Standard monomial bases, moduli of vector bundles, and invariant theory
Abstract
Consider the diagonal action of the special orthogonal group on the direct sum of a finite number of copies of the standard representation--the underlying field is assumed to be algebraically closed and of characteristic not equal to two. We construct a "standard monomial" basis for the ring of polynomial invariants for this action. We then deduce, by a deformation argument, our main result that this ring of polynomial invariants is Cohen-Macaulay. We give three applications of this result: (1) the first and second fundamental theorems of invariant theory for the above action; (2) Cohen-Macaulayness of the moduli space of equivalence classes of semi-stable vector bundles of rank two and degree zero on a smooth projective curve of genus at least three (for this application, characteristic three is also excluded); (3) a basis in terms of traces for the ring of polynomial invariants for the diagonal adjoint action of the special linear group SL(2) on a finite number of copies of its Lie algebra sl(2).
Keywords
Cite
@article{arxiv.math/0604321,
title = {Standard monomial bases, moduli of vector bundles, and invariant theory},
author = {V. Lakshmibai and K. N. Raghavan and P. Sankaran and P. Shukla},
journal= {arXiv preprint arXiv:math/0604321},
year = {2007}
}
Comments
latex; 35 pages