Representations and binomial coefficients

Abstract

For a root system R, a field K and a "choice of coefficients in K" we define a category of graded spaces with operators and study some of its properties. Then we assume that the coefficients are given by quantum binomials. We use basic arithmetic properties of binomial coefficients (such as q-versions of Lucas' theorem and the Pfaff-Saalsch\"utz identity) to construct a Frobenius pull back functor and prove a Steinberg's tensor product formula. Then we show that these results imply the corresponding results for reductive algebraic groups in positive characteristics and for quantum groups at roots of unity.