Chapter 28 The First Eigenspace of a \(Q\)-DRG

Monday, April 5, 1993

Lemma 28.1 Let \(\Gamma = (X,E)\) be distance-regular of diameter \(D\geq 3\) with standard module \(V\). Suppose \(\Gamma\) is \(Q\)-polynomial with respect to a primitive idempotent \(E_1\). Pick a vertex \(x\in X\). Then \[E_1V = \mathrm{Span}\{E_1\hat{y}\mid \partial(x,y)\leq 2\}.\] In particular, \[\dim E_1V \leq 1 + k_1 + k_2.\]

Proof. Let \(\Delta = \{E_1\hat{y}\mid \partial(x,y)\leq 2\}\).

\(E_1V \supseteq \mathrm{Span}\Delta\): clear.

\(E_1V \subseteq \mathrm{Span}\Delta\): Pick a vertex \(y\in X\). Show that \(E_1\hat{y}\in \mathrm{Span}\Delta\).

Induction on \(h = \partial(x,y)\).

Case \(h\leq 2\).

\(E_1\hat{y} \in \mathrm{Span}\Delta\) follows from construction.

Case \(h\geq 3\).

Pick a vertex \(x'\in X\) such that \[\partial(x,x') = h-3, \quad \partial(x',y) = 3.\] By Theorem 24.1. \[\sum_{z\in X, (x,z)\in R_1, (y,z)\in R_2}E_1\hat{z} - \sum_{z'\in X, (x,z')\in R_2, (y,z')\in R_1}E_1\hat{z'} = r^3_{12}(E_1\hat{x'}-E_1\hat{y}),\] \[r^3_{12} = \frac{c_3(\theta^*_1-\theta^*_2)}{\theta^*_0-\theta^*_3} \neq 0.\] So, \(E_1\hat{y}\in \mathrm{Span}\{f, g, E_1\hat{x'}\}\), where \[f = \sum_{z\in X, (x,z)\in R_1, (y,z)\in R_2}E_1\hat{z}, \quad g = \sum_{z'\in X, (x,z')\in R_2, (y,z')\in R_1}E_1\hat{z'}.\] Observe that each \(z\) in the \(f\)-sum satisfies \(\partial(x,z)=h-2\).

So, by induction hypothesis \[E_1\hat{z}\in \mathrm{Span}\Delta, \quad \text{or }\; f\in \mathrm{Span}\Delta.\]

Observe that each \(z'\) in the \(g\)-sum satisfies \(\partial(x,z')=h-1\).

So by induction hypothesis \[E_1\hat{z'}\in \mathrm{Span}\Delta, \quad \text{or }\; g\in \mathrm{Span}\Delta.\] Also \(\partial(x,x') = h-3\) implies \(E_1\hat{x'}\in \mathrm{Span}\Delta\).

Therefore \(E_1\hat{y} \in \mathrm{Span}\Delta\).

Note. Let \(\Gamma\), \(E_1\), \(x\) be as in Lemma 28.1.

Assume \(D\geq 4\).

Observe that there are many linear dependences among \[\{E\hat{y}\mid y\in \Delta\},\] where \(\Delta = \{y\in X\mid \partial(x,y)\leq 2\}\).

Take any \(y\in X\) such that \(\partial(x,y)\geq 4\).

More than one choice for \(x'\) in the proof of Lemma 28.1 implies

“more than one way to put \(E_1\hat{y}\in \mathrm{Span} E_1\Delta\).”

Open Problem:

\((i)\) Give a precise description of the linear dependences among

\[\{E_1\hat{y}\mid y\in \Delta\}.\]

\((ii)\) Find a subset \(\Delta'\subseteq \Delta\) such that

\[\{E_1\hat{y}\mid y\in \Delta'\}\] is a basis for \(E_1V\), (or find some other ‘nice’ basis for \(E_1V\)).

Conjecture 28.1 Let \(\Gamma\), \(E_1\), \(x\) be as in Lemma 28.1. Set \[\begin{align} \widetilde{X} & = \{y\in X\mid \partial(x,y)\leq 2\},\\ \tilde{\partial} & = \text{the restriction of the distance function $\partial$ to $\widetilde{X}$}. \end{align}\] Then \(\Gamma\) is determined by \(\widetilde{X}\) and \(\tilde{\partial}\).

(There should be some canonical way to reconstruct \(\Gamma\) from \(\widetilde{X}\) and \(\tilde{\partial}\).)