The path model, the quantum Frobenius map and standard monomial theory.

*(English)*Zbl 0938.14031
Carter, R. W. (ed.) et al., Algebraic groups and their representations. Proceedings of the NATO Advanced Study Institute on modular representations and subgroup structure of algebraic groups and related finite groups, Cambridge, UK, June 23-July 4, 1997. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 517, 175-212 (1998).

In the series ‘Geometry of \(G/P\)’, V. Lakshmibai, C. Musili and C. S. Seshadri [see Bull. Am. Math. Soc., New. Ser. 1, 432-435 (1979; Zbl 0466.14020)] developed a standard monomial theory for semisimple algebraic groups as a generalization of the Hodge-Young standard monomial theory for \(GL(n)\). Standard monomial theory consists in constructing explicit bases for spaces of sections of effective line bundles on the generalized flag variety. Standard monomial theory has led to very many important geometric and representation-theoretic consequences.

In this article, the author gives a different approach to standard monomial theory (which avoids the case by case consideration) through path models of representations and their associated bases. The bases are constructed using the theory of quantum groups at a root of unity. Using these bases, geometric and representation-theoretic consequences – vanishing theorems for higher cohomology for effetive line bundles on Schubert varieties, a proof of the Demazure character formula, projective normality of Schubert varieties, good filtration property – are deduced.

It should be added that the path model theory has also led to a Littlewood-Richardson rule for symmetrizable Kac-Moody algebras proved by the author [cf. P. Littelmann, Invent. Math. 116, No. 1-3, 329-346 (1994; Zbl 0805.17019)].

For the entire collection see [Zbl 0897.00029].

In this article, the author gives a different approach to standard monomial theory (which avoids the case by case consideration) through path models of representations and their associated bases. The bases are constructed using the theory of quantum groups at a root of unity. Using these bases, geometric and representation-theoretic consequences – vanishing theorems for higher cohomology for effetive line bundles on Schubert varieties, a proof of the Demazure character formula, projective normality of Schubert varieties, good filtration property – are deduced.

It should be added that the path model theory has also led to a Littlewood-Richardson rule for symmetrizable Kac-Moody algebras proved by the author [cf. P. Littelmann, Invent. Math. 116, No. 1-3, 329-346 (1994; Zbl 0805.17019)].

For the entire collection see [Zbl 0897.00029].

Reviewer: V.Lakshmibai (Boston)

##### MSC:

14M17 | Homogeneous spaces and generalizations |

20G05 | Representation theory for linear algebraic groups |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

14M15 | Grassmannians, Schubert varieties, flag manifolds |