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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]].
| | With so many schools and applications out there via the web an internet diploma program could also be harder to pick out than you assume! Nearly each sort of diploma is now supplied online. Colleges began launching extra on-line applications to stay competitive and increase pupil enrollments. There are numerous things to contemplate when doing a web-based search. On this article we are going to talk about a few of the main characteristics of a good online degree. After studying this you will be able to slim your search and hopefully select probably the most acceptable program that can suit your needs.<br><br>The very first thing you want to ask a school is their accreditation. There are lots of different types of accreditation, however you need to be certain the college is regionally accredited. That is the best stage of accreditation in keeping with the United States Division of Education. Regional accreditation permits a scholar to switch lessons from one college to another, teach at a University or faculty, and pursue higher education. As an illustration in the event you held a bachelor's diploma and wanted to apply to a grasp's program. Regional accreditation can also be what an employer will verify.<br><br>As soon as the varsity is regionally accredited ensure they have a bodily campus. You don't want to attend a strictly on-line school. Hopefully the varsity has a brink-and-mortar campus that gives the same program you'll be taking online. This ensures a top quality program as a result of the professors are most likely educating both on-line and on campus. Also, attempt to make sure the college doesn't hold a stigma of being an online for profit University. Employers have a tendency to take a look at that when they're interviewing potential candidates.<br><br>When talking with totally different advisors from schools be sure that to ask about graduation charges and job placement. Job placement is rare, however it is a nice characteristic for a scholar that is actively looking for employment. Many times a very good college may have a job board just for their graduates. This is a big benefit. You additionally wish to make certain the commencement price is high. Why would you wish to attend a faculty the place a number of the scholars do not graduate? Hopefully you wouldn't.<br><br>As you may see these are just some ideas when doing all your research. There are a lot of extra issues to think about like the price, size, and construction of the web degree. Additionally, see if the college or program is nationally ranked. It is a bonus when applying for various jobs. Pleased searching!<br><br>If you have any sort of questions pertaining to where and ways to use [http://takeonlinedegree.com/ Online Degree Courses], you can call us at the web-page. |
| The dimension of a Demazure module is a polynomial in the highest weight, called a '''Demazure polynomial'''. | |
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| ==Demazure modules==
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| 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.
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| 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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| 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''.
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| 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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| ==Demazure character formula==
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| ===History===
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| The Demazure character formula was introduced by {{harv|Demazure|1974b|loc=theorem 2}}.
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| [[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).
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| ===Statement===
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| The Demazure character formula is
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| :<math>\text{Ch}(F(w\lambda)) = \Delta_1\Delta_2\cdots\Delta_ne^\lambda</math>
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| Here:
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| *''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.
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| *λ is a lowest weight, and ''e''<sup>λ</sup> the corresponding element of the group ring of the weight lattice.
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| *Ch(''F''(''w''λ)) is the character of the Demazure module ''F''(''w''λ).
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| *''P'' is the weight lattice, and '''Z'''[''P''] is its group ring.
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| *Δ<sub>α</sub> for α a root is the endomorphism of the '''Z'''-module '''Z'''[''P''] defined by
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| :<math>\Delta_\alpha(u) = \frac{u-s_\alpha u}{1-e^{-\alpha}}</math>
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| :and Δ<sub>''j''</sub> is Δ<sub>α</sub> for α the root of ''s''<sub>''j''</sub>
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| ==References==
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| [[Category:Representation theory]]
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With so many schools and applications out there via the web an internet diploma program could also be harder to pick out than you assume! Nearly each sort of diploma is now supplied online. Colleges began launching extra on-line applications to stay competitive and increase pupil enrollments. There are numerous things to contemplate when doing a web-based search. On this article we are going to talk about a few of the main characteristics of a good online degree. After studying this you will be able to slim your search and hopefully select probably the most acceptable program that can suit your needs.
The very first thing you want to ask a school is their accreditation. There are lots of different types of accreditation, however you need to be certain the college is regionally accredited. That is the best stage of accreditation in keeping with the United States Division of Education. Regional accreditation permits a scholar to switch lessons from one college to another, teach at a University or faculty, and pursue higher education. As an illustration in the event you held a bachelor's diploma and wanted to apply to a grasp's program. Regional accreditation can also be what an employer will verify.
As soon as the varsity is regionally accredited ensure they have a bodily campus. You don't want to attend a strictly on-line school. Hopefully the varsity has a brink-and-mortar campus that gives the same program you'll be taking online. This ensures a top quality program as a result of the professors are most likely educating both on-line and on campus. Also, attempt to make sure the college doesn't hold a stigma of being an online for profit University. Employers have a tendency to take a look at that when they're interviewing potential candidates.
When talking with totally different advisors from schools be sure that to ask about graduation charges and job placement. Job placement is rare, however it is a nice characteristic for a scholar that is actively looking for employment. Many times a very good college may have a job board just for their graduates. This is a big benefit. You additionally wish to make certain the commencement price is high. Why would you wish to attend a faculty the place a number of the scholars do not graduate? Hopefully you wouldn't.
As you may see these are just some ideas when doing all your research. There are a lot of extra issues to think about like the price, size, and construction of the web degree. Additionally, see if the college or program is nationally ranked. It is a bonus when applying for various jobs. Pleased searching!
If you have any sort of questions pertaining to where and ways to use Online Degree Courses, you can call us at the web-page.