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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Kostant polynomial ： ウィキペディア英語版 Kostant polynomial 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. ==Background== 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 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 and independently as a tool to understand the Schubert calculus of the flag manifold. The Kostant polynomials are related to the Schubert polynomials defined combinatorially by for the classical flag manifold, when ''G'' = SL(n,C). Their structure is governed by difference operators associated to the corresponding root system. 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kostant polynomial」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.