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Standard Monomial
In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of the results have been extended to Kac–Moody algebras and their groups. There are monographs on standard monomial theory by and and survey articles by and One of important open problems is to give a completely geometric construction of the theory.M. Brion and V. Lakshmibai : A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680. History introduced monomials associated to standard Young tableaux. (see also ) used Young's monomials, which he called standard power products, named after standard tableaux, to give a basis for the homogeneous coordinate rings of complex Grassmannians. initiated a program, called standard monomial theory, to extend Hodge's work to varieties ''G''/''P'', for ''P'' any p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Base
A crystal base for a representation of a quantum group on a \Q(v)-vector space is not a base of that vector space but rather a \Q-base of L/vL where L is a \Q(v)-lattice in that vector spaces. Crystal bases appeared in the work of and also in the work of . They can be viewed as specializations as v \to 0 of the canonical basis defined by . Definition As a consequence of its defining relations, the quantum group U_q(G) can be regarded as a Hopf algebra over the field of all rational functions of an indeterminate ''q'' over \Q, denoted \Q(q). For simple root \alpha_i and non-negative integer n, define :\begin e_i^ = f_i^ &= 1 \\ e_i^ &= \frac \\ ptf_i^ &= \frac \end In an integrable module M, and for weight \lambda, a vector u \in M_ (i.e. a vector u in M with weight \lambda) can be uniquely decomposed into the sums :u = \sum_^\infty f_i^ u_n = \sum_^\infty e_i^ v_n, where u_n \in \ker(e_i) \cap M_, v_n \in \ker(f_i) \cap M_, u_n \ne 0 only if n + \frac \ge 0, and v_n \ne 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * * * * ISI Ale ... [...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 internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Filtration
In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraic group ''G'' such that the subquotients are isomorphic to the spaces of sections ''F''(λ) of line bundles λ over ''G''/''B'' for a Borel subgroup ''B''. In characteristic 0 this is automatically true as the irreducible modules are all of the form ''F''(λ), but this is not usually true in positive characteristic. showed that the tensor product of two modules ''F''(λ)⊗''F''(μ) has a good filtration, completing the results of who proved it in most cases and who proved it in large characteristic. showed that the existence of good filtrations for these tensor products also follows from standard monomial theory. References * * * *{{Citation , last1=Wang , first1=Jian Pan , title=Sheaf cohomology on G/B and tensor products of Weyl modules , doi=10.1016/0021-8693(82)90284-8 , mr=665171 , year=1982 , journal=Journal of Algebra ''Journal of Algebra'' (ISS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Littlewood–Richardson Rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say \lambda,\mu,\nu, of which \lambda and \mu describe the Schur functions being multiplied, and \nu gives the Schur function of which this is the coefficient in the linear combination; in other words they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kempf Vanishing Theorem
In algebraic geometry, the Kempf vanishing theorem, introduced by , states that the higher cohomology group ''H''''i''(''G''/''B'',''L''(λ)) (''i'' > 0) vanishes whenever λ is a dominant weight of ''B''. Here ''G'' is a reductive algebraic group over an algebraically closed field, ''B'' a Borel subgroup, and ''L''(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, ..., but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic. and found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism. References * * * * Algebraic groups Theorems in algebraic geometry {{algebraic-geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demazure's Conjecture
In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by . The conjecture implies that many of the results of his paper can be extended from complex algebraic groups to algebraic groups over fields of other characteristics or over the integers. showed that Demazure's conjecture (for classical groups) follows from their work on standard monomial theory In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of th ..., and Peter Littelmann extended this to all reductive algebraic groups. References * *{{cite journal , last1=Lakshmibai , first1=V. , last2=Musili , first2=C. , last3=Seshadri , first3=C. S. , title=Geometry of G/P , doi=10.1090/S0273-0979-1979-14631-7 , mr=520081 , year=1979 , journal=Bulletin of the America ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demazure Module
In mathematics, a Demazure module, introduced by , 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 , 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 ''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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
