Chapter 32 Irreducible Modules of Endpoint \(i\)

Monday, April 19, 1993

Lemma 32.1 Let \(\Gamma = (X, E)\) be any graph. With the notation of Lemma 31.1, the following hold.

\((i)\) Let \(W\) denote a thin irreducible \(T\)-module with endpoint \(r\), diameter \(d\). Pick \(i\) \((0\leq i\leq d)\), and pick \(v\in E^*_{r+i}W\). Then,

\[R^{-1}R v = \begin{cases} v & \text{if } i<d,\\ 0 & \text{if } i = d. \end{cases}\]

\((ii)\) Assume \(\Gamma\) is distance regular and thin with respect to \(x\). Pick \(t\) \((0\leq i < D/2)\), and pick \(v\in E^*_tV\). Then

\[R^{-1}R^iv = R^{i-1}v\quad (1\leq i\leq D-2t).\] In particular, \(R^{-1}R v = v\).

\((iii)\) Assume \(\Gamma\) is distance regular and thin with respect to \(x\). Then

\[R: E^*_iV \to E^*_{i+1}V \quad (0\leq i< D/2)\] is one-to-one.

Proof.

\((i)\) Let \(w_0, w_1, \ldots, w_d\) be a basis for \(W\) and \(w_i\in E^*_{r+i}W\),

\[Rw_i = w_{i+1} \quad (0\leq i< d), \quad Lw_i = x_i(W)w_{i-1}\quad (1\leq i\leq d).\] So, \[RLw_i = x_i(W)w_i \quad (1\leq i\leq d).\] (See Lemma 9.1.)

We want to find \(R^{-1}Rw_i\).

If \(i=d\), \(R^{-1}Rw_d = 0\).

If \(0\leq i<d\), \[\begin{align} R^{-1}Rw_i & = R^{-1}w_{i+1}\\ & = x_{i+1}(W)^{-1}R^{-1}RLw_{i+1}\\ & = x_{i+1}(W)^{-1}Lw_{i+1}\\ & = x_{i+1}(W)^{-1}x_{i+1}(W)w_i\\ & = w_i. \end{align}\] Thus, we have \((i)\).

HS MEMO

\[\begin{align} RLw_i & = Rx_i(W)w_{i-1} = x_i(W)w_i,\\ LRw_i & = Lw_{i+1} = x_{i+1}(W)w_i,\\ [L,R]w_i & = (x_{i+1}(W)-x_i(W))w_i, \quad (0\leq i\leq d),\\ & \qquad x_0(W) = 0, \quad x_{d+1}(W) = 0, \\ [L,R]|_W & = \sum_{i=0}^d(x_{i+1}-x_i(W))E^*_{r+i}|_W. \end{align}\]

\((ii)\) Let

\[V = \sum W \quad \text{orthogonal direct sum of thin irreducible $T$-modules.}\] Then, \[E^*_tV = \sum_{r(W)\leq t}E^*_tW\quad (\text{orthognal direct sum}).\] Without loss of generality, we may assume \[v\in E^*_t W\] for some thin irreducible \(T\)-module with endpoint at most \(t\).

Now if \(i\leq D-2t\), then \[\begin{align} t+i & \leq D-t \\ & \leq D-r(W)\\ & \leq r(W) + d(W) \quad (D\leq 2r+d), \end{align}\] by Lemma 14.1 \((iii)\).

So \[t+i-1\leq r(W) + d(W) -1.\] Hence, \[\begin{align} R^{-1}R^iv & = R^{-1}R(R^{i-1}v) \quad (R^{i-1}v\in E^*_{t+i-1}W)\\ & = R^{i-1}v \quad \text{by $(i)$.} \end{align}\]

\((iii)\) Suppose \(Rv = 0\) for some \(v\in E^*_iV\) \((0\leq i< D/2)\). Then

\[0 = R^{-1}Rv = v,\] by \((ii)\) with \(t = i\) and \(i=1\).

Definition 32.1 Let \(\Gamma = (X, E)\) denote any graph with the standard module \(V\). Fix a vertex \(x\in X\). Write \(E^*_i \equiv E^*_i(x)\), \(T \equiv T(x)\), \(L\equiv L(x)\).

1. For every \(i\) \((0\leq i\leq D)\), define subspace \(V_i := V_i(x) \subseteq V\) by \[V_i = \sum W,\] where the sum begin over irreducible \(T\)-modules \(W\) with endpoint \(i\).

Observe: \[V = V_0 + V_1 + \cdots + V_D \quad (\text{orthogonal direct sum.})\] \(V_0\) is the trivial \(T\)-module.

