The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two.(English)Zbl 1136.05076

Let $$\Gamma$$ be a distance-regular graph with diameter $$D$$, distance matrices $$A_i$$ and eigenvalues $$\theta_0>\theta_1>\dots >\theta_D$$. Fix a base vertex $$x$$ of $$\Gamma$$. Let $$T=T(x)$$ denote the subalgebra of $$Mat_X({\mathbb C})$$ generated by $$A,E_0^*,\dots ,E_D^*$$, where $$A=A_1$$ and $$E_i^*$$ denotes the projection onto the $$i$$th subconstituent of $$\Gamma$$ with respect to $$x$$. $$T$$ is called the Terwilliger algebra of $$\Gamma$$ with respect to $$x$$. There exist the unique irreducible $$T$$-module $$V_0$$ with endpoint 0 ($$V_1$$ with endpoint 1). Both $$V_0$$ and $$V_1$$ are thin. Assume $$\Gamma$$ is bipartite and $$W$$ is thin irreducible $$T$$-module with endpoint 2. Then $$E_2^*W$$ is a one-dimensional eigenspace for $$E_2^*A_2E_2^*$$ with eigenvalue $$\eta$$ and $$\tilde \theta_1\leq \eta\leq \tilde \theta_d$$, where $$\tilde \theta_i=-1-b_2b_3(\theta_i^2-b_2)^{-1}$$ and $$d=[D/2]$$. To describe the structure of $$W$$ we distinguish four cases:
(i)
$$\eta=\tilde \theta_1$$,
(ii)
$$D$$ is odd and $$\eta=\tilde \theta_d$$,
(iii)
$$D$$ is even and $$\eta=\tilde \theta_d$$,
(iv)
$$\tilde \theta_1<\eta<\tilde \theta_d$$.
Cases $$(i)$$ and $$(ii)$$ were investigated earlier. It is showed that $${\text{ dim}}W=D-1-e$$ ($$e=1$$ in the case $$(iii)$$, $$e=0$$ in the case $$(iv)$$), $$W$$ has orthogonal basis $$E_iv (i\in S)$$, where $$v$$ – a nonzero vector of $$E_2^*W$$, $$E_i$$ – primitive idempotent $$A$$, associated with $$\theta_i$$, and $$S=\{1,2,\dots D-1\}-\{d\}$$ in the case $$(iii)$$, $$S=\{1,2,\dots D-1\}$$ in the case $$(iv)$$, moreover $$W$$ has orthogonal basis $$E_{i+2}^*A_iv (0\leq i\leq D-2-e)$$. It is founded square-norm of each vector of two bases and transition matrix relating this two bases.

MSC:

 05E30 Association schemes, strongly regular graphs 20D15 Finite nilpotent groups, $$p$$-groups
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References:

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