Chapter 30 \(R, F, L\) Matrices

Monday, April 12, 1993

Let \(\Gamma = (X, E)\) be distance regular of diameter \(D\geq 3\) with standard module \(V\).

Assume \(\Gamma\) is \(Q\)-polynomial with respect to the ordering \[E_0, E_1, \ldots, E_D\] of primitive idempotents. Let \(A_i\) be an \(i\)-th adjacency matrix, and \(A = A_1\). \[A = \sum_{i=0}^D\theta_i A_i, \quad E_i = |X|^{-1}\sum_{i=0}^D\theta^*_i A_i.\]

Fix a vertex \(x\in X\), write \[E^*_i \equiv E^*_i(x), \quad A^*_i\equiv A^*_i(x), \quad A^* \equiv A^*_1, \quad T\equiv T(x).\] Then \[A^* = \sum_{i=0}^D \theta^*_i E^*_i.\]

By Theorem 29.1, there exist \(\beta, \gamma, \gamma^*, \delta, \delta^*\in \mathbb{R}\) such that \[\begin{align} 0 & = [A, A^2A^*-\beta AA^*A + A^*A^2 - \gamma(AA^*+A^*A) - \delta A^*]\\ 0 & = [A^*, {A^*}^2A-\beta^* A^*AA^* + A{A^*}^2 - \gamma^*(A^*A+AA^*) - \delta^* A] \end{align}\]

Recall raising matrix \[R = \sum_{i=0}^D E^*_{i+1}AE^*_i\] satisfies \[R(E^*_iV) \subseteq E^*_{i+1}V \quad (0\leq i\leq D), \quad E^*_{D+1}V = 0,\] lowering matrix \[L = \sum_{i=0}^D E^*_{i-1}AE^*_i\] satisfies \[L(E^*_iV) \subseteq E^*_{i-1}V \quad (0\leq i\leq D), \quad E^*_{-1}V = 0,\] and flat matrix \[F = \sum_{i=0}^D E^*_{i}AE^*_i\] satisfies \[F(E^*_iV) \subseteq E^*_{i}V \quad (0\leq i\leq D).\] Also, \[A = R + F + L.\]

Theorem 30.1 With the above notation and assumptions,

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

\[g^-_iFL^2 + LFL + g^+_iL^2F - \gamma L^2)E^*_i = O,\] where \[\begin{align} g^+_i & = \frac{\theta^*_{i-2}-(\beta+1)\theta^*_{i-1}+\beta\theta^*_i}{\theta^*_{i-2}-\theta^*_i}\\ g^-_i & = \frac{\theta^*_{i-2}+(\beta+1)\theta^*_{i-1}-\theta^*_i}{\theta^*_{i-2}-\theta^*_i}. \end{align}\]

\((ii)\) For all \(i\) \((0\leq i\leq D)\),

\[[F, LR - h_iRL]E^*_i = O,\] where \[\begin{align} h_i & = \frac{\theta^*_{i-1}-\theta^*_i}{\theta^*_i-\theta^*_{i+1}} \quad (1\leq i\leq D-1), \end{align}\] and \(h_0, h_D\) are indeterminants.

\((iii)\) For all \(i\) \((1\leq i\leq D)\),

\[(e^-_iRL^2 + (\beta+2)LRL + e^+_iL^2R + LF^2 - \beta FLF + F^2L - \gamma(LF+FL) - \delta L)E^*_i = O,\] where \[\begin{align} e^+_i & = \frac{\theta^*_{i-1}-(\beta+2)\theta^*_{i}+(\beta+1)\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i} \quad (1\leq i\leq D)\\ e^-_i & = \frac{-(\beta+1)\theta^*_{i-2}+(\beta+2)\theta^*_{i-1}-\theta^*_i}{\theta^*_{i-1}-\theta^*_i} \quad (2\leq i\leq D), \end{align}\] and \(e^+_0, e^-_1\) are indeterminants.

Proof. We have \[O = A^3A^*-A^*A^3 - (\beta+1)(A^2A^*A-AA^*A^2)-\gamma(A^2A^*-A^*A^2)-\delta(AA^*-A^*A).\]

\((i)\) Fix \(i\) \((2\leq i\leq D)\), and multply above on the left by \(E^*_{i-2}\), and on the right by \(E^*_i\). Now reduce.

