Discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems

Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in $1+1$ dimension and in $2+1$ dimension. Especially in the $(2+1)$-dimensional case, the corresponding system can be extended to $2\times 2$ matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations.