The Terwilliger Algebra of a Distance-Regular Graph of Negative Type

Let $\mathrm{\Gamma }$ denote a distance-regular graph with diameter $D\ge 3$. Assume $\mathrm{\Gamma }$ has classical parameters $(D,b,\alpha ,\beta )$ with $b<-1$. Let $X$ denote the vertex set of $\mathrm{\Gamma }$ and let $A\in MX$ denote the adjacency matrix of $\mathrm{\Gamma }$. Fix $x\in X$ and let $A^{\ast }\in MX$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A, A^$\mathrm{.}Wecall$T$the\text{em}Terwilligeralgebraof$\Gamma$withrespectto$x$\mathrm{.}Weshowthatuptoisomorphismthereexistexactlytwoirreducible$T$-moduleswithendpoint1;theirdimensionsare$D$and$2D-2$\mathrm{.}Forthese$T$-moduleswedisplayabasisconsistingofeigenvectorsfor$A^$, and for each basis we give the action of $A$