2. \((E^*_iV)_{new} \equiv E^*_iV_i \quad (0\leq i\leq D).\)

In general, \[(E^*_iV)_{new} \subseteq \mathrm{Ker}L \cap E^*_iV \subseteq \mathrm{Ker}L \cap E^*_iV \subseteq \mathrm{Ker}(LE^*_i).\]

If each irreducible \(T\)-module with endpoint strictly less than \(i\) is thin, \[(E^*_iV)_{new} = \mathrm{Ker}L \cap E^*_iV \subseteq \mathrm{Ker}(L\cdot E^*_i).\] We have the assertion.

HS MEMO

\[E^*_iV = \sum_{j<i} V_j + V_i.\] For \(V_j\) part, take \(w_{i-j}\in W\) irreducible with endpoint \(j<i\). Then, \[Lw_{i-j} = x_{i-j}(W)w_{i-j-1}\neq 0,\] and \[L|_{\sum_{j<i}E^*_iV_j}: \sum_{j<i}E^*_iV_j \to V\] is one to one.

Lemma 32.2 Let \(\Gamma = (X, E)\) be distance regular of diameter \(D\geq 3\). Fix a vertex \(x\in X\), \(R\equiv R(x)\). \(L\equiv L(x)\), \(F \equiv F(x)\). Pick \(v\in (E^*_1V)_{new}\). Then,

\((i)\) \(RE^*_iA_{i-1}v = c_iE^*_{i+1}A_iv\quad (1\leq i\leq D)\).

\((ii)\) \(FE^*_iA_{i-1}v = RE^*_{i-1}A_iV + (a_{i-1}-c_i+c_{i-1})E^*_iA_{i-1}v + c_{i}E^*_{i}A_{i+1}v\) \((1\leq i\leq D)\).

\((iii)\) \(LE^*_iA_{i-1}v = FE^*_{i-1}A_iV + (a_{i-1}-c_i+c_{i-1})E^*_{i-1}A_{i}v + b_{i-1}E^*_{i-1}A_{i-2}v\) \((2\leq i\leq D)\).

\((iv)\) \(LE^*_iA_{i+1}v = b_iE^*_{i-1}A_iv \quad (1\leq i\leq D-1)\).

Proof.

\((i)\) Let

\[v = \sum_{y\in X, \partial(x,y)=1}\alpha_y \hat{y} \quad \text{for some }\{\alpha_g\}\subseteq \mathbb{C}.\] Then \[Lv = \left(\sum_{y\in X, \partial(x,y)=1}\alpha_y\right)\hat{x} = 0.\] So, \[\sum_{y\in X, \partial(x,y)=1}\alpha_y = 0.\] Thus, \[v = \sum_{y\in X, \partial(x,y)=1}\alpha_y(\hat{y}-\hat{x}).\] Let \[\tilde{A}_i = A_0 + A_1 + \cdots + A_i \quad (0\leq i\leq D).\] Then \[\begin{align} \tilde{A}_iv & = \sum_{y\in X, \partial(x,y)=1}\alpha_y \tilde{A}_i(\hat{y}-\hat{x})\\ & = \sum_{y\in X, \partial(x,y)=1}\alpha_y\left(\sum_{z\in X, \partial(y,z)=i, \partial(x,z)=i+1}\hat{z}-\sum_{z'\in X, \partial(y,z')=i+1, \partial(x,z')=i}\hat{z}'\right)\\ & = \sum_{y\in X, \partial(x,y)=1}\alpha_y(E^*_{i+1}A_i\hat{y}- E^*_iA_{i+1}\hat{y})\\ & = E^*_{i+1}A_iv - E^*_{i+1}A_{i+1}v. \end{align}\]

Recall (Claim 1 in the proof of Theorem 16.1.) \[A\tilde{A}_i = c_{i+1}\tilde{A}_{i+1}+ (a_i-c_{i+1}+c_i)\tilde{A}_i + b_i\tilde{A}_{i-1}\quad (0\leq i\leq D-1).\] (This is valid for \(i=0\) as \(A\tilde{A}_0 = AI = c_1\tilde{A}- \tilde{A}_0 = A\) by setting \(\tilde{A}_{i-1} = O\).)

Now \((i)-(iv)\) are obtained by applying this to \(v\) on the right and multiplied by \(E^*_j\) \((0\leq j\leq D)\) on the left.

HS MEMO

\(A\tilde{A}_{i-1}v = AE^*_iA_{i-1}v - AE^*_{i-1}A_iv\). For \(1\leq i\leq D\), \[\begin{align} & (c_i\tilde{A}_i + (a_{i-1}-c_i+c_{i-1})\tilde{A}_{i-1}+b_{i-1}\tilde{A}_{i-2})v\\ & \quad = c_i E^*_{i+1}A_iv - c_iE^*_iA_{i+1}v\\ & \qquad + (a_{i-1}-c_i+c_{i-1})E^*_iA_{i-1}v - (a_{i-1}-c_i+c_{i-1})E^*_{i-1}A_iv\\ & \qquad + b_{i-1}E^*_{i-1}A_{i-2}v - b_{i-1}E^*_{i-2}A_{i-1}v. \end{align}\]

\((i)\) \(RE^*_iA_{i-1}v = E^*_{i+1}AE^*_iA_{i-1}v = c_iE^*_{i+1}A_iv\) \((1\leq i\leq D)\).

