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List Of Things Named After Erich Hecke
{{Short description, none These are things named after Erich Hecke, a German mathematician. * Hecke algebra ** Hecke algebra of a locally compact group **Hecke algebra of a finite group **Hecke algebra of a pair ** Hecke polynomial **Iwahori–Hecke algebra * Affine Hecke algebra ** Double affine Hecke algebra * Hecke algebra (other) * Hecke character * Hecke congruence subgroup * Hecke correspondence * Hecke eigenform * Hecke group * Hecke L-function (other) In mathematics, a Hecke ''L''-function may refer to: * an ''L''-function of a modular form * an ''L''-function of a Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to constr ... * Hecke operator * Hecke ring Hecke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erich Hecke
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] |
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] |
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] |
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] |
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] |
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] |
Double Affine Hecke Algebra
In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald polynomials. Infinitesimal Cherednik algebras have significant implications in representation theory, and therefore have important applications in particle physics and in chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions .... References * * *A. A. KirilloLectures on affine Hecke algebras and Macdonald's conjectures Bull. Amer. Math. Soc. 34 (1997), 251–292. * Macdonald, I. G. ''Affine Hecke algebras and orthogonal polynomials.'' Cambridge Tracts in Mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function. A name sometimes used for ''Hecke character'' is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.). Definition using ideles A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map. This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the unit circle in C ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible matrix, invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are ''even''. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer. The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite Index of a subgroup, index are essentially congruence subgroups. Congruence subgroups of 2×2 matrices are fundamental objects in the classical the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Correspondence
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations. History used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by . Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form, : \Delta(z)=q\left(\prod_^(1-q^n)\right)^= \sum_^ \tau(n)q^n, \quad q=e^, is a multiplicative function: : \tau(mn)=\tau(m)\tau(n) \quad \text (m,n)=1. The idea goes back to earlier work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators. Mathematical description Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer some func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |