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Λ-ring
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ''n'' on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (). λ-rings were introduced by . For more about λ-rings see , , and . Motivation If ''V'' and ''W'' are finite-dimensional vector spaces over a field ''k'', then we can form the direct sum ''V'' ⊕ ''W'', the tensor product ''V'' ⊗ ''W'', and the ''n''-th exterior power of ''V'', Λ''n''(''V''). All of these are again finite-dimensional vector spaces over ''k''. The same three operations of direct sum, tensor product and exterior power are also available when working with ''k''-linear representations of a finite group, when working with vector bundles over some topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Symmetric Functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number ''n'' of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative. Symmetric polynomials The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More forma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Exact Sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics), image of one morphism equals the kernel (algebra), kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Symmetric Polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree in variables for each positive integer , and it is formed by adding together all distinct products of distinct variables. Definition The elementary symmetric polynomials in variables , written for , are defined by :\begin e_1 (X_1, X_2, \dots, X_n) &= \sum_ X_a,\\ e_2 (X_1, X_2, \dots, X_n) &= \sum_ X_a X_b,\\ e_3 (X_1, X_2, \dots, X_n) &= \sum_ X_a X_b X_c,\\ \end and so forth, ending with : e_n (X_1, X_2, \dots,X_n) = X_1 X_2 \cdots X_n. In general, for we define : e_k (X_1 , \ldots , X_n )=\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Ring
In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representations of the group. Elements of the representation ring are sometimes called virtual representations. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed fields of characteristic ''p'' where the Sylow ''p''-subgroups are cyclic is also theoretically approachable. Formal definition Given a group ''G'' and a field ''F'', the elements of its representation ring ''R''''F''(''G'') are the formal differences of isomorphism classes of finite-dimensional ''F''-representations of ''G''. For the ring structure, addition is given by the direct sum of representations, and multiplication by their tensor product over ''F''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological K-theory
In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological -theory is due to Michael Atiyah and Friedrich Hirzebruch. Definitions Let be a compact Hausdorff space and k= \R or \Complex. Then K_k(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X). The remaining discussion is focused on complex -theory. As a first example, note that the -theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the nat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams Operation
In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in symmetric function theory, at the level of vector bundles or other representing object in more abstract theories. Adams operations can be defined more generally in any λ-ring. Adams operations in K-theory Adams operations ψ''k'' on K theory (algebraic or topological) are characterized by the following properties. # ψ''k'' are ring homomorphisms. # ψ''k''(l)= lk if l is the class of a line bundle. # ψ''k'' are functorial. The fundamental idea is that for a vector bundle ''V'' on a topological space ''X'', there is an analogy between Adams operators and exterior powers, in which :ψ''k''(''V'') is to Λ''k''(''V'') as :the power sum Σ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Ring
In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...s :\binom = \frac for ''x'' in the ring and ''n'' a positive integer. Binomial rings were introduced by . showed that binomial rings are essentially the same as λ-rings for which all Adams operations are the identity. References * * *{{citation, mr=2649360 , last=Yau, first= Donald , title=Lambda-rings, publisher= World Scientific Publishing Co. Pte. Ltd., place= Hackensack, NJ, year= 2010 , isbn= 978-981-4299-09-1 , url=https://books.google.com/books?id=d7vKnjxyvxQC Ring theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivation
Motivation is an mental state, internal state that propels individuals to engage in goal-directed behavior. It is often understood as a force that explains why people or animals initiate, continue, or terminate a certain behavior at a particular time. It is a complex phenomenon and its precise definition is disputed. It contrasts with #Amotivation and akrasia, amotivation, which is a state of apathy or listlessness. Motivation is studied in fields like psychology, neuroscience, motivation science, and philosophy. Motivational states are characterized by their direction, Motivational intensity, intensity, and persistence. The direction of a motivational state is shaped by the goal it aims to achieve. Intensity is the strength of the state and affects whether the state is translated into action and how much effort is employed. Persistence refers to how long an individual is willing to engage in an activity. Motivation is often divided into two phases: in the first phase, the indi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Symmetric Polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree in variables for each positive integer , and it is formed by adding together all distinct products of distinct variables. Definition The elementary symmetric polynomials in variables , written for , are defined by :\begin e_1 (X_1, X_2, \dots, X_n) &= \sum_ X_a,\\ e_2 (X_1, X_2, \dots, X_n) &= \sum_ X_a X_b,\\ e_3 (X_1, X_2, \dots, X_n) &= \sum_ X_a X_b X_c,\\ \end and so forth, ending with : e_n (X_1, X_2, \dots,X_n) = X_1 X_2 \cdots X_n. In general, for we define : e_k (X_1 , \ldots , X_n )=\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
