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Standard monomial theory. (English) Zbl 0785.14028

Ramanan, S. (ed.), Proceedings of the Hyderabad conference on algebraic groups held at the School of Mathematics and Computer/Information Sciences of the University of Hyderabad, India, December 1989. Madras: Manoj Prakashan. 279-322 (1991).
Let \(G_{r,n}\) denote the Grassmannian of \(r\)-dimensional linear subspaces of an \(n\)-dimensional vector space (say over \(\mathbb{C})\). We have a canonical imbedding of \(G_{r,n}\) in a projective space, namely the one given by the Plücker coordinates (called the Plücker imbedding). In the 1940’s Hodge gave a nice basis of the homogeneous coordinate ring \(R\) of \(G_{r,n}\), as well as its Schubert subvarieties. He called them standard monomials.
The aim of standard monomial theory (written briefly as SMT) which we have pursued in a series of papers (along with Musili) has been to generalize the work of Hodge in the context of groups which are more general than \(SL(n)\).
Recall the Borel-Weil theorem, namely, when \(G\) is a semisimple, simply- connected algebraic group (over \(\mathbb{C})\), every finite dimensional irreducible \(G\)-module is \(G\)-isomorphic to \(H^ 0(G/B,L)\), where \(L\) is a line bundle on \(G/B\). Hence giving nice bases of finite dimensional irreducible \(G\)-modules is a part of SMT. Our result on SMT provide a very satisfactory generalization of the work of Hodge when \(G\) is a classical group in a series with title “Geometry of G/P”, I–V. Such a generalization has also been done for the exceptional groups \(G_ 2\) and \(E_ 6\) [cf. V. Lakshmibai, J. Algebra 98, 281-318 (1986; Zbl 0605.14041) and V. Lakshmibai and K. N. Rajeswari in Invariant theory, Proc. AMS Spec. Sess., Denton 1986, Contemp. Math. 88, 449-578 (1989; Zbl 0682.14035)]. A beginning has also been done for Kac-Moody groups.
For the entire collection see [Zbl 0777.00047].

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14M17 Homogeneous spaces and generalizations
14L99 Algebraic groups
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