Standard monomial bases, moduli spaces of vector bundles, and invariant theory

Abstract

Consider the diagonal action of SO_n(K) on the affine space X=V^⊕m where V=Kⁿ, K an algebraically closed field of characteristic ≠ 2. We construct a "standard monomial" basis for the ring of invariants K[X]^SO_n(K)*. As a consequence, we deduce that K[X]^SO_n(K) is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for SO_n(K)-actions. As the second application, assuming that the characteristic of K is ≠ 2,3, we give a characteristic-free proof of the Cohen-Macaulayness of the moduli space M₂ of equivalence classes of semi-stable, rank 2, −1760 vector bundles on a smooth projective curve of genus 7gt; 2. As the third application, we describe a K-basis for the ring of invariants for the adjoint action of SL₂(K) on m copies of sl₂(K) in terms of traces.