Chapter 14 Parameters of Thin Modules, I

Friday, February 19, 1993

Summary.

Definition 14.1 Assume $\Gamma = (X, E)$ is distance-regular with respect to every vertex $x\in X$.

Notation: Let $x\in X$. The data of the trivial $T(x)$-module.

\[\begin{array}{|c|c|c|} \hline & \text{Case DR} & \text{Case DBR} \\ \hline \text{valency} k_x & k & \begin{cases} k^+ & \text{ if } x\in X^+\\ k^- & \text{ if } x\in X^-\end{cases}\\ x\text{-diameter } D_x & D & \begin{cases} D^+ & \text{ if } x\in X^+\\ D^- & \text{ if } x\in X^-\end{cases}\\ \text{measure $m_x$} & m & \begin{cases} m^+ & \text{ if } x\in X^+\\ m^- & \text{ if } x\in X^-\end{cases}\\ \text{int. number }c_i(x) & c_i & \begin{cases} c_i^+ & \text{ if } x\in X^+\\ c_i^- & \text{ if } x\in X^-\end{cases}\\ \text{int. number }b_i(x) & b_i & \begin{cases} b_i^+ & \text{ if } x\in X^+\\ b_i^- & \text{ if } x\in X^-\end{cases}\\ \text{int. number }a_i(x) & a_i & 0\\ \hline \end{array}\]

Call $m$, $m^{\pm 1}$ the measure of $\Gamma$.

Assume $\Gamma = (X, E)$ is distance-regular.

To what extent do $a_i$’s, $b_i$’s and $c_i$’s determine the structure of irreducible $T(x)$-modules? In general, the following hold.

Lemma 14.1 Assume $\Gamma = (X, E)$ is distance-regular. Pick $x\in X$. Let $W$ be a thin irreducible $T(x)$-module with endpoint $r$, diameter $d$ and measure $m_W$.

$(i)$ There is a unique polynomial $f_W \in \mathbb{C}[\lambda]$ with the following properties.

$\quad (ia)$ $\deg f_W \leq D$ (diameter of $\Gamma$).

$\quad (ib)$ $m_W(\theta) = m(\theta)f_W(\theta)$ for every $\theta\in \mathbb{R}$, where $m$ is the measure of $\Gamma$.

Moreover, $f_W\in \mathbb{R}[\lambda]$, and

$(ii)$ $\deg f_W \leq 2r$.
$(iii)$ For all eigenvalues $\theta_i$ of $\Gamma$, $\lambda - \theta_i$ is a factor of $f_W$ whenever, $E_iW = 0$.

In particular, $2r-D+d\geq 0$.

Proof. Let $\theta_0, \ldots, \theta_D$ denote distinct eigenvalues of $\Gamma$. Then $m(\theta_i) \neq 0$ $(0\leq i\leq D)$ by Proposition 13.1.

There exists a unique $f_W\in \mathbb{C}[\lambda]$ with $\deg f_W\leq D$ such that \[f_W(\theta_i) = \frac{m_W(\theta_i)}{m(\theta_i)} \quad (0\leq i\leq D)\] by polynomial interpolation.

$f_W\in \mathbb{R}[\lambda]$ since \[\theta_0, \ldots, \theta_D\in \mathbb{R} \quad \text{and}\quad f_W(\theta_0), \ldots, f_W(\theta_D) \in \mathbb{R}.\]

$(ii)$ Without loss of generality, we may assume $r < D/2$, else trivial.

Pick $0\neq w \in E^*_r(x)W$. \[w = \sum_{y\in W, \partial(x,y) = r}\alpha_y\hat{y} \quad \text{ for some } \; \alpha_y\in \mathbb{C}.\] Pick $y\in X$ such that $\alpha_y\neq 0$.

Set $W'$ be the trivial $T(y)$-module. ($\langle w, \hat{y}\rangle \neq 0, \text{ as } W\not\bot W'$.) \[r' = 0, \quad m' = m, \quad \Delta = r.\]

Apply Theorem 12.1, we have \[\begin{align} \deg p & \leq \Delta - r' + r = 2r, \quad p\neq 0\\ \deg p' & \leq \Delta - r + r' = 0 , \quad p'\neq 0. \end{align}\] \[m_W(\theta)\overline{p'(\theta)} = m(\theta)p(\theta) \quad (\text{ for all } \theta \in \mathbb{R}).\] So, \[\deg p/\bar{p}' \leq 2r,\] and $p/\bar{p}'$ satisfies the conditions of $f_W$. \[\left(\frac{p(\theta)}{\bar{p}'(\theta)} = \frac{m_W(\theta)}{m(\theta)}\right)\]

$(iii)$ \[E_iW = 0 \Rightarrow m_W(\theta_i) = 0 \Rightarrow f_W(\theta_i) = 0.\] that is, $E_iW = 0$. Hence $\theta_i$ is a root of $f_W(\lambda) = 0$. So, \[2r \geq \deg f_W \geq |\{\theta_i\mid E_iW = 0\}| = D-d.\] Hence, \[2r-D + d \geq 0.\] This proves the assertions.

Lemma 14.2 Let $\Gamma = (X, E)$ be any distance-regular graph with valency $k$, diameter $D$ $(D\geq 2)$, measure $m$, and eigenvalues \[k = \theta_0 > \theta_1 > \cdots > \theta_D.\] Pick $x\in X$. Let $W$ be a thin irreducible $T(x)$-module with endpoint $r = 1$, diameter $d$ and measure $m_W = mf_W$. Then one fo the following cases $(i)-(iv)$ occurs.

\[\begin{array}{|c|c|c|c|} \hline \text{Case} & d & f_W(\lambda) & a_0(W) \\ \hline (i) & D-2 & \frac{(\lambda - k)(\lambda - \theta_1)}{k(\theta_1 + 1)} & -\frac{b_1}{\theta_1 + 1} -1\\ (ii) & D-2 & \frac{(\lambda - k)(\lambda - \theta_D)}{k(\theta_D + 1)} & -\frac{b_1}{\theta_D + 1} -1\\ (iii) & D-1 & \frac{k - \lambda}{k} & -1\\ (iv) & D-1 & \frac{(\lambda - k)(\lambda - \beta)}{k(\beta + 1)} & -\frac{b_1}{\beta + 1} -1\\ \hline \end{array}\] for some $\beta\in \mathbb{R}$ with $\beta\in (-\infty, \theta_D) \cup (\theta_1, \infty)$. Moreover, the isomorphism class of $W$ is determined by $a_0(W)$.

Note. By $(iii)$, the possible “shapes” of a thin irreducible $T(x)$-modules are: \[\begin{align} r = 0 & \quad d = D,\\ r = 1 & \quad d = D-1,\\ r = 1 & \quad d = D-2. \end{align}\]