Demazure module: Difference between revisions

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In mathematics, thea '''Demazure character formulamodule''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year1=1974a|year2=1974b}}, is a generalization[[submodule]] of thea finite-dimensional representation generated by an extremal [[WeylWeight character(representation formulatheory)|weight]] forspace under the charactersaction of finite dimensional representations ofa [[semisimple LieBorel algebrasubalgebra]]s. The '''Demazure character formula''', introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year=1974b|loc=theorem 2}}., Demazure's formula gives the characters of '''Demazure modules''', theand submodules ofis a finitegeneralization dimensional representation generated by an extremal weight underof the action[[Weyl ofcharacter a Borel subalgebraformula]].

The dimension of a Demazure module is a polynomial in the highest weight, called a '''Demazure polynomial'''.

==Demazure modules==

==History of the proof==

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.

[[Victor Kac]] pointed out that the original proof of the character formula in {{harv|Demazure|1974b}} has a serious gap, as it depends on {{harvtxt|Demazure|1974a|loc=Proposition 11, section 2}}, which is false. {{harvtxt|Anderson|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).

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''.

==Statement==

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 of the proof== =

The Demazure character formula was introduced by {{harv|Demazure|1974b|loc=theorem 2}}.

[[Victor Kac]] pointed out that the originalDemazure's proof of the character formula in {{harv|Demazure|1974b}} has a serious gap, as it depends on {{harvtxtharv|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===

The Demazure character formula is

:<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''&nbsp;=&nbsp;''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.

*<math>\rho</math> is the sum of fundamental weights and the dot 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 | url=http://dx.doi.org/10.1007/BF01388527 | doi=10.1007/BF01388527 | idmr={{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 | urlmr=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 | id={{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 | idmr={{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 | urlmr=http://www.numdam.org/item?id=ASENS_1985_4_18_3_389_0 | id={{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 | url=http://dx.doi.org/10.1215/S0012-7094-93-07131-1 | doi=10.1215/S0012-7094-93-07131-1 | idmr={{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 | url=http://dx.doi.org/10.1006/jabr.1995.1175 | doi=10.1006/jabr.1995.1175 | idmr={{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 | url=http://dx.doi.org/10.2307/1971368 | doi=10.2307/1971368 | idmr={{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 | url=http://dx.doi.org/10.1007/BF01388970 | doi=10.1007/BF01388970 | idmr={{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]]