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The Terwilliger algebra of the hypercube. (English) Zbl 0997.05097

Let \(Q_D\) be a hypercube of dimension \(D\) with the vertex set \(X\). Fix a vertex \(x\in X\), and let \(T=T(x)\) denote the associated Terwilliger algebra. It is proved that (1) \(T\) is the subalgebra of \(\text{Mat}_X({\mathbf C})\) generated by the adjacency matrix \(A\) and a diagonal matrix \(A^*\) with \(yy\) entry \(D-\partial(x,y)\) for all \(y\in X\) (Lemma 3.8); (2) \(A^2A^*-2AA^*A+A^*A^2=4A^*\), \((A^*)^2A-2A^*AA^*+A(A^*)^2=4A\) (Theorem 4.2); (3) \(\dim(T)=(D+3)(D+2)(D+1)/6\) (Corollary 14.15); (4) let \(\phi=(1/8)(A(A^*)^2A+A^*A^2A^*-(AA^*)^2-(A^*A)^2+4A^2+4(A^*)^2)\), \(\gamma=\lfloor D/2\rfloor\), \(\alpha_r=(D-2r)(D-2r+2)/2\) (\(0\leq r\leq \gamma\), \(\phi_r=f_r(\phi)\), where \(f_r\in {\mathbf C}[\lambda]\) is given by \(f_r=\prod_{i\in \{0,\dots ,\gamma\}-r}(\lambda-\alpha_i)/(\alpha_r-\alpha_i))\) then (i) \(\phi_0+\cdots +\phi_\gamma=I\), (ii) \(\phi_r\phi_s=\delta_{rs}\phi_r\), and (iii) \(\phi_0,\dots ,\phi_\gamma\) is a basis for the center of \(T\) (Theorem 14.10).

MSC:

05E30 Association schemes, strongly regular graphs
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