Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra
Abstract
Let
Γ
denote a distance-regular graph with diameter
D≥3
and Bose-Mesner algebra
M
. For
θ∈C∪∞
we define a 1 dimensional subspace of
M
which we call
M(θ)
. If
θ∈C
then
M(θ)
consists of those
Y
in
M
such that
(A−θI)Y∈C
A
D
, where
A
(resp.
A
D
) is the adjacency matrix (resp.
D
th distance matrix) of
Γ.
If
θ=∞
then
M(θ)=C
A
D
. By a {\it pseudo primitive idempotent} for
θ
we mean a nonzero element of
M(θ)
. We use pseudo primitive idempotents to describe the irreducible modules for the Terwilliger algebra, that are thin with endpoint one.