Krawtchouk polynomials and Krawtchouk matrices

Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed Sylvester-Hadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is presented. Our approach is used to solve Kac' formulation of the Ehrenfest urn model. Connections with quantum and classical random walks are shown as well as various extensions of the classical polynomials.