Chapter 36 Dual Endpoint

Wednesday, April 28, 1993

Let $\Gamma = (X, E)$ be distance regular of diameter $D\geq 3$, $Q$-polynomial with respect to $E_0, E_1, \ldots, E_D$. Fix a vertex $x\in X$, write $E^*_i\equiv E^*_i(x)$, $T\equiv T(x)$.

Let $W$ be an irreducible $T$-module of diameter $d$.

Recall that the endpoint \[r(W) = \min\{i\mid 0\leq i\leq D, E^*_iW \neq 0\}.\]

Definition 36.1 The dual endpoint (with respect to above ordering $E_0, E_1, \ldots, E_D$), \[r^*(W) = \min\{i\mid 0\leq i\leq D, E_iW \neq 0\}.\] \[r(W) = 0 \leftrightarrow r^*(W) = 0 \leftrightarrow W: \text{ trivial $T$-module},\] (by Lemma 10.1).

Suppose $W$ is thin. Then $W$ is dual thin. (See Corollary 9.1.)

Moreover, $\{i\mid E_iW \neq 0\}$ is a subinterval of $\{0, 1, \ldots, D\}$. (same proof as for distance regular)

HS MEMO

Dual version of Lemma 4.1.

Lemma 4.1’. Let $A^* \equiv A^*_1(x)$, $W$ an irreducible $T$-moduoe, and $d^* = \{i\mid E_iW\neq 0\}|-1$.

$(i)$ $E_iA^*E_j = 0 \; \text{ if }\; |i-j|>1, \; E_iA^*E_j\neq 0 \; \text{ if }\; |i-j| = 1, \quad 0\leq i,j\leq d^*(x)$.
$(ii)$ $A^*E_jW \subseteq E_{j-1}W + E_jW + E_{j+1}W$, $0\leq j \leq d^*(x)$. $(E_iW = 0 \; \text{ if } i<j$ or $i > d^*(x)$.)
$(iii)$ $E_jW \neq 0$ if $r^*\leq j \leq r^*+d^*$, $E_jW=0$ if $0\leq j\leq r^*$ or $r^*+d^* < j \leq d^*(x)$.
$(iv)$ $E_iA^*E_jW \neq 0$, if $|i-j| = 1$ $(r^* \leq i,j \leq r^*+d^*)$.

Proof of 4.1’

$(i)$ By Lemma 20.3,

\[E_iA^*E_j = O \leftrightarrow q^j_{i1} = 0.\] By Lemma 22.2, \[\begin{align} \Gamma\text{: $Q$-polynomial} &\leftrightarrow q^j_{i1} \;\begin{cases} = 0 & \text{if }|j-i|>1,\\ \neq 0 & \text{if }|j-i|=1.\end{cases}\\ & \leftrightarrow E_iA^*E_j \;\begin{cases} = O & \text{if }|j-i|>1,\\ \neq O & \text{if }|j-i|=1.\end{cases} \end{align}\]

$(ii)$ We have

\[\begin{align} A^*E_jW & = \left(\sum_{i=0}^{D}E_i\right)A^*E_jW\\ & = E_{j-1}A^*E_jW + E_jA^*E_jW + E_{j+1}A^*E_jW\\ & \subseteq E_{j-1}W + E_jW + E_{j+1}W. \end{align}\]

$(iii)$ Suppose $E_jW = 0$ for some $j\in \{r^*, \ldots, r^*+d^*\}$. Then $r^* < j$ by the definition of $r^*$. Set

\[\widetilde{W} = E_{r^*}W + E_{r^*+1}W + \cdots + E_{j-1}W.\] Observe $0\subsetneq \widetilde{W} \subsetneq W$. Also $A\widetilde{W} \subseteq \widetilde{W}$ by $(ii)$, and $E_i^*\widetilde{W} \subseteq \widetilde{W}$ for every $i$ by construction.

Thus, $T\widetilde{W} \subseteq \widetilde{W}$, contradicting $W$ being irreducible.

$(iv)$ Suppose $E_{j+1}A^*E_jW = 0$ for some $j\in \{r^*, \ldots, r^*+d^*-1\}$. Then,

\[\widetilde{W} = E_{r^*}W + E_{r^*+1}W + \cdots + E_{j}W\] is $T$-invariant. If $E_{j-1}A^*E_jW = 0$ for some $j\in \{r^*+1, \ldots, r^*+d^*\}$, then \[\widetilde{W} = E_{j}W + E_{j+1}W + \cdots + E_{r^*+d^*}W\] is $T$-invariant. Moreover, $0 \subsetneq \widetilde{W} \subsetneq W$ in both cases. A contradiction.

Definition. Let $W$ be an irreducible dual thin $T$-module with dual endpoint $r^*$ and diameter $d^*$.

