In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.
In mathematics, a finite field or Galois field 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 p when p is a prime number.
An irreducible fraction is a fraction in which the numerator and denominator are integers that have no other common divisors than 1. In other words, a fraction a⁄b is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every positive rational number can be represented as an irreducible fraction in exactly one way.
In mathematics, a unique factorization domain (UFD) is an integral domain in which every non-zero non-unit element can be written as a product of prime elements, uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.
In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as
if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals.
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group
or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.
In mathematics, a Young tableau is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley.
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles.
In mathematics, and specifically in algebraic geometry, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation
Arthur Cohn's irreducibility criterion is a sufficient condition for a polynomial to be irreducible in
—that is, for it to be unfactorable into the product of lower-degree polynomials with integer coefficients.
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For example,
is absolutely irreducible, but while
is irreducible over the integers and the reals, it is reducible over the complex numbers as
and thus not absolutely irreducible.
This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups. It also covers terms related to their Lie algebras, their representation theory, and various geometric, algebraic and combinatorial structures that occur in connection with the development of what is a central theory of contemporary mathematics.
David Alexander Vogan, Jr. is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups.
In mathematics, the Langlands classification is a description of the irreducible representations of a reductive Lie group G, suggested by Robert Langlands (1973). There are two slightly different versions of the Langlands classification. One of these describes the irreducible admissible (g,K)-modules,
for g a Lie algebra of a reductive Lie group G, with maximal compact subgroup K, in terms of tempered representations of smaller groups. The tempered representations were in turn classified by Anthony Knapp and Gregg Zuckerman. The other version of the Langlands classification divides the irreducible representations into L-packets, and classifies the L-packets in terms of certain homomorphisms of the Weil group of R or C into the Langlands dual group.
In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space
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 the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these 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 is matrix multiplication.
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allow one to make general conclusions about representations of a finite group G without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.
In representation theory, a branch of mathematics, the theorem of the highest weight states that the irreducible representations of semisimple Lie algebras or compact Lie groups are classified by their highest weights: given a simply-connected compact Lie group G with Lie algebra
, there is a bijection
