## 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

### Keywords:

standard monomial theory; Schubert subvarieties

### Citations:

Zbl 0605.14041; Zbl 0682.14035; Zbl 0759.22022