Standard monomial bases, moduli of vector bundles, and invariant theory
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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).
latex; 35 pages
latex; 35 pages