Hypercube and tetrahedron algebra

Abstract

Let D be an integer at least 3 and let H(D, 2) denote the hypercube. It is known that H(D, 2) is a Q-polynomial distance-regular graph with diameter D, and its eigenvalue sequence and its dual eigenvalue sequence are all {D − 2i} _i=0 ^D, Suppose that ⊠ denotes the tetrahedron algebra. In this paper, the authors display an action of ⊠ on the standard module V of H(D, 2). To describe this action, the authors define six matrices in Mat _X(ℂ), called $A,A^* ,B,B^* ,K,K^* .$ Moreover, for each matrix above, the authors compute the transpose and then compute the transpose of each generator of ⊠ on V.

Keywords

Tetrahedron algebra / Hypercube / Distance-regular graph / Onsager algebra

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