Demazure module: Difference between revisions

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In mathematics, a '''Demazure module''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year1=1974a|year2=1974b}}, is a [[submodule]] of a finite-dimensional representation generated by an extremal [[Weight (representation theory)|weight]] space under the action of a [[Borel subalgebra]]. The '''Demazure character formula''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year=1974b|loc=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'''.

==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.

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.

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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|1985|loc=section 4}} for Kac's counterexample. {{harvtxt|AndersonAndersen|1985}} gave a proof of Demazure's character formula using the work on the geometry of [[Schubert varieties]] by {{harvtxt|Ramanan|Ramanathan|1985}} and {{harvtxt|Mehta|Ramanathan|1985}}. {{harvtxt|Joseph|1985}} gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques. {{harvtxt|Kashiwara|1993}} proved a refined version of the Demazure character formula that {{harvtxt|Littelmann|1995}} conjectured (and proved in many cases).

===Statement===

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*''w'' is an element of the Weyl group, with reduced decomposition ''w'' = ''s''₁...''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''(''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<submath>α\rho</submath> for α a root is the endomorphismsum of fundamental weights and the '''Z'''-moduledot '''Z'''[''P'']action is defined by <math>w\cdot u=w(u+\rho)-\rho</math>.

*Δ_α 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_''j'' is Δ_α for α the root of ''s''_''j''

==References==

*{{Citation | last1=Andersen | first1=H. H. | title=Schubert varieties and Demazure's character formula | doi=10.1007/BF01388527 | mr=782239 | year=1985 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=79 | issue=3 | pages=611–618| bibcode=1985InMat..79..611A | s2cid=121295084 }}

*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Désingularisation des variétés de Schubert généralisées | url=http://www.numdam.org/item?id=ASENS_1974_4_7_1_53_0 | series=Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I | mr=0354697 | year=1974a | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=7 |series=Série 4 | pages=53–88| doi=10.24033/asens.1261 | doi-access=free }}<!-- Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday-->

*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Une nouvelle formule des caractères | mr=0430001 | year=1974b| journal=Bulletin des Sciences Mathématiques. |series=2e Série | issn=0007-4497 | volume=98 | issue=3 | pages=163–172}}

*{{Citation | last1=Joseph | first1=Anthony | title=On the Demazure character formula | url=http://www.numdam.org/item?id=ASENS_1985_4_18_3_389_0 | mr=826100 | year=1985 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième |series=Série 4 | issn=0012-9593 | volume=18 | issue=3 | pages=389–419| doi=10.24033/asens.1493 | doi-access=free }}

*{{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | title=The crystal base and Littelmann's refined Demazure character formula | doi=10.1215/S0012-7094-93-07131-1 | mr=1240605 | year=1993 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=71 | issue=3 | pages=839–858}}

*{{Citation | last1=Littelmann | first1=Peter | title=Crystal graphs and Young tableaux | doi=10.1006/jabr.1995.1175 | mr=1338967 | year=1995 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=175 | issue=1 | pages=65–87| doi-access=free }}

*{{Citation | last1=Mehta | first1=V. B. | last2=Ramanathan | first2=A. | title=Frobenius splitting and cohomology vanishing for Schubert varieties | doi=10.2307/1971368 | mr=799251 | year=1985 | journal=[[Annals of Mathematics|Annals of Mathematics.]] |series=Second Series]] | issn=0003-486X | volume=122 | issue=1 | pages=27–40| jstor=1971368 }}

*{{Citation | last1=Ramanan | first1=S. | last2=Ramanathan | first2=A. | title=Projective normality of flag varieties and Schubert varieties | doi=10.1007/BF01388970 | mr=778124 | year=1985 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=79 | issue=2 | pages=217–224| bibcode=1985InMat..79..217R | s2cid=123105737 }}

[[Category:Representation theory]]