\((ii)\) For \(1\leq i\leq D\),

\[\begin{align} FE^*_iA_{i-1}v & = E^*_{i}AE^*_iA_{i-1}v\\ & = RE^*_{i-1}A_iv - c_iE^*_iA_{i+1}v + (a_{i-1}-c_i+c_{i-1})E^*_iA_{i-1}v. \end{align}\]

\((iii)\) For \(2\leq i\leq D\),

\[\begin{align} LE^*_iA_{i-1}v & = E^*_{i-1}AE^*_{i}A_{i-1}v\\ & = FE^*_{i-1}A_iv - (a_{i-1}-c_i+c_{i-1})E^*_{i-1}A_i v + b_{i-1}E^*_{i-1}A_{i-2}v. \end{align}\] (Even if \(i=1\), this is valid by setting \(A_{i-2}=O\).)

\((iv)\) For \(1\leq i\leq D-1\), \(LE^*_iA_{i+1}v = E^*_{i-1}AE^*_i A_{i+1} = b_iE^*_{i-1}A_iv\).

Lemma 32.3 Let \(\Gamma = (X, E)\) be distance regular of diameter \(D\geq 3\). Fix a vertex \(x\in X\), \(T\equiv T(x)\), \(E^*_i\equiv E^*_i(x)\), \(R = R(x)\), \(F = F(x)\), \(L = L(x)\).

For every \(v\in (E^*_1V)_{new}\), the following are equivalent.

\((i)\) \(E^*_iA_{i-1}v\), \(E^*_{i}A_{i+1}v\) are linearly dependent for every \(i\) \((1\leq i\leq D-1)\).

\((ii)\) There exists a thin irreducible \(T\)-module \(W\) with endpoint \(1\) that contains \(v\).

If \((i)\), \((ii)\) hold then \[W = \mathrm{Span}(E^*_1A_0v, E^*_2A_1v, \ldots, E^*_DA_{i-1}v).\]

Proof. \((ii)\to(i)\). Clear as \[E^*_iA_{i-1}v, \; E^*_iA_{i+1}v \in E^*_iW = \mathrm{Span}(w_{i-1}).\]

\((i)\to(ii)\) Consider the sequence

\[E^*_1A_{0}v, E^*_2A_1v, E^*_3A_2v, \ldots, E^*_{D+1}A_Dv.\] The first term is nonzero and the last term is \(0\). So there exists \[n := \mathrm{min}\{i\mid 1\leq i\leq D, \; E^*_{i+1}A_iv = 0\}.\] Now \[\begin{equation} E^*_{j+1}A_jv = 0 \quad (n\leq j\leq D). \tag{32.1} \end{equation}\]

HS MEMO

Use induction and Lemma 32.2 \((i)\), \[E^*_{j+1}A_jv \in \mathrm{Span}(RE^*_jA_{j-1}v) \quad (j\geq 1).\]

By our assumption \((i)\), and the definition of \(n\), \[E^*_jA_{j+1}v \in \mathrm{Span}(E^*_jA_{j-1}v)\neq 0 \quad (1\leq j\leq n).\]

By Lemma 32.2 \((i)\), \[RE^*_jA_{j-1}v \in \mathrm{Span}(E^*_{j+1}A_jv) \quad (1\leq j\leq n).\]

By Lemma 32.2 \((ii)\), \[\begin{align} FE^*_jA_{j-1}v & \in \mathrm{Span}(RE^*_{j-1}A_jv, E^*_jA_{j-1}v, E^*_jA_{j+1}v)\\ & \subseteq \mathrm{Span}(RE^*_{j-1}A_{j-2}v, E^*_jA_{j-1}v)\\ & \mathrm{Span}(E^*_{j-1}A_{j-1}v) \quad (1\leq j\leq n). \end{align}\]

By Lemma 32.2 \((iii)\), \[\begin{align} FE^*_jA_{j-1}v & \in \mathrm{Span}(FE^*_{j-1}A_j v, E^*_{j-1}A_{j}v, E^*_{j-1}A_{j-2}v)\\ & \subseteq \mathrm{Span}(FE^*_{j-1}A_{j-2}v, E^*_{j-1}A_{j-2}v)\\ & \subseteq \mathrm{Span}(E^*_{j-1}A_{j-2}v) \quad (2\leq j\leq n). \end{align}\]

Hence, \[W = \mathrm{Span}(E^*_1A_0v, E^*_2A_1v, \ldots, E^*_nA_{n-1}v).\] is \(R\), \(F\), \(L\) invariant.

Therefore \(W\) is a thin \(T\)-module with endpoint \(1\) that contains \(v\).