Let $a^*_i = a^*_i(W)\in\mathbb{C}$ satisfying \[E_{r^*+i}A^*E_{r^*+i}|_{E_{r^*+i}W} = a^*_i\cdot 1|_{E_{r^*+i}W}.\] Let $x^*_i = x^*_i(W)\in \mathbb{C}$ satisfying \[E_{r^*+i-1}A^*E_{r^*+i}A^*E_{r^*+i-1}|_{E_{r^*+i-1}W} = x^*_i\cdot 1||_{E_{r^*+i-1}W}.\]

Lemma 9.1’. With above notation, the following hold.

$(i)$ $a^*_i\in \mathbb{R}$ for all $i\in \{0, \ldots, d^*\}$.

$(ii)$ $x^*_i\in \mathbb{R}^{>0}$ for all $i\in \{1, \ldots, d^*\}$.

$(iii)$ Pick $0\neq w^*_0\in E^*_{r^*}W$. Set $w^*_i = E_{r^*+i}{A^*}^iw^*_0$ for all $i$. Then

$\quad (iiia)$ $w^*_0, w^*_1, \ldots, w^*_{d^*}$ is a basis for $W$, $w^*_{-1} = w^*_{d^*+1} = 0$.

$\quad (iiib)$ $A^*w^*_i = w^*_{i+1} + a^*_iw_{i} + x^*_iw^*_{i-1}$ for all $i\in \{0, \ldots, d^*\}$.

$(iv)$ Define $p^*_0, p^*_1, \ldots, p^*_{d^*+1}\in \mathbb{R}[\lambda]$ by

\[p^*_0 = 1, \quad \lambda p^*_i = p^*_{i+1} + a^*_i p^*_i + x^*_i p^*_{i-1} \quad \text{ for all } i\in \{0, \ldots, d^*\}, \quad p^*_{-1} = 0.\]

$\quad (iva)$ $p^*_i(A^*)w^*_0 = w^*_i$, for all $i\in \{0, \ldots, d^*+1\}$.

$\quad (ivb)$ $p^*_{d^*+1}$ is the minimal polynomial of $A^*|_W$.

Proof of Lemma 9.1’

$(i)$ Recall

\[A^* = \sum_{j=0}^D\theta^*_jE^*_j, \quad \theta^*_j = q_1(j) = |X|(E_1)_{xy}\in \mathbb{R}, \; \partial(x,y)=j.\] $a_i^*$ is an eigenvalue of a real symmetric matrix $E_{r^*+i}A^*E_{r^*+i}$.

$(ii)$ Let $B = E^*_{r^*+i}AE^*_{r^*+i-1}$.

Then, $x^*_i$ is an eigenvalue of a real symmetrix matrix $B^\top B$. Let $\mathrm{Span}\{v_{i-1}\} = E_{r^*+i-1}W$, and $Bv_{i-1}\neq 0$ by Lemma 4.1’ $(iv)$ for $i\in \{1, \ldots, d^*\}$. So, $x_i\in \mathbb{R}^{>0}$ for all $i\in \{1, \ldots, d^*\}$.

$(iiia)$ Observe

\[w^*_i = E_{r^*+i}A^*E^*_{r^*+i-1}w^*_{i-1} \quad \text{ for all }i\in \{1, \ldots, d^*\}.\] So $w^*_i \neq 0$ for all $i\in \{1, \ldots, d^*\}$ by Lemma 4.1’ $(iv)$.

Hence, \[W = \mathrm{Span}(w^*_0, \ldots, w^*_d)\] by Lemma 4.1’ $(iii)$.

$(iiib)$ We have that \[\begin{align} A^*w^*_i & = E_{r^*+i+1}A^*w^*_i + E_{r^*+i}A^*w^*_i + E_{r^*+i-1}A^*w^*_i\\ & = w^*_{i+1} + E_{r^*+i}A^*E_{r^*+i}w^*_i + E_{r^*+i-1}A^*E_{r^*+i}A^*E_{r^*+i-1}w_{i-1}\\ & = w^*_{i+1} + a^*_iw^*_{i} + x^*_iw^*_{i-1}. \end{align}\]

$(iva)$ Clear for $i=0$. Assume it is valid for $0, \ldots, i$. \[p^*_{i+1}(A^*)w^*_0 = (A^*-a^*_iI)w^*_i - x^*_iw^*_{i-1} = w^*_{i+1}.\]

$(ivb)$ By definition, \[p^*_{d^*+1}(A^*)w^*_0 = 0.\] Since $W = \{p(A^*)w^*_0\mid p\in \mathbb{C}[\lambda]\}$, $p^*_{d^*+1}(A^*)W = 0$, and $p^*_{d^*+1}$ is a minimal polynomial, as $w^*_0, w^*_1, \ldots, w^*_{d^*}$ is a basis of $W$.

Corollary 9.1’. With the notation above, let $W$ be a dualthin irreducible $T$-module with dual endpoint $r^*(W)$, and dual diameter $d^*$. Then,

$(i)$ $W$ is thin,

$(ii)$ $d^* = d = |\{i\mid E^*_iW\neq 0\}| -1$.

