Tridiagonal pairs of Krawtchouk type arising from finite-dimensional irreducible so4-modules

Abstract

Let $F$ be an algebraically closed field with $\mathrm{char}\left(F\right)=0$. The special linear algebra ${\mathrm{sl}}_2$ is the $F$-Lie algebra with Chevalley basis $\{e,h,f\}$. Since the special orthogonal algebra ${\mathrm{so}}_4$ is isomorphic to ${\mathrm{sl}}_2\oplus {\mathrm{sl}}_2$, ${\mathrm{so}}_4$ is viewed as the $F$-Lie algebra with Chevalley basis $\{e_1,h_1,f_1,e_2,h_2,f_2\}$. In [21, Lemma 3.1], there is an automorphism $⁎:{\mathrm{so}}_4\to {\mathrm{so}}_4$ so that $\{e_1^{⁎},h_1^{⁎},f_1^{⁎},e_2^{⁎},h_2^{⁎},f_2^{⁎}\}$ is another Chevalley basis of ${\mathrm{so}}_4$. In [21, Section 5], there is a simple construction of a finite-dimensional irreducible ${\mathrm{so}}_4$-module V on which ${\mathrm{so}}_4$ acts by derivation. In this paper, we construct four tridiagonal pairs (or TD pairs) on V via the action of the Chevalley bases of ${\mathrm{so}}_4$. We prove that these TD pairs are of Krawtchouk type and are not necessarily pairwise isomorphic based on their associated Drinfel'd polynomials. Consequently, we display four Lie algebra homomorphisms from the tetrahedron algebra ⊠ to ${\mathrm{so}}_4$ and via these homomorphisms, we describe how the generators of ⊠ act on V. Finally, we show that the irreducible ${\mathrm{so}}_4$-module V is isomorphic to a tensor product of two evaluation modules in view of [16, Definition 4.11].