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

V. Lakshmibai; K. N. Raghavan; P. Sankaran; P. Shukla

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

V. Lakshmibai·K. N. Raghavan·P. Sankaran·P. Shukla

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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).

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BibTeX RIS

V. Lakshmibai, K. N. Raghavan, P. Sankaran, P. Shukla. 2006-04-13. Standard monomial bases, moduli of vector bundles, and invariant theory. https://arxiv.org/abs/math/0604321

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