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| In [[mathematics]], the '''Kostant polynomials''', named after [[Bertram Kostant]], provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the [[finite reflection group]] of a [[root system]].
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. If you have any kind of inquiries relating to where and ways to make use of [http://decarboni.se/users/clemmie-clemmie wordpress backup plugin], you could call us at the page. Medical word press themes give you the latest medical designs. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. If you are using videos on your site then this is the plugin to use. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br>The Internet is a vast open market where businesses and consumers congregate. If you are a positive thinker businessman then today you have to put your business online. It sorts the results of a search according to category, tags and comments. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. After age 35, 18% of pregnancies will end in miscarriage. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Have you heard about niche marketing and advertising. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. The company gains commission from the customers' payment. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products and services for your customers. Here's a list of some exciting Word - Press features that have created waves in the web development industry:. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Your topic is going to be the basis of your site's name. |
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| ==Background==
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| If the reflection group ''W'' corresponds to the [[Weyl group]] of a compact semisimple group ''K'' with maximal torus ''T'', then the Kostant polynomials describe the structure of the [[de Rham cohomology]] of the generalized [[flag manifold]] ''K''/''T'', also isomorphic to ''G''/''B'' where ''G'' is the [[complex Lie group|complexification]] of ''K'' and ''B'' is the corresponding [[Borel subgroup]]. [[Armand Borel]] showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by the invariant homogeneous polynomials of positive degree. This ring had already been considered by [[Claude Chevalley]] in establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with [[André Weil]], [[Jean-Louis Koszul]] and [[Henri Cartan]]; the existence of such a basis was used by Chevalley to prove that the ring of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by {{harvtxt|Bernstein|Gelfand|Gelfand|1973}} and independently {{harvtxt|Demazure|1973}} as a tool to understand the [[Schubert calculus]] of the flag manifold. The Kostant polynomials are related to the [[Schubert polynomial]]s defined combinatorially by {{harvtxt|Lascoux|Schützenberger|1982}} for the classical flag manifold, when ''G'' = SL(n,'''C'''). Their structure is governed by [[difference operator]]s associated to the corresponding [[root system]].
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| {{harvtxt|Steinberg|1975}} defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of the weight lattice. If ''K'' is [[simply connected]], this ring can be identified with the [[representation ring]] ''R''(''T'') and the ''W''-invariant subring with ''R''(''K''). Steinberg's basis was again motivated by a problem on the topology of homogeneous spaces; the basis arises in describing the ''T''-[[equivariant K-theory]] of ''K''/''T''.
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| ==Definition==
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| Let Φ be a [[root system]] in a finite-dimensional real inner product space ''V'' with [[Weyl group]] ''W''. Let Φ<sup>+</sup> be a set of positive roots and Δ the corresponding set of simple roots. If α is a root, then ''s''<sub>α</sub> denotes the corresponding reflection operator. Roots are regarded as linear polynomials on ''V'' using the inner product α(''v'') = (α,''v''). The choice of Δ gives rise to a [[Bruhat order]] on the Weyl group
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| determined by the ways of writing elements minimally as products of simple root reflection. The minimal length for an elenet ''s'' is denoted
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| <math>\ell(s)</math>. Pick an element ''v'' in ''V'' such that α(''v'') > 0 for every positive root.
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| If α<sub>''i''</sub> is a simple root with reflection operator ''s''<sub>''i''</sub>
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| :<math> s_i x= x- 2{(x,\alpha_i)\over (\alpha_i,\alpha_i)}\alpha_i,</math>
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| then the corresponding '''divided difference operator''' is defined by | |
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| :<math> \delta_i f = {f-f\circ s_i\over \alpha_i}.</math>
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| If <math>\ell(s)=m</math> and ''s'' has reduced expression
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| :<math>s=s_{i_1}\cdots s_{i_m},</math>
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| then
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| :<math>\delta_s=\delta_{i_1}\cdots \delta_{i_m}</math>
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| is independent of the reduced expression. Moreover | |
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| :<math>\displaystyle \delta_s\delta_t=\delta_{st}</math>
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| if <math>\ell(st)=\ell(s)+\ell(t)</math> and 0 otherwise.
