Standard monomial bases, Moduli spaces of vector bundles, and Invariant theory

Abstract

Consider the diagonal action of \({\rm SO}_n(K)\) on the affine space \(X=V^{\oplus m}\) where \(V=K^n,\,K\) an algebraically closed field of characteristic \(\not= 2.\) We construct a "standard monomial" basis for the ring of invariants \(K[X]^{{\rm SO}_n(K)}.\) As a consequence, we deduce that \(K[X]^{{\rm SO}_n(K)}\) is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for \({\rm SO}_n(K)\)-actions. As the second application, assuming that the characteristic of K is \(\neq 2,3,\) we give a characteristic-free proof of the Cohen-Macaulayness of the moduli space \(\mathcal{M}_2\) of equivalence classes of semi-stable, rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2. As the third application, we describe a K-basis for the ring of invariants for the adjoint action of \({\rm SL}_2(K)\) on m copies of \(sl_2(K)\) in terms of traces.