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θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field. introduced θ10 for the symplectic group Sp4(F''q'') over a finite field F''q'' of order ''q'', and showed that in this case it is ''q''(''q'' – 1)2/2-dimensional. The subscript 10 in θ10 is a historical accident that has stuck: Srinivasan arbitrarily named some of the characters of Sp4(F''q'') as θ1, θ2, ..., θ13, and the tenth one in her list happens to be the cuspidal unipotent character. θ10 is the only cuspidal unipotent representation of Sp4(F''q''). It is the simplest example of a cuspidal unipotent representation of a reductive group, and also the simplest example of a degenerate cuspidal representation (one without a Whittaker model). General linear groups have no cuspidal unipotent representations and no degenerate cuspidal representations, so θ10 exhibits properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuspidal Unipotent Representation
In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a Langlands dual group, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group. Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense. Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction. Finite fields Over finite fields, the unipotent representations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whittaker Model
In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named after E. T. Whittaker even though he never worked in this area, because pointed out that for the group SL2(R) some of the functions involved in the representation are Whittaker functions. Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation ''θ''10 of the symplectic group Sp4 is the simplest example of a degenerate representation. Whittaker models for GL2 If ''G'' is the algebraic group ''GL''2 and F is a local field, and is a fixed non-trivial character of the additive group of F and is an irreducible representation of a general linear group ''G''(F), then the Whittaker model for is a representation on a space of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name " symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The ... [...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 (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), 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 the integers mod n, integers mod p when p 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 p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, 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 that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact topological field with respect to a non-discrete topology. The real numbers R, and the complex numbers C (with their standard topologies) are Archimedean local fields. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Field
In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function field: The function field of an irreducible algebraic curve over a finite field, equivalently, a finite extension of \mathbb_q(T), the field of rational functions in one variable over the finite field with q=p^n elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s. Formal definitions A ''global field'' is one of the following: ;An algebraic number field An algebraic number field ''F'' is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus ''F'' is a field that contains Q and has finite dimension when considered as a vector space over Q. ;The function field of an irreducible algebraic curve over a finite field A fun ... [...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 that has a finite kernel and 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Ramanujan Conjecture
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig (mathematician), Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