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| If ''w''<sub>0</sub> is the [[Coxeter element]] of ''W'', the element of greatest length or equivalently the element sending Φ<sup>+</sup> to
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| −Φ<sup>+</sup>, then
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| :<math> \delta_{w_0}f= {\sum_{s \in W} {\rm det} \, s \, f\circ s\over \prod_{\alpha>0} \alpha}.</math>
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| More generally
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| :<math>\delta_{s}f={{\rm det}\, s \, f\circ s + \sum_{t<s} a_{s,t} \,f\circ t\over \prod_{\alpha>0, \, s^{-1}\alpha<0} \alpha}</math>
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| for some constants ''a''<sub>''s'',''t''</sub>.
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| Set
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| :<math>\displaystyle d= |W|^{-1}\prod_{\alpha>0} \alpha.</math>
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| and
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| :<math>\displaystyle P_s=\delta_{s^{-1}w_0} d.</math>
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| Then ''P''<sub>s</sub> is a homogeneous polynomial of degree <math>\ell(s)</math>.
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| These polynomials are the '''Kostant polynomials'''.
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| ==Properties==
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| '''Theorem'''. ''The Kostant polynomials form a free basis of the ring of polynomials over the W-invariant polynomials.''
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| In fact the matrix
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| :<math>\displaystyle N_{st} =\delta_s (P_t) </math>
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| is unitriangular for any total order such that ''s'' ≥ ''t'' implies <math>\ell(s)\ge \ell(t)</math>.
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| Hence
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| :<math>\displaystyle {\rm det}\, N=1.</math>
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| Thus if
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| :<math> \displaystyle f =\sum_s a_s P_s</math>
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| with ''a''<sub>''s''</sub> invariant under ''W'', then
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| :<math>\displaystyle \delta_t(f) = \sum_s \delta_t(P_s) a_s.</math>
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| Thus
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| :<math> \displaystyle a_s = \sum_t M_{s,t} \delta_t(f),</math>
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| where
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| :<math>\displaystyle M=N^{-1}</math>
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| another unitriangular matrix with polynomial entries.
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| It can be checked directly that ''a''<sub>''s''</sub> is invariant under ''W''.
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| In fact δ<sub>''i''</sub> satisfies the [[Derivation (abstract algebra)|derivation]] property
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| :<math>\delta_i(fg)=\delta_i(f)g + (f\circ s_i)\delta_i(g).</math>
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| Hence
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| :<math>\delta_i\delta_s(f) = \sum_t \delta_i( \delta_s(P_t))a_t) = \sum_t (\delta_s(P_t)\circ s_i)\delta_i(a_t) + \sum_t \delta_i\delta_s(P_t)a_t.</math>
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| Since
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| :<math>\delta_i\delta_s=\delta_{s_is}</math>
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| or 0, it follows that
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| :<math> \sum_t \delta_s(P_t)\,\delta_i(a_t)\circ s_i=0</math>
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| so that by the invertibility of ''N''
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| :<math>\displaystyle\delta_i(a_t)=0</math>
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| for all ''i'', i.e. ''a''<sub>''t''</sub> is invariant under ''W''.
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| ==Steinberg basis==
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| As above let Φ be a [[root system]] in a real inner product space ''V'', and Φ<sup>+</sup> a subset of positive roots. From these data we obtain the subset Δ = { α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>''n''</sub>} of the simple roots, the coroots
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| :<math>\displaystyle \alpha_i^\vee=2(\alpha_i,\alpha_i)^{-1}\alpha_i,</math>
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| and the fundamental weights λ<sub>1</sub>, λ<sub>2</sub>, ..., λ<sub>''n''</sub> as the dual basis of the coroots.
