|
Hecke Algebra (other)
In mathematics, a Hecke algebra is classically the algebra of Hecke operators studied by Erich Hecke. It may also refer to one of several algebras (some of which are related to the classical Hecke algebra): * Iwahori–Hecke algebra of a Coxeter group. * Hecke algebra of a pair (''g'',''K'') where ''g'' is the Lie algebra of a Lie group ''G'' and ''K'' is a compact subgroup of ''G''. *Hecke algebra of a locally compact group ''H''(''G'',''K''), for a locally compact group ''G'' with respect to a compact subgroup ''K''. ** Hecke algebra of a finite group, the algebra spanned by the double cosets ''HgH'' of a subgroup ''H'' of a finite group ''G''. ** Spherical Hecke algebra, when ''K'' is a maximal open compact subgroup of a general linear group. *Affine Hecke algebra In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. Definition Let V be a Euclidean space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators. Properties The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators ''T''''n'' with ''n'' coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators. Each of these basic forms possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime ''p'' is the reciprocal of the Hecke polynomial, a quadratic polynomial in ''p''−''s''. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of ''τ''(''n''). See al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwahori–Hecke Algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. Hecke algebras are quotients of the group rings of Artin braid groups. This connection found a spectacular application in Vaughan Jones' construction of new invariants of knots. Representations of Hecke algebras led to discovery of quantum groups by Michio Jimbo. Michael Freedman proposed Hecke algebras as a foundation for topological quantum computation. Hecke algebras of Coxeter groups Start with the following data: * (''W'', ''S'') is a Coxeter system with the Coxeter matrix ''M'' = (''m''''st''), * ''R'' is a commutative ring with identity. * is a family of units of ''R'' such that ''qs'' = ''qt'' whenever ''s'' and ''t'' are conjugate in ''W'' * ''A'' is the ring of Laurent polynomials over Z with indeterminates ''qs'' (and the above restriction that ''qs'' = ''qt'' whenever ''s'' and ''t'' are conjugate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra Of A Pair
In mathematical representation theory, the Hecke algebra of a pair (''g'',''K'') is an algebra with an approximate identity, whose approximately unital modules are the same as ''K''-finite representations of the pairs (''g'',''K''). Here ''K'' is a compact subgroup of a Lie group with Lie algebra ''g''. Definition The Hecke algebra of a pair (''g'',''K'') is the algebra of ''K''-finite distributions on ''G'' with support in ''K'', with the product given by convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' .... References * Representation theory {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra Of A Locally Compact Group
In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution. Definition Let (''G'',''K'') be a pair consisting of a unimodular locally compact topological group ''G'' and a closed subgroup ''K'' of ''G''. Then the space of bi-''K''-invariant continuous functions of compact support :''C'' 'K''\''G''/''K'' can be endowed with a structure of an associative algebra under the operation of convolution. This algebra is denoted :''H''(''G''//''K'') and called the Hecke ring of the pair (''G'',''K''). If we start with a Gelfand pair then the resulting algebra turns out to be commutative. Examples SL(2) In particular, this holds when :''G'' = ''SL''''n''(''Q''''p'') and ''K'' = ''SL''''n''(''Z''''p'') and the representations of the corresponding commutative Hecke ring were studied by Ian G. Macdonald. GL(2) On the other hand, in the case :''G'' = ''GL''2(Q) and ''K'' = ''GL''2(Z) we have the classical Hecke alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra Of A Finite Group
The Hecke algebra of a finite group is the algebra spanned by the double cosets ''HgH'' of a subgroup ''H'' of a finite group ''G''. It is a special case of a Hecke algebra of a locally compact group. Definition Let ''F'' be a field of characteristic zero, ''G'' a finite group and ''H'' a subgroup of ''G''. Let F /math> denote the group algebra of ''G'': the space of ''F''-valued functions on ''G'' with the multiplication given by convolution. We write F /H/math> for the space of ''F''-valued functions on G/H. An (''F''-valued) function on ''G''/''H'' determines and is determined by a function on ''G'' that is invariant under the right action of ''H''. That is, there is the natural identification: :F /H= F H. Similarly, there is the identification :R := \operatorname_G(F /H = F given by sending a ''G''-linear map ''f'' to the value of ''f'' evaluated at the characteristic function of ''H''. For each double coset HgH, let T_g denote the characteristic function of it. Then those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Hecke Algebra
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Hecke Algebra
In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. Definition Let V be a Euclidean space of a finite dimension and \Sigma an affine root system on V. An affine Hecke algebra is a certain associative algebra that deforms the group algebra \mathbb /math> of the Weyl group W of \Sigma (the affine Weyl group). It is usually denoted by H(\Sigma,q), where q:\Sigma\rightarrow \mathbb is multiplicity function that plays the role of deformation parameter. For q\equiv 1 the affine Hecke algebra H(\Sigma,q) indeed reduces to \mathbb /math>. Generalizations Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-called double affine Hecke algebra (usually referred to as DAHA). Using this he was able to give a proof of Macdonald's constant term conjecture for Macdonald polynomials (building on work of Eric Opdam). Another main inspiration for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Hecke Algebra
Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable. Parabolic may refer to: *In mathematics: **In elementary mathematics, especially elementary geometry: **Parabolic coordinates **Parabolic cylindrical coordinates ** parabolic Möbius transformation **Parabolic geometry (other) ** Parabolic spiral **Parabolic line **In advanced mathematics: ***Parabolic cylinder function ***Parabolic induction ***Parabolic Lie algebra ***Parabolic partial differential equation *In physics: **Parabolic trajectory *In technology: **Parabolic antenna **Parabolic microphone **Parabolic reflector **Parabolic trough - a type of solar thermal energy collector **Parabolic flight - a way of achieving weightlessness ** Parabolic action, or parabolic bending curve - a term often used to refer to a progressive bending curve in fishing rods. *In commodities and stock markets: **Parabolic SAR - a chart pattern in which prices rise or fall with an increasingly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parahoric Hecke Algebra
In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a nonarchimedean local field that is analogous to a Borel subgroup of an algebraic group. A parahoric subgroup is a proper subgroup that is a finite union of double cosets of an Iwahori subgroup, so is analogous to a parabolic subgroup of an algebraic group. Iwahori subgroups are named after Nagayoshi Iwahori, and "parahoric" is a portmanteau of "parabolic" and "Iwahori". studied Iwahori subgroups for Chevalley groups over ''p''-adic fields, and extended their work to more general groups. Roughly speaking, an Iwahori subgroup of an algebraic group ''G''(''K''), for a local field ''K'' with integers ''O'' and residue field ''k'', is the inverse image in ''G''(''O'') of a Borel subgroup of ''G''(''k''). A reductive group over a local field has a Tits system (''B'',''N''), where ''B'' is a parahoric group, and the Weyl group of the Tits system is an affine Coxeter group. Definition More precisely, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke 代數
Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms. Biography Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He obtained his doctorate in Göttingen under the supervision of David Hilbert. Kurt Reidemeister and Heinrich Behnke were among his students. In 1933 Hecke signed the '' Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State''. Hecke died in Copenhagen, Denmark. André Weil, in the foreword to his text Basic Number Theory says: "To improve upon Hecke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task", referring to Hecke's book "Lectures on the Theory of Algebraic Numbers." Research His early work included establishing the functional equation for the Dedekind zeta function, with a proof based on theta functions. The method extended to the L-functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
