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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 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'''.


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==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.
 
==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|1985|loc=section 4}} for Kac's counterexample. {{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).
 
===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''<sub>1</sub>...''s''<sub>''n''</sub> as a product of reflections of simple roots.
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 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 | doi=10.1007/BF01388527 | mr=782239 | year=1985 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=79 | issue=3 | pages=611–618}}
*{{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 | pages=53–88}}
*{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Une nouvelle formule des caractères | mr=0430001 | year=1974b| journal=Bulletin des Sciences Mathématiques. 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 Série | issn=0012-9593 | volume=18 | issue=3 | pages=389–419}}
*{{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}}
*{{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. Second Series]] | issn=0003-486X | volume=122 | issue=1 | pages=27–40}}
*{{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}}
 
[[Category:Representation theory]]

Revision as of 17:34, 30 December 2013

In mathematics, a Demazure module, introduced by Template:Harvs. 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 Template:Harvs, 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 wλ 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 wλ, 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 algebras, 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 Template:Harv. Victor Kac pointed out that Demazure's proof has a serious gap, as it depends on Template:Harv, which is false; see Template:Harv for Kac's counterexample. Template:Harvtxt gave a proof of Demazure's character formula using the work on the geometry of Schubert varieties by Template:Harvtxt and Template:Harvtxt. Template:Harvtxt gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques. Template:Harvtxt proved a refined version of the Demazure character formula that Template:Harvtxt conjectured (and proved in many cases).

Statement

The Demazure character formula is

Here:

  • w is an element of the Weyl group, with reduced decomposition w = s1...sn as a product of reflections of simple roots.
  • λ is a lowest weight, and eλ 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.
  • Δα for α a root is the endomorphism of the Z-module Z[P] defined by
and Δj is Δα for α the root of sj

References

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    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010