## 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:

 5e+30 Association schemes, strongly regular graphs

### Keywords:

association schemes; Terwilliger algebra; hypercube
Full Text:

### References:

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