For example, \[E^*_{i-2}A^3A^*E^*_i = \theta^*_i E^*_{i-2}A^3 E^*_i,\] where \[\begin{align} E^*_{i-2}A^3 E^*_i & = E^*_{i-2}A\left(\sum_{r=0}^D E^*_r\right)A\left(\sum_{s=0}^D E^*_s\right)AE^*_i\\ & = \sum_{r,s}E^*_{i-2}AE^*_r AE^*_s AE^*_i\\ & = \sum_{r,s, |i-2-r|\leq 1, |r-s|\leq 1, |s-i|\leq 1}E^*_{i-2}AE^*_r AE^*_s AE^*_i\\ & = E^*_{i-2}AE^*_{i-2}AE^*_{i-1}AE^*_i + E^*_{i-2}AE^*_{i-1}AE^*_{i-1}AE^*_i + E^*_{i-2}AE^*_{i-1}AE^*_{i}AE^*_i\\ & = (FL^2 + LFL + L^2F)E^*_i. \end{align}\] Reducing the other terms in a similar manner, and simplifying, we obtain \((i)\).

HS MEMO

\[\begin{align} E^*_{i-2}A^*A^3E^*_i & = \theta^*_{i-2}E^*_{i-2}A^3E^*_i\\ & = \theta^*_{i-2}(FL^2+LFL + L^2F)E^*_i,\\ E^*_{i-2}A^2A^*AE^*_i & = (\theta^*_{i-1}(FL^2+LFL) + \theta^*_iL^2F)E^*_i\\ E^*_{i-2}AA^*A^2E^*_i & = (\theta^*_{i-2}FL^2 + \theta^*_{i-1}(LFL+L^2F))E^*_i,\\ E^*_{i-2}(A^2A^*-A^*A^2)E^*_i & = (\theta^*_i-\theta^*_{i-2})L^2E^*_i,\\ E^*_{i-2}(AA^*-A^*A)E^*_i & = O. \end{align}\] Then we have \[\begin{align} O &= ((\theta^*_i-\theta^*_{i-2})(FL^2+LFL+L^2F)\\ & \quad -(\beta+1)(\theta^*_{i-1}(FL^2+LFL) + \theta^*_i L^2F - \theta^*_{i-2}FL^2 - \theta^*_{i-1}(LFL+L^2F)) \\ & \quad -\gamma(\theta^*_i-\theta^*_{i-2})L^2)E^*_i\\ & = ((\theta^*_i-\theta^*_{i-2}-(\beta+1)(\theta^*_{i-1}-\theta^*_{i-2}))FL^2 + (\theta^*_i-\theta^*_{i-2})LFL\\ & \quad + (\theta^*_i-\theta^*_{i-2}-(\beta+1)(\theta^*_i-\theta^*_{i-1}))L^2F - \gamma(\theta^*_i-\theta^*_{i-2})L^2)E^*_1\\ & = -(\theta^*_{i-2}-\theta^*_i)\left(\left(\frac{-\beta \theta^*_{i-2}+(\beta+1)\theta^*_{i-1}-\theta^*_i}{\theta^*_{i-2}-\theta^*_i}\right)FL^2+LFL\right.\\ & \quad + \left. \left(\frac{\theta^*_{i-2}-(\beta+1)\theta^*_{i-1}+\beta \theta^*_i}{\theta^*_{i-2}-\theta^*_i}\right)L^2F - \gamma L^2\right)E^*_i\\ & = (\theta^*_i-\theta^*_{i-2})(g^-_iFL^2 + LFL + g^+_iL^2F - \gamma L^2)E^*_i. \end{align}\]

\((ii)\), \((iii)\) are obtained in a similar manner replacing \(i-2\) by \(i\) (resp. \(i-1\)).

HS MEMO

\((ii)\) We have

\[O = E^*_i(A^3A^*-A^*A^3-(\beta+1)(A^2A^*A-AA^*A^2)-\gamma(A^2A^*-A^*A^2)-\delta(AA^*-A^*E))E^*_i.\] Since \(\beta+1\neq 0\), by (29.20) if \(D\geq 3\), \[\begin{align} O &= E^*_i(A^2A^*A - AA^*A^2)E^*_i\\ & = ((\theta^*_i-\theta^*_{i-1})RLF + (\theta^*_i-\theta^*_{i+1})LRF)+(\theta^*_{i-1}-\theta^*_i)FRL+(\theta^*_{i+1}-\theta^*_i)FLR)E^*_i\\ & = [F, (\theta^*_{i-1}-\theta^*_i)RL - (\theta^*_i-\theta^*_{i+1})LR]E^*_i\\ & = (\theta^*_{i+1}-\theta^*_i)\left[F, LR - \frac{\theta^*_{i-1}-\theta^*_i}{\theta^*_i-\theta^*_{i+1}}RL\right]E^*_i\\ & = (\theta^*_{i+1}-\theta^*_i)[F, LR - h_i RL]E^*_i. \end{align}\]

