On central elements in the Terwilliger algebra of hamming graph

• Main Author: Arcilla, Abigail P.
• Format: text
• Language: English
• Published: Animo Repository 2018
• Subjects: Graph theory; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/554
• Institution: De La Salle University

Let n and D be positive integers with n 3, and let H(D n) denote the Hamming graph. Recall the graph H(D n) is distance-regular of diameter D. Let X denote the vertex set of H(D n), and let MatX(C) denote the C-algebra of matrices with rows and columns indexed by X. Let A 2 MatX(C) denote the adjacency matrix and let @ denote the path-length distance in H(D n). Fix x 2 X. For all i (0 i D), let E i = E i (x) 2 MatX(C) denote the diagonal matrix with yy-entry equal to 1 if @(x y) = i and 0 otherwise, for y 2 X. Let T = T(x) denote the subalgebra of MatX(C) generated by A and E 0 E 1 : : : E D of H(D n). We call T the Terwilliger algebra of H(D n) with respect to x. It is known that A and A generate T, where A = PD i=0 i E i and i = (n {u100000} 1)D {u100000} ni (0 i D). By the center of T, denoted Z(T), we mean the subalgebra of T consisting of elements that commute with all elements of T. This study focuses on describing all elements C 2 Z(T) of H(D n) satisfying the property that for all y z 2 X with @(y z) 2, the yz-entry of C is 0. We show that C = XD i=0 iE i AE i + XD i=0 iE i for some i i 2 C for 0 i D. We determine the scalars i and i. Finally, we prove the conjecture of Terwilliger that the space of all central elements satisfying the given property has basis f Ig where = n(n {u100000} 2)A + n2A {u100000} A2A + 2AA A {u100000} A A2 2n(n {u100000} 1) and I is the identity matrix in MatX(C).