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| For each element ''s'' in ''W'', let Δ<sub>''s''</sub> be the subset of Δ consisting of the simple roots satisfying ''s''<sup>−1</sup>α < 0, and put
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| :<math>\lambda_s = s^{-1}\sum_{\alpha_i \in \Delta_s} \lambda_i,</math>
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| where the sum is calculated in the weight lattice ''P''.
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| The set of linear combinations of the exponentials ''e''<sup>μ</sup> with integer coefficients for μ in ''P'' becomes a ring over '''Z''' isomorphic to the group algebra of ''P'', or equivalently to the representation ring
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| ''R''(''T'') of ''T'', where ''T'' is a maximal torus in ''K'', the simply connected, connected compact semisimple Lie group with root system Φ. If ''W'' is the Weyl group of Φ, then the representation ring ''R''(''K'') of ''K'' can be identified with ''R''(''T'')<sup>''W''</sup>.
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| '''Steinberg's theorem'''. ''The exponentials'' λ<sub>''s''</sub> (''s'' ''in'' ''W'') ''form a free basis for the ring of exponentials over the subring of'' ''W''-''invariant exponentials.''
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| Let ρ denote the half sum of the positive roots, and ''A'' denote the antisymmetrisation operator
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| :<math>A(\psi)=\sum_{s\in W} (-1)^{\ell(s)} s\cdot \psi.</math>
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| The positive roots β with ''s''β positive can be seen as a set of positive roots for a root system on a subspace of ''V''; the roots are the ones orthogonal to s.λ<sub>''s''</sub>. The corresponding Weyl group equals the stabilizer of λ<sub>''s''</sub> in ''W''. It is generated by the simple reflections ''s''<sub>''j''</sub> for which ''s''α<sub>''j''</sub> is a positive root.
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| Let ''M'' and ''N'' be the matrices
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| :<math>M_{ts}=t(\lambda_s),\,\,N_{st}= (-1)^{\ell(t)}\cdot t(\psi_s),</math>
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| where ψ<sub>''s''</sub> is given by the weight ''s''<sup>−1</sup>ρ - λ<sub>''s''</sub>. Then the matrix
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| :<math>B_{s,s^\prime}=\Omega^{-1}(NM)_{s,s^\prime}={A(\psi_s\lambda_{s^\prime})\over \Omega}</math>
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| is triangular with respect to any total order on ''W'' such that ''s'' ≥ ''t'' implies <math>\ell(s)\ge \ell(t)</math>.
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| Steinberg proved that the entries of ''B'' are ''W''-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1. Hence its inverse ''C'' has the same form. Define
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| :<math>\varphi_s=\sum C_{s,t}\psi_t.</math>
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| If χ is an arbitrary exponential sum, then it follows that
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| :<math>\chi=\sum_{s\in W} a_s \lambda_s</math>
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| with ''a''<sub>''s''</sub> the ''W''-invariant exponential sum
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| :<math> a_s ={A(\varphi_s\chi)\over \Omega}.</math>
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| Indeed this is the unique solution of the system of equations
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| :<math>t\chi=\sum_{s\in W} t(\lambda_s)\,\,a_s=\sum_s M_{t,s}a_s.</math>
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| ==References==
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| *{{citation|last=Bernstein|first= I. N.|last2= Gelfand|first2=I. M.|authorlink2=I. M. Gelfand |last3=Gelfand|first3= S. I.
