|
Alvis–Curtis Duality
In mathematics, the Alvis–Curtis duality is a duality operation on the characters of a reductive group over a finite field, introduced by and studied by his student . introduced a similar duality operation for Lie algebras. Alvis–Curtis duality has order 2 and is an isometry on generalized characters. discusses Alvis–Curtis duality in detail. Definition The dual ζ* of a character ζ of a finite group ''G'' with a split BN-pair is defined to be :\zeta^*=\sum_(-1)^\zeta^G_ Here the sum is over all subsets ''J'' of the set ''R'' of simple roots of the Coxeter system of ''G''. The character ζ is the truncation of ζ to the parabolic subgroup ''P''''J'' of the subset ''J'', given by restricting ζ to ''P''''J'' and then taking the space of invariants of the unipotent radical of ''P''''J'', and ζ is the induced representation of ''G''. (The operation of truncation is the adjoint functor of parabolic induction In mathematics, parabolic induction is a method of constructi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, ''linear algebra duality'' corresponds in this way to bilinear maps from pairs of vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the unit circle); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if \chi_1,\chi_2, \ldots , \chi_n are different cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductive Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BN-pair
In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems. Definition A (''B'', ''N'') pair is a pair of subgroups ''B'' and ''N'' of a group ''G'' such that the following axioms hold: * ''G'' is generated by ''B'' and ''N''. * The intersection, ''T'', of ''B'' and ''N'' is a normal subgroup of ''N''. *The group ''W'' = ''N/T'' is generated by a set ''S'' of elements of order 2 such that **If ''s'' is an element of ''S'' and ''w'' is an element of ''W'' then ''sBw'' is contained in the union of ''BswB'' and ''BwB''. **No element of ''S'' normalizes ''B''. The set ''S'' is uniquely determined by ''B'' and ''N'' and the pair (''W'',''S'') is a Coxeter sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Induction
In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If ''G'' is a reductive algebraic group and P=MAN is the Langlands decomposition of a parabolic subgroup ''P'', then parabolic induction consists of taking a representation of MA, extending it to ''P'' by letting ''N'' act trivially, and inducing the result from ''P'' to ''G''. There are some generalizations of parabolic induction using cohomology, such as cohomological parabolic induction and Deligne–Lusztig theory. Philosophy of cusp forms The ''philosophy of cusp forms'' was a slogan of Harish-Chandra, expressing his idea of a kind of reverse engineering of automorphic form theory, from the point of view of representation theory. The discrete group Γ fundamental to the classical theory disappears, superficially. What remains is the basic idea that representations in general are to be constructed by parabolic induction o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinberg Character
In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by ''St'', is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1. For groups over finite fields, these representations were introduced by , first for the general linear groups, then for classical groups, and then for all Chevalley groups, with a construction that immediately generalized to the other groups of Lie type that were discovered soon after by Steinberg, Suzuki and Ree. Over a finite field of characteristic ''p'', the Steinberg representation has degree equal to the largest power of ''p'' dividing the order of the group. The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation. , , and defined analogous Steinberg representations (somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuspidal Character
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term ''cuspidal'' is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups. When the group is the general linear group \operatorname_2, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform ( newform) corresponds to a cuspidal representation. Formulation Let ''G'' be a reductive algebraic group over a number field ''K'' and let A denote the adeles of ''K''. The group ''G''(''K'') embeds diagonally in the group ''G''(A) by sending ''g'' in ''G''(''K'') to the tuple (''g''''p'')''p'' in ''G''(A) with ''g'' = ''g''''p'' for all (finite and infinite) primes ''p''. Let ''Z'' d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
