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''.

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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.

*<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''

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==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 | 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 | 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 | mr=826100 | year=1985 | journal=Annales Scientifiques de l'École Normale Supérieure |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]] |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]]