\((iii)\) We have

\[\begin{align} O & = E^*_{i-1}(A^3A^*-A^*A^3-(\beta+1)(A^2A^*A-AA^*A^2)-\gamma(A^2A^*-A^*A^2)-\delta(AA^*-A^*A))E^*_i\\ & = ((\theta^*_i-\theta^*_{i-1})(RL^2+ LRL + L^2R + LF^2 + FLF + F^2L))\\ & \quad -(\beta+1)((\theta^*_{i-1}-\theta^*_{i-2})RL^2 + (\theta^*_{i-1}-\theta^*_i)LRL + (\theta^*_{i+1}-\theta^*_i)L^2R\\ & \quad + (\theta^*_i-\theta^*_{i-1})FLF\\ & \quad - \gamma(\theta^*_i-\theta^*_{i-1})(LF+FL)\\ & \quad - \delta(\theta^*_i-\theta^*_{i-1})L)E^*_i\\ & = ((\theta^*_i-\theta^*_{i-1})-(\beta+1)(\theta^*_{i-1}-\theta^*_{i-2}))RL^2 \\ & \quad + ((\theta^*_i-\theta^*_{i-1})-(\beta+1)(\theta^*_{i-1}-\theta^*_{i}))LRL\\ & \quad + ((\theta^*_i-\theta^*_{i-1})-(\beta+1)(\theta^*_{i+1}-\theta^*_{i}))L^2R\\ & \quad + (\theta^*_{i}-\theta^*_{i-1})LF^2 + (\theta^*_i-\theta^*_{i-1})F^2L\\ & \quad + (\theta^*_i - \theta^*_{i-1}-(\beta+1)(\theta^*_i-\theta^*_{i-1}))FLF\\ & \quad - \gamma(\theta^*_i-\theta^*_{i-1})(LF+FL)\\ & \quad - \delta (\theta^*_i-\theta^*_{i-1})L)E^*_i\\ & = (\theta^*_i-\theta^*_{i-1})\biggl(\frac{-(\beta+1)\theta^*_{i-2}+(\beta+2)\theta^*_{i-2}-\theta^*_i}{\theta^*_{i-1}-\theta^*_i}RL^2+(\beta+2)LRL\\ & \qquad + \frac{\theta^*_{i-1}-(\beta+2)\theta^*_i+(\beta+1)\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i}L^2R + LF^2 - \beta FLF + F^2L\\ & \qquad -\gamma(LF+FL) - \delta L\biggr)E^*_i\\ & = (e^-_iRL^2 + (\beta+2)LRL + e^+_iL^2R + LF^2 - \beta FLF + F^2L-\gamma(LF+FL)-\delta L)E^*_i\\ & = O. \end{align}\]

Lemma 30.1 With the notation of Theorem 30.1, \[\begin{align} e^+_i & = \frac{\theta^*_i-\theta^*_{i+2}}{\theta^*_i-\theta^*_{i-1}} \quad (1\leq i\leq D-2)\\ e^-_i & = \frac{\theta^*_{i-1}-\theta^*_{i-3}}{\theta^*_{i-1}-\theta^*_{i}} \quad (3\leq i\leq D)\\ g^+_i & = \frac{\theta^*_i-\theta^*_{i+1}}{\theta^*_i-\theta^*_{i-2}} \quad (2\leq i\leq D-1)\\ g^-_i & = \frac{\theta^*_{i-2}-\theta^*_{i-3}}{\theta^*_{i-2}-\theta^*_{i}} \quad (3\leq i\leq D). \end{align}\] In particular, \(e^\pm_i\), \(g^\pm_i\) are non-zero for the range of \(i\) given above.

Proof. In each case, equate the above expression with the corresponding expression in Theorem 30.1. The resulting equation is equal to (29.7).

