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Matrix valued orthogonal polynomials related to SU$(N+1)$, their algebras of differential operators and the corresponding curves

F. Alberto Grünbaum, Manuel D. De la Iglesia

TL;DR

The work studies algebras of matrix-valued differential operators tied to matrix-valued orthogonal polynomials arising from representations of $SU(N{+}1)$. It develops two explicit examples (one-step and two-step) by conjugating Casimir-derived operators to hypergeometric form, yielding complete eigenbases and rich operator algebras that are commutative in the first case and noncommutative in the second. The polynomials are constructed via matrix hypergeometric functions, with eigenvalues encoding the action of the differential operators; explicit relations among generators resemble Burchnall-Chaundy curves. The results illuminate the deep link between representation theory, matrix-valued orthogonality, and differential-operator algebras, and suggest a structured, finitely-generated framework for these families with substantial algebraic and geometric structure.

Abstract

We give two examples of algebras of differential operators associated to families of matrix valued orthogonal polynomials arising from representations of SU$(N+1)$. The first one gives a commutative algebra and the second one a non--commutative one.

Matrix valued orthogonal polynomials related to SU$(N+1)$, their algebras of differential operators and the corresponding curves

TL;DR

The work studies algebras of matrix-valued differential operators tied to matrix-valued orthogonal polynomials arising from representations of . It develops two explicit examples (one-step and two-step) by conjugating Casimir-derived operators to hypergeometric form, yielding complete eigenbases and rich operator algebras that are commutative in the first case and noncommutative in the second. The polynomials are constructed via matrix hypergeometric functions, with eigenvalues encoding the action of the differential operators; explicit relations among generators resemble Burchnall-Chaundy curves. The results illuminate the deep link between representation theory, matrix-valued orthogonality, and differential-operator algebras, and suggest a structured, finitely-generated framework for these families with substantial algebraic and geometric structure.

Abstract

We give two examples of algebras of differential operators associated to families of matrix valued orthogonal polynomials arising from representations of SU. The first one gives a commutative algebra and the second one a non--commutative one.
Paper Structure (12 sections, 187 equations)

This paper contains 12 sections, 187 equations.

Table of Contents

  1. Introduction and outline of the paper
  2. From the Casimir operator to some examples of ordinary differential operators with matrix coefficients
  3. The one step example
  4. Generating the polynomial eigenfunctions of the differential operator
  5. The algebra of operators
  6. The two steps example
  7. Generating the polynomial eigenfunctions of the differential operator
  8. The algebra of operators
  9. Appendix
  10. Explicit expression of $\{P_n\}_{n\geq0}$ for the one step example
  11. Some constants
  12. Explicit expression of $\{P_n\}_{n\geq0}$ for the two steps example