A Linear Map Acts as a Leonard Pair with Each of the Generators of Usl2

Abstract EN

Let ℱ denote an algebraically closed field with a characteristic not two.

Fix an integer d≥3; let x, y, and z be the equitable basis of sl2 over ℱ.

Let V denote an irreducible sl2-module with dimension d+1; let A∈EndV.

In this paper, we show that if each of the pairs A,x, A,y, and A,z acts on V as a Leonard pair, then these pairs are of Krawtchouk type.

Moreover, A is a linear combination of 1, x, y, and z.