Standard Monomial Theory and applications

In these notes, we explain how one can construct Standard Monomial Theory for reductive algebraic groups by using the path models of their representations and quantum groups at a root of unity. As applications, we obtain a combinatorial proof of the Demazure character formula and representation theoretic proofs of geometrical properties of Schubert varieties, such as normality, vanishing theorems, ideal theory and so on. Further applications of Standard Monomial Theory are made to prove geometrical properties of certain ladder determinantal varieties and certain quiver varieties. We sketch at the end an extension of the theory to Bott-Samelson varieties and configuration varieties.

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