Enveloping algebra U(gl(3)) and orthogonal polynomials in several discrete indeterminates

TL;DR

This paper constructs multivariable orthogonal polynomials related to the enveloping algebra U(gl(3)) by generalizing known one-variable polynomials, expanding the algebraic and combinatorial understanding of such functions.

Contribution

It extends the construction of orthogonal polynomial bases from U(sl(2)) to U(gl(3)), producing multivariable analogs of Hahn polynomials for specific maximal ideals.

Findings

Derived multivariable Hahn-type polynomials from U(gl(3))

Established orthogonality relations for these polynomials

Connected algebraic structures with multivariable orthogonal polynomials

Abstract

Let A be an associative complex algebra and L an invariant linear functional on it (trace). Let i be an involutive antiautomorphism of A such that L(i(a))=L(a) for any a in A. Then A admits a symmetric invariant bilinear form (a, b)=L(a i(b)). For A=U(sl(2))/m, where m is any maximal ideal of U(sl(2)), Leites and I have constructed orthogonal basis whose elements turned out to be, essentially, Chebyshev and Hahn polynomials in one discrete variable. Here I take A=U(gl(3))/m for the maximal ideals m which annihilate irreducible highest weight gl(3)-modules of particular form (generalizations of symmetric powers of the identity representation). In this way we obtain multivariable analogs of Hahn polynomials.