HS MEMO

By Corollary 26.1 and Therem 29.1, \[e^+_i = \frac{\theta^*_{i-1}-(\beta+2)\theta^*_i + (\beta+1)\theta^*_{i+1}}{\beta^*_{i-1}-\theta^*_i},\] and \[\beta + 1 = \frac{\theta^*_{j-1}-\theta^*_j+\theta^*_{j+1}-\theta^*_{j+2}}{\theta^*_j-\theta^*_{j+1}}+1 = \frac{\theta^*_{j-1}-\theta^*_{j+2}}{\theta^*_j-\theta^*_{j+1}}.\] Hence, \[\begin{align} e^+_i & = \frac{1}{\theta^*_{i-1}-\theta^*_i}(\theta^*_{i-1}-\theta^*_i-(\beta+1)(\theta^*_i-\theta^*_{i+1}))\\ & = \frac{1}{\theta^*_{i-1}-\theta^*_i}(\theta^*_{i-1}-\theta^*_i - (\theta^*_{i-1}-\theta^*_{i+2}))\\ & = \frac{\theta^*_i-\theta^*_{i+2}}{\theta^*_{i}-\theta^*_{i-1}},\\ e^-_i & = \frac{1}{\theta^*_{i-1}-\theta^*_i}(-(\beta+1)\theta^*_{i-2}+(\beta+2)\theta^*_{i-1}-\theta^*_i)\\ & = \frac{1}{\theta^*_{i-1}-\theta^*_i}(\theta^*_{i-1}-\theta^*_i - \theta^*_{i-3}+\theta^*_{i})\\ & = \frac{\theta^*_{i-1}-\theta^*_{i-3}}{\theta^*_{i-1}-\theta^*_{i}},\\ g^+_i & = \frac{1}{\theta^*_{i-2}-\theta^*_i}(\theta^*_{i-2}-(\beta+1)\theta^*_{i-1}+\beta\theta^*_{i})\\ & = \frac{1}{\theta^*_{i}-\theta^*_{i-2}}(\theta^*_{i}-\theta^*_{i-2} + \theta^*_{i-2}-\theta^*_{i+1})\\ & = \frac{\theta^*_i-\theta^*_{i+1}}{\theta^*_{i}-\theta^*_{i-2}},\\ g^-_i & = \frac{1}{\theta^*_{i-2}-\theta^*_i}(-\beta\theta^*_{i-2}+(\beta+1)\theta^*_{i-1}-\theta^*_i)\\ & = \frac{1}{\theta^*_{i-2}-\theta^*_i}(\theta^*_{i-2}-\theta^*_i + \theta^*_{i}-\theta^*_{i-3})\\ & = \frac{\theta^*_{i-2}-\theta^*_{i-3}}{\theta^*_{i-2}-\theta^*_{i}}. \end{align}\]

Corollary 30.1 Let \(\Gamma = (X, E)\) be dostance-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 \(E^*_i \equiv E^*_i(x), R\equiv R(x), L\equiv L(x), F\equiv F(x)\). Then the following hold.

\((i)\) \(FR^2E^*_j \in \mathrm{Span}(RFRE^*_j, R^2FE^*_j, R^2E^*_j)\), \((0\leq j\leq D-3)\).
\((ii)\) \(R^2FE^*_j \in \mathrm{Span}(RFRE^*_j, FR^2E^*_j, R^2E^*_j)\), \((1\leq j\leq D-2)\).
\((iii)\) \(LR^2E^*_j \in \mathrm{Span}(RLRE^*_j, R^2LE^*_j, F^2RE^*_j, FRFE^*_j, RF^2E^*_j, RFE^*_j, FRE^*_j, RE^*_j)\), \((0\leq j\leq D-3)\).
\((iv)\) \(R^2LE^*_j \in \mathrm{Span}(RLRE^*_j, LR^2E^*_j, F^2RE^*_j, FRFE^*_j, RF^2E^*_j, RFE^*_j, FRE^*_j, RE^*_j)\), \((1\leq j\leq D)\).

Proof. Immediate from Theorem 30.1, and Lemma 30.1.

HS MEMO

By Theorem 30.1, and Lemma 30.1, we have the following, but similarly we can obtain above.

\((i)\) \(FL^2E^*_j \in \mathrm{Span}(LFLE^*_j, L^2FE^*_j, L^2E^*_j)\), \((3\leq j\leq D)\).
\((ii)\) \(L^2FE^*_j \in \mathrm{Span}(LFLE^*_j, FL^2E^*_j, L^2E^*_j)\), \((2\leq j\leq D-1)\).
\((iii)\) \(RL^2E^*_j \in \mathrm{Span}(LRLE^*_j, L^2RE^*_j, F^2LE^*_j, FLFE^*_j, LF^2E^*_j, LFE^*_j, FLE^*_j, LE^*_j)\), \((3\leq j\leq D)\).
\((iv)\) \(L^2RE^*_j \in \mathrm{Span}(LRLE^*_j, RL^2E^*_j, F^2LE^*_j, FLFE^*_j, LF^2E^*_j, LFE^*_j, FLE^*_j, LE^*_j)\), \((2\leq j\leq D)\).