Content deleted Content added
→History: ref |
Link suggestions feature: 3 links added. Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit Newcomer task Suggested: add links |
||
| (22 intermediate revisions by 14 users not shown) | |||
Line 1:
In mathematics, a '''Demazure module''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year1=1974a|year2=1974b}}
The dimension of a Demazure module is a polynomial in the highest weight, called a '''Demazure polynomial'''.
==Demazure modules==
Suppose that ''g'' is a complex [[semisimple Lie algebra]], with a [[Borel subalgebra]] ''b'' containing a [[Cartan subalgebra]] ''h''. An irreducible finite-dimensional representation ''V'' of ''g'' splits as a sum of eigenspaces of ''h'', and the highest weight space is 1-dimensional and is an eigenspace of ''b''. The [[Weyl group]] ''W'' acts on the weights of ''V'', and the conjugates ''w''λ of the highest weight vector λ under this action are the extremal weights, whose weight spaces are all 1-dimensional.
==Demazure character formula==▼
A Demazure module is the ''b''-submodule of ''V'' generated by the weight space of an extremal vector ''w''λ, so the Demazure submodules of ''V'' are parametrized by the Weyl group ''W''.
There are two extreme cases: if ''w'' is trivial the Demazure module is just 1-dimensional, and if ''w'' is the element of maximal length of ''W'' then the Demazure module is the whole of the irreducible representation ''V''.
Demazure modules can be defined in a similar way for highest weight representations of [[Kac–Moody algebra]]s, except that one now has 2 cases as one can consider the submodules generated by either the Borel subalgebra ''b'' or its opposite subalgebra. In the finite-dimensional these are exchanged by the longest element of the Weyl group, but this is no longer the case in infinite dimensions as there is no longest element.
▲==Demazure character formula==
===History===
The Demazure character formula was introduced by {{harv|Demazure|1974b|loc=theorem 2}}.
[[Victor Kac]] pointed out that Demazure's proof has a serious gap, as it depends on {{harv|Demazure|1974a|loc=Proposition 11, section 2}}, which is false; see {{harv|Joseph|
===Statement===
Line 16 ⟶ 23:
:<math>\text{Ch}(F(w\lambda)) = \Delta_1\Delta_2\cdots\Delta_ne^\lambda</math>
Here:
*''w'' is an element of the Weyl group, with reduced decomposition ''w'' = ''s''<sub>1</sub>...''s''<sub>''n''</sub> as a product of reflections of simple roots.
*λ is a lowest weight, and ''e''<sup>λ</sup> the corresponding element of the [[group ring]] of the weight lattice.
*Ch(''F''(''w''λ)) is the character of the Demazure module ''F''(''w''λ).
*''P'' is the weight lattice, and '''Z'''[''P''] is its group ring.
*
*Δ<sub>α</sub> for α a root is the [[endomorphism]] of the '''Z'''-module '''Z'''[''P''] defined by
:<math>\Delta_\alpha(u) = \frac{u-s_\alpha \cdot u}{1-e^{-\alpha}}</math>
:and Δ<sub>''j''</sub> is Δ<sub>α</sub> for α the root of ''s''<sub>''j''</sub>
==References==
*{{Citation | last1=Andersen | first1=H. H. | title=Schubert varieties and Demazure's character formula
*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Désingularisation des variétés de Schubert généralisées |
*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Une nouvelle formule des caractères |
*{{Citation | last1=Joseph | first1=Anthony | title=On the Demazure character formula |
*{{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | title=The crystal base and Littelmann's refined Demazure character formula
*{{Citation | last1=Littelmann | first1=Peter | title=Crystal graphs and Young tableaux
*{{Citation | last1=Mehta | first1=V. B. | last2=Ramanathan | first2=A. | title=Frobenius splitting and cohomology vanishing for Schubert varieties
*{{Citation | last1=Ramanan | first1=S. | last2=Ramanathan | first2=A. | title=Projective normality of flag varieties and Schubert varieties
[[Category:Representation theory]]
| |||