1 - Polynomial functions and combinatorics

Summary

Introductory remarks

This chapter opens with a brief summary of Schur's work on the complex polynomial representations of the general linear group GL_n(C) of n × n nonsingular matrices with entries in C; we then proceed to analyse ways of extending his ideas to representations over an infinite field K of arbitrary characteristic. Thus, Schur considered a polynomial matrix representation of M ∊ GL_n>(C), i.e. a group homomorphism T : GL_n(C) → GL_m(C) such that each entry of T(M) could be expressed as a polynomial with complex coefficients in the entries of M; however we shall study a KGL_n(K)-module V whose coefficient space lies in the bialgebra A_K(n) generated by the n² coordinate functions. In this way V affords a matrix representation whose entries are polynomials over K in the original entries. We shall then take the rth homogeneous component, A_K(n, r), of A_K(n) and consider the category of all representations having coefficient space lying in this component. Basic properties of the coalgebra A_K(n, r) are catalogued in 1.3.

Throughout the text the methods employed use a blend of the standard combinatorial machinery. Nowadays I can safely assume that the combinatorics of partitions and their associated Young diagrams is well known, so no more than a cursory treatment is given in 1.4. This leads to one definition of a weight (as a certain n-tuple of integers) and we connect this idea with the algebraic group concept of weight (arising from the action of the torus) in 1.5.