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| |title=Schubert cells, and the cohomology of the spaces G/P|journal=Russian Math. Surveys|volume= 28 |year=1973|pages= 1–26|doi=10.1070/RM1973v028n03ABEH001557}}
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| *{{citation|last=Billey|first= Sara C.|authorlink=Sara Billey|title=Kostant polynomials and the cohomology ring for G/B.|journal=Duke Math. J. |volume=96 |year=1999|pages= 205–224|doi=10.1215/S0012-7094-99-09606-0}}
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| *{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Groupes et algèbres de Lie, Chapitres 4, 5 et 6|publisher=Masson|isbn=2-225-76076-4|year=1981}}
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| *{{citation|last=Cartan|first=Henri|authorlink=Henri Cartan|title=Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie|journal= Colloque de topologie (espaces fibrés), Bruxelles|year= 1950|pages= 15–27}}
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| *{{citation|last=Cartan|first=Henri|authorlink=Henri Cartan|title=La transgression dans un groupe de Lie et dans un espace fibré principal|
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| journal= Colloque de topologie (espaces fibrés), Bruxelles|year= 1950|pages= 57–71}}
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| *{{citation|last=Chevalley|first= Claude|authorlink=Claude Chevalley|title=Invariants of finite groups generated by reflections|journal=Amer. J. Math.|volume= 77|year=1955|pages= 778–782|doi=10.2307/2372597|jstor=2372597|issue=4|publisher=The Johns Hopkins University Press}}
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| *{{citation|last=Demazure|first= Michel|authorlink=Michel Demazure|title=Invariants symétriques entiers des groupes de Weyl et torsion|
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| journal=Invent. Math.|volume= 21 |year=1973|pages= 287–301|doi=10.1007/BF01418790}}
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| *{{citation|last=Greub|first= Werner|last2=Halperin|first2= Stephen|last3= Vanstone|first3= Ray|title=Connections, curvature, and cohomology. Volume III: Cohomology of principal bundles and homogeneous spaces|series= Pure and Applied Mathematics|volume= 47-III|publisher= Academic Press |year = 1976}}
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| *{{citation|title=Introduction to Lie Algebras and Representation Theory|first= James E.|last= Humphreys|edition=2nd|publisher=Springer|year= 1994
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| |isbn=0-387-90053-5}}
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| *{{citation|last=Kostant|first= Bertram|authorlink=Bertram Kostant|title=Lie algebra cohomology and generalized Schubert cells|
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| journal=Ann. Of Math.|volume= 77|year= 1963|pages= 72–144|doi=10.2307/1970202|jstor=1970202|issue=1|publisher=Annals of Mathematics}}
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| *{{citation|last=Kostant|first= Bertram|authorlink=Bertram Kostant|title=Lie group representations on polynomial rings|journal=Amer. J. Math.|volume= 85|year= 1963 |pages=327–404|doi=10.2307/2373130|jstor=2373130|issue=3|publisher=The Johns Hopkins University Press}}
| |
| *{{citation|last=Kostant|first= Bertram|authorlink=Bertram Kostant|last2= Kumar|first2= Shrawan|
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| title=The nil Hecke ring and cohomology of G/P for a Kac–Moody group G.|journal=Proc. Nat. Acad. Sci. U.S.A.|volume= 83 |year=1986|pages= 1543–1545|doi=10.1073/pnas.83.6.1543}}
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| *{{citation|first=Lascoux|last= Alain|last2= Schützenberger|first2= Marcel-Paul|author1-link=Alain Lascoux|authorlink2=Marcel-Paul Schützenberger|
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| title=Polynômes de Schubert [Schubert polynomials]|journal=C. R. Acad. Sci. Paris Sér. I Math.|volume= 294 |year=1982|pages=447–450}}
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| *{{citation|last=McLeod|first=John|title=The Kunneth formula in equivariant K-theory |pages= 316–333|
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| series=Lecture Notes in Math.|volume= 741|publisher= Springer|year= 1979}}
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| *{{citation|last=Steinberg|first= Robert|authorlink=Robert Steinberg|title=On a theorem of Pittie|journal=Topology|volume= 14|year=1975|pages= 173–177|doi=10.1016/0040-9383(75)90025-7}}
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| [[Category:Invariant theory]]
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| [[Category:Topology of homogeneous spaces]]
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| [[Category:Algebraic groups]]
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