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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] |