In algebraic geometry, standard monomial theory describes the sections of a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

over a

generalized flag variety In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...

Schubert variety In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using line ...

of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

by giving an explicit basis of elements called standard monomials. Many of the results have been extended to

Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ge ...

s and their groups. There are monographs on standard monomial theory by and and survey articles by and One of important open problems is to give a completely geometric construction of the theory.M. Brion and V. Lakshmibai : A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680.

History

introduced monomials associated to standard

Young tableaux In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and ...

. (see also ) used Young's monomials, which he called standard power products, named after standard tableaux, to give a basis for the homogeneous coordinate rings of complex

Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...

s. initiated a program, called standard monomial theory, to extend Hodge's work to varieties ''G''/''P'', for ''P'' any

parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

of any

in any characteristic, by giving explicit bases using standard monomials for sections of line bundles over these varieties. The case of Grassmannians studied by Hodge corresponds to the case when ''G'' is a special linear group in characteristic 0 and ''P'' is a maximal parabolic subgroup. Seshadri was soon joined in this effort by V. Lakshmibai and

Chitikila Musili Chitikila Musili was an Indian mathematician at the University of Hyderabad who developed standard monomial theory in collaboration with his PhD supervisor C. S. Seshadri Conjeevaram Srirangachari Seshadri (29 February 1932 – 17 July 2020) ...

. They worked out standard monomial theory first for minuscule representations of ''G'' and then for groups ''G'' of classical type, and formulated several conjectures describing it for more general cases. proved their conjectures using the

Littelmann path model In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities ''without overcounting'' in the representation theory of symmetrisable Kac–Moody algebras. Its most important application ...

, in particular giving a uniform description of standard monomials for all reductive groups. and and give detailed descriptions of the early development of standard monomial theory.

Applications

*Since the sections of line bundles over generalized flag varieties tend to form irreducible representations of the corresponding algebraic groups, having an explicit basis of standard monomials allows one to give character formulas for these representations. Similarly one gets character formulas for

Demazure module In mathematics, a Demazure module, introduced by , is a submodule of a finite-dimensional representation generated by an extremal weight space under the action of a Borel subalgebra. The Demazure character formula, introduced by , gives the charac ...

s. The explicit bases given by standard monomial theory are closely related to

crystal base A crystal base for a representation of a quantum group on a \Q(v)-vector space is not a base of that vector space but rather a \Q-base of L/vL where L is a \Q(v)-lattice in that vector spaces. Crystal bases appeared in the work of and also in the ...

s and

s of representations. *Standard monomial theory allows one to describe the singularities of Schubert varieties, and in particular sometimes proves that Schubert varieties are normal or Cohen–Macaulay. . *Standard monomial theory can be used to prove

Demazure's conjecture In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by . The conjecture implies that many of the results of his paper can be extended from complex algebraic groups to algebraic gro ...

. *Standard monomial theory proves the

Kempf vanishing theorem In algebraic geometry, the Kempf vanishing theorem, introduced by , states that the higher cohomology group ''H'i''(''G''/''B'',''L''(λ)) (''i'' > 0) vanishes whenever λ is a dominant weight of ''B''. Here ''G'' is a reductive a ...

and other vanishing theorems for the higher cohomology of effective line bundles over Schubert varieties. *Standard monomial theory gives explicit bases for some rings of invariants in

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

. *Standard monomial theory gives generalizations of the

Littlewood–Richardson rule In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural number ...

about decompositions of tensor products of representations to all reductive algebraic groups. *Standard monomial theory can be used to prove the existence of

good filtration In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraic group ''G'' such that the subquotients are isomorphic to the spaces of sections ''F''(λ) of line bundles λ over ''G''/''B'' for ...

s on some representations of reductive algebraic groups in positive characteristic.

Notes

References

* * * * * * * * * * * * *{{Citation , last1=Young , first1=Alfred , author1-link=Alfred Young (mathematician) , title=On Quantitative Substitutional Analysis , doi=10.1112/plms/s2-28.1.255 , year=1928 , journal= Proc. London Math. Soc. , volume=28 , issue=1 , pages=255–292, url=https://zenodo.org/record/1447746 Algebraic geometry Invariant theory