Demazure module

In mathematics, a Demazure module, , introduced by Demazure (1974a, 1974b). 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 Demazure (1974b, theorem 2), gives the characters of Demazure modules, and is a generalization of the Weyl character formula. The dimension of a Demazure module is a polynomial in the highest weight, called a Demazure polynomial.

History of the proof

Victor Kac pointed out that the original proof of the character formula in (Demazure 1974b) has a serious gap, as it depends on Demazure (1974a, Proposition 11, section 2), which is false. Anderson (1985) gave a proof of Demazure's character formula using the work on the geometry of Schubert varieties by Ramanan & Ramanathan (1985) and Mehta & Ramanathan (1985). Joseph (1985) gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques. Kashiwara (1993) proved a refined version of the Demazure character formula that Littelmann (1995) conjectured (and proved in many cases).

Statement

The Demazure character formula is

${\text{Ch}}(F(w\lambda ))=\Delta _{1}\Delta _{2}\cdots \Delta _{n}e^{\lambda }$

Here:

• w is an element of the Weyl group, with reduced decomposition w=s₁...s_n as a product of reflections of simple roots.
• λ is a lowest weight, and e^λ the corresponding element of the group ring of the weight lattice.
• Ch(F(wλ)) is the character of the Demazure module F(wλ).
• P is the weight lattice, and Z[P] is its group ring.
• Δ_α for α a root is the endomorphism of the Z-module Z[P] defined by

$\Delta _{\alpha }(u)={\frac {u-s_{\alpha }u}{1-e^{-\alpha }}}$

and Δ_j is Δ_α for α the root of s_j

References

• Andersen, H. H. (1985), "Schubert varieties and Demazure's character formula", Inventiones Mathematicae, 79 (3): 611–618, doi:10.1007/BF01388527, ISSN 0020-9910, MR782239
• Demazure, Michel (1974a), "Désingularisation des variétés de Schubert généralisées", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I, 7: 53–88, ISSN 0012-9593, MR0354697
• Demazure, Michel (1974b), "Une nouvelle formule des caractères", Bulletin des Sciences Mathématiques. 2e Série, 98 (3): 163–172, ISSN 0007-4497, MR0430001
• Joseph, Anthony (1985), "On the Demazure character formula", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 18 (3): 389–419, ISSN 0012-9593, MR826100
• Kashiwara, Masaki (1993), "The crystal base and Littelmann's refined Demazure character formula", Duke Mathematical Journal, 71 (3): 839–858, doi:10.1215/S0012-7094-93-07131-1, ISSN 0012-7094, MR1240605
• Littelmann, Peter (1995), "Crystal graphs and Young tableaux", Journal of Algebra, 175 (1): 65–87, doi:10.1006/jabr.1995.1175, ISSN 0021-8693, MR1338967
• Mehta, V. B.; Ramanathan, A. (1985), "Frobenius splitting and cohomology vanishing for Schubert varieties", Annals of Mathematics. Second Series, 122 (1): 27–40, doi:10.2307/1971368, ISSN 0003-486X, MR799251
• Ramanan, S.; Ramanathan, A. (1985), "Projective normality of flag varieties and Schubert varieties", Inventiones Mathematicae, 79 (2): 217–224, doi:10.1007/BF01388970, ISSN 0020-9910, MR778124