Terwilliger algebra T (x) of a finite connected simple graph Γ with respect to a vertex x is the complex semisimple matrix algebra generated by the adjacency matrix A of Γ and the diagonal matrices Ei∗(x) = diag(vi) (i = 0, 1, 2, … ), where vi denotes the characteristic vector of the set of vertices at distance i from x. The twisted Grassmann graph Jq (2D +1, D) discovered by Van Dam and Koolen in 2005 has two orbits of the automorphism group on its vertex set, and it is known that one of the orbits has the property that T (x) is thin whenever x is chosen from it, i.e., every irreducible T (x)-module W satisfies dim Ei∗(x)W ≤ 1 for all i. In this paper, we determine all the irreducible T (x)-modules of Jq (2D + 1, D) for this “thin” case.
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U2 - 10.37236/9873
DO - 10.37236/9873
M3 - Article
AN - SCOPUS:85095448716
VL - 27
SP - 1
EP - 22
JO - Electronic Journal of Combinatorics
JF - Electronic Journal of Combinatorics
SN - 1077-8926
IS - 4
M1 - P4.15
ER -