nLab standard monomial

Idea

Standard monomials are elements of particular (vector space) bases of homogeneous coordinate rings and of the spaces of sections of line bundles over generalized flag varieties and Schubert varieties.

Historically first case is the homogeneous coordinate ring of complex Grassmannians, where the standard monomials in Plücker coordinates are described by William Hodge via parametrization by standard Young tableaux. Hodge also described the related straightening algorithm. Other cases are also parametrized by combinatorial data.

There are also quantum group versions of the theory.

References

• Wikipedia: standard monomial theory

Description of the standard monomial bases of the coordinate rings of complex Grassmannians is in

• W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Camb. Phil. Soc. 39 (1943) 22–30

and in vol. 2 of

• W. V. D. Hodge, Daniel Pedoe, Methods of algebraic geometry, 3 vols. (reviewed by Coxeter in Bull. Amer. Math. Soc. 55, 3, part 1 (1949) 315–316, euclid)

More general case is treated in

• Venkatramani Lakshmibai, C. Musili, C. S. Seshadri, Geometry of $G/P$, Bull. Amer. Math. Soc., N. S. 1 (2): 432–435 (1979) doi

• Venkatramani Lakshmibai, Komaranapuram N. Raghavan, Standard monomial theory – invariant theoretic approach, Encyclopaedia of Mathematical Sciences 137, (series: Invariant Theory and Algebraic Transformation Groups VIII), Springer 2008 doi

• C. S. Seshadri, Introduction to the theory of standard monomials, Texts and Readings in Mathematics 46, 2nd. ed., Springer 2016, Hundustani Book Agency 2015

• Peter Littelmann, Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (3): 551–567 (1998) doi

Quantum group case (type A)

• V. Lakshmibai, N. Reshetikhin, Quantum deformations of $SL_n/B$ and its Schubert varieties, In: M. Kashiwara, T. Miwa (eds) ICM-90 Satellite Conference Proceedings. Springer, Tokyo 1991 (doi, preprint version pdf)

category: algebra