  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.