Proof of Corollary 9.1’

Set as in Lemma {4.1}’. \[w^*_i = p^*_i(A^*)w^*_0 \in E_{r^*+i}W.\] Then, $w^*_0, w^*_1, \ldots, w^*_{d^*}$ is a basis for $W$. We have $W = M^*w^*_0$.

So, $E^*_iW = E^*_iM^*w^*_0 = \mathrm{Span}(E^*_iw^*_0)$.

Thus, $W$ is thin, and so, we have $(ii)$.

Suppose $r(W) = 1$. Then $d(W) = D-2$ or $D-1$ by Lemma 14.1 $(iii)$. See also Lemma 14.2.

Case $d(W) = D-2$. Then

$\quad E_1W = 0$ implies $r^*(W) = 2$.

$\quad E_1W\neq 0$ implies $r^*(W) = 1$.

Case $d(W) = D-1$. Then

$\quad r^*(W) = 1$.

Up to isomorphism,

there are at most $3$ thin irreducible $T$-modules with $r(W) =1$ and $r^*(W)=1$,

there are at most $1$ thin irreducible $T$-module with $r(W) = 1$ and $r^*(W)=2$,

there are none thin irreducible $T$-modules with $r(W) = 1$ and $r^*(W) > 2$.

By dual argument,

there are at most $3$ thin irreducible $T$-modules with $r^*(W) =1$ and $r(W)=1$,

there are at most $1$ thin irreducible $T$-module with $r^*(W) = 1$ and $r(W)=2$,

there are none thin irreducible $T$-modules with $r%*(W) = 1$ and $r(W) > 2$.

Conjecture 36.1 Let $\Gamma = (X, E)$ be a thin distance regular graph of diameter $D\geq 3$. Let $E_1$ be any primitive idempotent not equal to $E_0$.

Then the following are equivalent.

$(i)$ For every vertex $x\in X$, there is no irreducible $T$-module $W$ with $r(W)>2$, and $E_1W\neq 0$, there exists at most $1$ irreducible $T$-module with $r(W) =2$, and $E_1W \neq 0$, and there exist at most $3$ irreducible $T$-modules $W$ with $r(W)=1$, and $E_1W\neq 0$.

$(ii)$ $\Gamma$ is $Q$-polynomial with respect to $E_1$.

Conjecture 36.2 Let $\Gamma = (X, E)$ be distance regular of diameter $D\geq 3$, $Q$-polynomial with respect to $E_0, E_1, \ldots, E_D$. Fix a vertex $x\in X$, and write $E^*_i\equiv E^*_i(x)$, $T\equiv T(x)$. Let $W$ denote an irreducible $T$-module with endpoint $r$, dual endpoint $r^*$, diameter $d$ and dual diameter $d^*$.

Then the following hold.

$(i)$ $d = d^*$.

$(ii)$ there exists $s \in \{r, \ldots, r+d\}$ such that

\[1 = \dim E^*_rW \leq \dim E^*_{r+1}W \leq \cdots \leq \dim E^*_sW \geq \cdots \geq \dim E^*_{r+d}W.\]

$(iii)$ there exists $s^* \in \{r^*, \ldots, r^*+d^*\}$ such that

\[1 = \dim E_{r^*}W \leq \dim E_{r^*+1}W \leq \cdots \leq \dim E_{s^*}W \geq \cdots \geq \dim E_{r^*+d^*}W.\]

Conjecture 36.3 The following are equivalent.

$(i)$ The sequence $\dim E^*_1W, \dim E^*_2W, \ldots, E^*_DW$ equals

\[1, 2, 2, \ldots, 2, 1.\]

$(ii)$ $v, Av, A_2v, \ldots, A_{D-2}v$, $v^*, A^*v^*, A^*_2v^*, \ldots, A^*_{D-2}v^*$ is a basis for $W$, where

\[0\neq v\in E^*_iW, \text{ and } v^* = |X|E_1v.\]

$(iv)$ $v^+_1, v^+_2, \ldots, v^+_D, v^-_2, v^-_3, \ldots, v^-_{D-1}$ is a basis for $W$, where

\[v^+_i = E^*_iA_{i-1}v, \quad v^-_i = E^*_iA_{i+1}v.\]

Problem. Let $B$ denote the orthogonal basis for $W$ obtained by applying the Gram-Schemidt procedure to be basis in $(iv)$.

Find the matrix representation $A$ with respect to this basis.

I believe the entries are necely foctorable expressions in the basic variables, \[q, s, s^*, r_1, r_1.\] (Hint: use Theorem 35.1.)

If not, find some nice basis for $W$, and find the matrices representing $A$, $A^*$ with respect to this basis.

Perhaps, some orthogonal basis based on $(iii)$.

Algebraically, everything is determined by the intersection numbers and $a_0(W)$.

Combinatorically, certain quantities mulst be nonnegative integers. Does this give some new bounds, or other information on $a_0(W)$?