The Path Model, the Quantum Frobenius Map and Standard Monomial Theory

Abstract

The aim of this article is to give an introduction to the theory of path models of representations and their associated bases. The starting point for the theory was a series of articles in which Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space H ⁰(G/B, $\mathcal{L}$ _λ) with some particularly nice geometric properties. Here we suppose that G is a reductive algebraic group defined over an algebraically closed field k, B is a fixed Borel subgroup, and $\mathcal{L}$ _λ is the line bundle on the flag variety G/B associated to a dominant weight λ. The purpose of the program is to extend the Hodge-Young standard monomial theory for the group GL(n) to the case of any semisimple linear algebraic group and, more generally, to Kac-Moody algebras. Apart from the independent interest of such a construction, the results have important applications to the combinatorics of representations as well as to the geometry of Schubert varieties. For the geometric applications note that standard monomial theory provides proofs of the vanishing theorems for the higher cohomology of effective line bundles on Schubert varieties, explicit bases for the rings of invariants in classical invariant theory, a proof of Demazure’s conjecture, a proof of the normality of Schubert varieties, another proof of the good filtration property, a determination of the singular locus of Schubert varieties [9], a deformation of SL(n)/B into a toric variety [2], etc.