# Standard Monomial Theory and applications

@inproceedings{Lakshmibai1998StandardMT, title={Standard Monomial Theory and applications}, author={Venkatramani Lakshmibai and Peter Littelmann and Peter M. Magyar}, year={1998} }

In these notes, we explain how one can construct Standard Monomial Theory for reductive algebraic groups by using the path models of their representations and quantum groups at a root of unity. As applications, we obtain a combinatorial proof of the Demazure character formula and representation theoretic proofs of geometrical properties of Schubert varieties, such as normality, vanishing theorems, ideal theory and so on. Further applications of Standard Monomial Theory are made to prove… Expand

#### 39 Citations

Standard monomial theory for flag algebras of GL(n) and Sp(2n)

- Mathematics
- 2008

Let G be the general linear group or the symplectic group over the complex numbers, and U be its maximal unipotent subgroup. We study standard monomial theory for the ring of regular functions on… Expand

Affine Weyl Groups in K-Theory and Representation Theory

- Mathematics
- 2003

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our… Expand

Standard Paths and Standard Monomials Fixed by a Diagram Automorphism

- Mathematics
- 2002

Abstract We show that the standard paths and the standard monomials fixed by a diagram automorphism for a Kac–Moody algebra g can be identified with the standard paths and the standard monomials for… Expand

Pieri algebras and Hibi algebras in representation theory

- Mathematics
- 2014

A class of algebras that unify a variety of calculations in the representation theory of classical groups is discussed. Because of their relation to the classical Pieri Rule, these algebras are… Expand

Richardson varieties and equivariant K-theory

- Mathematics
- 2002

We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get… Expand

The Path Model, the Quantum Frobenius Map and Standard Monomial Theory

- Mathematics
- 1998

The aim of this article is to give an introduction to the theory of path models of representations and their associated bases. The starting point for the theory was a series of articles in which… Expand

The ring of sections of a complete symmetric variety

- Mathematics
- 2002

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semisimple group and H is the fixed subgroup for an involutorial… Expand

Equations Defining Symmetric Varieties and Affine

- 2007

Suppose σ be a simple involution of a semisimple algebraic group G, and suppose H is the subgroup of G of points fixed by σ . If the restricted root system is of type A, C, or BC and G is simply… Expand

LS algebras and application to Schubert varieties

- Mathematics
- 2000

In this paper we introduce LS algebras. We study their general properties and apply these results to Schubert varieties. Our main achievement is that any Schubert variety admits a flat deformation to… Expand

Deformation and Cohen-Macaulayness of the multicone over the flag variety

- Mathematics
- 2001

Abstract. A general theory of LS algebras over a multiposet is developed. As a main result, the existence of a flat deformation to discrete algebras is obtained. This is applied to the multicone over… Expand

#### References

SHOWING 1-10 OF 64 REFERENCES

Chern class formulas for quiver varieties

- Mathematics
- 1999

In this paper a formula is proved for the general degeneracy locus associated to an oriented quiver of type A_n. Given a finite sequence of vector bundles with maps between them, these loci are… Expand

Introduction to Quantum Groups

- Mathematics
- 1998

We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra… Expand

Standard Monomial Theory for Bott-Samelson Varieties of GL(n)

- Mathematics
- 1998

We construct an explicit basis for the coordinate ring of the Bott-Samelson variety Z{ associated to G=GL(n) and an arbitrary sequence of simple reflections i. Our basis is parametrized by certain… Expand

Standard Monomial Theory for Bott–Samelson Varieties

- Mathematics
- 1997

Bott–Samelson varieties are an important tool in geometric representation theory [1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the… Expand

On Tangent Spaces to Schubert Varieties, II

- Mathematics
- 2000

Abstract We prove the results on the tangent spaces to Schubert varieties announced in [V. Lakshmibai, Math. Res. Lett.2 (1995), 473–477] for G classical. We give two descriptions of the tangent… Expand

On Zariski tangent spaces of Schubert varieties, and a proof of a conjecture of Deodhar

- Mathematics
- 1994

Abstract Two points are discussed in this note. First, it is shown that in the case of finite Weyl groups, a proof of a conjecture of Deodhar concerning Bruhat intervals can be derived from… Expand

Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras

- Mathematics
- 1998

Let G be a reductive algebraic group defined over an algebraically closed field k. We fix a Borel subgroup B, and for a dominant weight λ let Lλ be the associated line bundle on the generalized flag… Expand

Singular Loci of Ladder Determinantal Varieties and Schubert Varieties

- Mathematics
- 2000

Abstract We relate certain ladder determinantal varieties (associated to one-sided ladders) to certain Schubert varieties in SL ( n )/ Q , for a suitable n and a suitable parabolic subgroup Q , and… Expand

Schubert Varieties and the Variety of Complexes

- Mathematics
- 1983

De Concini and Strickland [s] have obtained interesting results on the “variety of complexes” (e.g. the Cohen-Macaulay nature of its irreducible components) by introducing certain “standard… Expand

Tangent spaces to Schubert varieties

- Mathematics
- 1995

In this note, we announce a criterion for smoothness of a Schubert variety in the ag variety G=B. Let G be a semi simple, simply connected algebraic group, which we assume for simplicity to be de ned… Expand