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Bruhat Order
In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties. History The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by , and the analogue for more general semisimple algebraic groups was studied by . started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat. The left and right weak Bruhat orderings were studied by . Definition If (''W'', ''S'') is a Coxeter system with generators ''S'', then the Bruhat order is a partial order on the group ''W''. Recall that a reduced word for an element ''w'' of ''W'' is a minimal length expression of ''w'' as a product of elements of ''S'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter System
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate Analytics, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed subject sectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazhdan–Lusztig Polynomial
In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P_(q) is a member of a family of integral polynomials introduced by . They are indexed by pairs of elements ''y'', ''w'' of a Coxeter group ''W'', which can in particular be the Weyl group of a Lie group. Motivation and history In the spring of 1978 Kazhdan and Lusztig were studying Springer representations of the Weyl group of an algebraic group on \ell-adic cohomology groups related to unipotent conjugacy classes. They found a new construction of these representations over the complex numbers . The representation had two natural bases, and the transition matrix between these two bases is essentially given by the Kazhdan–Lusztig polynomials. The actual Kazhdan–Lusztig construction of their polynomials is more elementary. Kazhdan and Lusztig used this to construct a canonical basis in the Hecke algebra of the Coxeter group and its representations. In their first paper Kazhdan and Lusztig men ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Order Of Permutations
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order. Definitions Inversion Let \pi be a permutation. There is an inversion of \pi between i and j if i \pi(j). The inversion is indicated by an ordered pair containing either the places (i, j) or the elements \bigl(\pi(i), \pi(j)\bigr). The inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the inverse permutation's inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two components of each ordered pair exchanged. Inversions are usually defined for permutations, but may also be defined for sequences:Let S be a sequence (or multiset permutation). If i S(j), either the pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
François Bruhat
François Georges René Bruhat (; 8 April 1929 – 17 July 2007) was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him. He was the son of physicist (and associate director of the École Normale Supérieure during the occupation) Georges Bruhat, and brother of physicist Yvonne Choquet-Bruhat. See also *Hadamard space In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete ... References * 1929 births 2007 deaths École Normale Supérieure alumni Members of the French Academy of Sciences Nicolas Bourbaki 20th-century French mathematicians 21st-century French mathematicians {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruhat Decomposition
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this. More generally, any group with a (''B'', ''N'') pair has a Bruhat decomposition. Definitions *''G'' is a connected, reductive algebraic group over an algebraically closed field. *''B'' is a Borel subgroup of ''G'' *''W'' is a Weyl group of ''G'' corresponding to a maximal torus of ''B''. The Bruhat decomposition of ''G'' is the decomposition :G=BWB =\bigsqcup_BwB of ''G'' as a disjoint union of double cosets of ''B'' parameterized by the elements of the Weyl group ''W''. (Note that al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl group W of \Phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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