Krawtchouk-Griffiths Systems I: Matrix Approach

TL;DR

This paper introduces a matrix-based approach to multivariate Krawtchouk-Griffiths systems, revealing their algebraic structure, orthogonality properties, and connections to quantum observables and Lie algebras.

Contribution

It develops a novel matrix framework for KG-systems, extending classical Krawtchouk polynomials to multivariate cases with quantum and algebraic interpretations.

Findings

Matrix construction of KG-systems satisfying the K-condition

Representation of recurrence relations via multiplication operators

Connection between associated random walks and Lie algebra representations

Abstract

We call Krawtchouk-Griffiths systems, or KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Our approach is to work directly with matrices comprising the values of the polynomials at points of a discrete grid based on the possible counting values of the underlying multinomial distribution. The starting point for the construction of a KG-system is a generating matrix satisfying the K-condition, orthogonality with respect to the basic probability distribution associated to an individual step of the multinomial process. The variables of the polynomials corresponding to matrices may be interpreted as quantum observables in the real case, or quantum variables in the complex case. The structure of the recurrence relations for the orthogonal polynomials is presented with multiplication operators as the matrices corresponding to the quantum variables. An interesting feature is that the associated random walks correspond to the Lie algebra of the representation of symmetric tensor powers of matrices.