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Littlewood–Richardson Rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say \lambda,\mu,\nu, of which \lambda and \mu describe the Schur functions being multiplied, and \nu gives the Schur function of which this is the coefficient in the linear combination; in other words they ... [...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] |
Young Tableau
In mathematics, a Young tableau (; plural: tableaux) 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. Definitions ''Note: this article uses the English convention for displaying Young diagrams and tableaux''. Diagrams A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pieri's Formula
In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function. In terms of Schur functions ''s''λ indexed by partitions λ, it states that :\displaystyle s_\mu h_r=\sum_\lambda s_\lambda where ''h''''r'' is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding ''r'' elements, no two in the same column. By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule for multiplying an elementary symmetric polynomial with a Schur polynomial: :\displaystyle s_\mu e_r=\sum_\lambda s_\lambda The sum is now taken over all partitions λ obtained from μ by adding ''r'' elements, no two in the same ''row''. Pieri's formula implies Giambelli's formula. The Littlewood–Richardson rule In mathematics, the Lit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picture (mathematics)
In combinatorics, combinatorial mathematics, a picture is a bijection between Young tableau#Skew tableaux, skew diagrams satisfying certain properties, introduced by in a generalization of the Robinson–Schensted correspondence and the Littlewood–Richardson rule. References * *{{Citation , authorlink=Andrei Zelevinsky , last1=Zelevinsky , first1=A. V. , title=A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence , doi=10.1016/0021-8693(81)90128-9 , mr=613858 , year=1981 , journal=Journal of Algebra , issn=0021-8693 , volume=69 , issue=1 , pages=82–94, doi-access=free Algebraic combinatorics Combinatorial algorithms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeu De Taquin
In the mathematical field of combinatorics, jeu de taquin is a construction due to which defines an equivalence relation on the set of skew standard Young tableaux. A jeu de taquin slide is a transformation where the numbers in a tableau are moved around in a way similar to how the pieces in the fifteen puzzle move. Two tableaux are jeu de taquin equivalent if one can be transformed into the other via a sequence of such slides. "Jeu de taquin" (literally "teasing game") is the French name for the fifteen puzzle. Definition of a jeu de taquin slide Given a skew standard Young tableau ''T'' of skew shape \lambda / \mu, pick an adjacent empty cell ''c'' that can be added to the skew diagram \lambda\setminus\mu; what this means is that ''c'' must share at least one edge with some cell in ''T'', and \ \cup \lambda\setminus\mu must also be a skew diagram. There are two kinds of slide, depending on whether ''c'' lies to the upper left of ''T'' or to the lower right. Suppose to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Module
Schur is a German or Jewish surname. Notable people with the surname include: * Alexander Schur (born 1971), German footballer * Dina Feitelson-Schur (1926–1992), Israeli educator * Friedrich Schur (1856-1932), German mathematician * Fritz Schur (born 1951), Danish businessman * Issai Schur (1875–1941), Lithuanian-German-Israeli mathematician * Max Schur (1897–1969), Austrian physician * Michael Schur (born 1975), American television producer and writer * Philipp Johann Ferdinand Schur Philipp Johann Ferdinand Schur (February 18, 1799 – May 27, 1878) was a German-Austrian pharmacist and botanist born in Königsberg. He obtained his education in Königsberg and Berlin, later serving as a director of a chemical factory in Liesin ..., German-Austrian botanist, 1799-1878 * Täve Schur (born 1939), German cyclist See also * Schor (other) {{surname, Schur Surnames from nicknames German-language surnames Jewish surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Reciprocity
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is named for Ferdinand Georg Frobenius, the inventor of the representation theory of finite groups. Statement Character theory The theorem was originally stated in terms of character theory. Let be a finite group with a subgroup , let \operatorname_H^G denote the restriction of a character, or more generally, class function of to , and let \operatorname_H^G denote the induced class function of a given class function on . For any finite group , there is an inner product \langle -,-\rangle_A on the vector space of class functions A\to\mathbb (described in detail in the article Schur orthogonality relations). Now, for any class functions \psi:H\to\mathbb and \varph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Variety
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces ''V'', specified using linear algebra, inside a fixed vector subspace ''W''. Here ''W'' may be a vector space over an arbitrary field, though most commonly over the complex numbers. A typical example is the set ''X'' whose points correspond to those 2-dimensional subspaces ''V'' of a 4-dimensional vector space ''W'', such that ''V'' non-trivially intersects a fixed (reference) 2-dimensional subspace ''W''2: :X \ =\ \. Over the real number field, this can be pictured in usual ''xyz''-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of \mathbb(W), we obtain an open subset ''X''° ⊂ ''X''. This is isomorphic to the set of all lines ''L'' (not necessarily through the origin) which m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact space, compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Schur Function
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions a_ (x_ ... [...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 formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Search Tree
In computer science, a search tree is a tree data structure used for locating specific keys from within a set. In order for a tree to function as a search tree, the key for each node must be greater than any keys in subtrees on the left, and less than any keys in subtrees on the right. The advantage of search trees is their efficient search time given the tree is reasonably balanced, which is to say the leaves at either end are of comparable depths. Various search-tree data structures exist, several of which also allow efficient insertion and deletion of elements, which operations then have to maintain tree balance. Search trees are often used to implement an associative array. The search tree algorithm uses the key from the key–value pair to find a location, and then the application stores the entire key–value pair at that particular location. Types of Trees Binary search tree A Binary Search Tree is a node-based data structure where each node contains a key and two sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
