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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] |
Mario Pieri
Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore Pincherle. Obtaining a scholarship, Pieri transferred to ''Scuola Normale Superiore'' in Pisa. There he took his degree in 1884 and worked first at a technical secondary school in Pisa. When an opportunity arose at the military academy in Turin to teach projective geometry, Pieri moved there and, by 1888, he was also an assistant instructor in the same subject at the University of Turin. By 1891, he had become ''libero docente'' at the university, teaching elective courses. Pieri continued to teach in Turin until 1900 when, through competition, he was awarded the position of ''extraordinary professor'' at University of Catania on the island of S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Cycle
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schubert Calculus
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties. Even more generally, "Schubert calculus" is often understood to encompass the study of analogous questions in generalized cohomology theories. The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomial
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_1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition (number Theory)
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. Examples The seven partitions of 5 are: * 5 * 4 + 1 * 3 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Homogeneous Symmetric Polynomial
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials. Definition The complete homogeneous symmetric polynomial of degree in variables , written for , is the sum of all monomials of total degree in the variables. Formally, :h_k (X_1, X_2, \dots,X_n) = \sum_ X_ X_ \cdots X_. The formula can also be written as: :h_k (X_1, X_2, \dots,X_n) = \sum_ X_^ X_^ \cdots X_^. Indeed, is just the multiplicity of in the sequence . The first few of these polynomials are :\begin h_0 (X_1, X_2, \dots,X_n) &= 1, \\0pxh_1 (X_1, X_2, \dots,X_n) &= \sum_ X_j, \\ h_2 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k, \\ h_3 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k X_l. \end Thus, for each nonnegative integer , there exists exactly one complete homogeneous symmetric polynomi ... [...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_j,\\ e_2 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k,\\ e_3 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k X_l,\\ \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 )=\sum_ X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giambelli's Formula
In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes in terms of special Schubert classes, or Schur functions in terms of complete symmetric functions. It states :\displaystyle \sigma_\lambda= \det(\sigma_)_ where σλ is the Schubert class of a partition λ. Giambelli's formula is a consequence of Pieri's formula. The Porteous formula is a generalization to morphisms of vector bundles over a variety. See also * Schubert calculus In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of ... - includes examples References * * Symmetric functions {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Monk's Formula
In mathematics, Monk's formula, found by , is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold. Write ''t''ij for the transposition ''(i j)'', and ''s''i = ''t''i,i+1. Then 𝔖sr = ''x''1 + ⋯ + ''x''r, and Monk's formula states that for a permutation ''w'', \mathfrak_ \mathfrak_w = \sum_ \mathfrak_, where \ell(w) is the length of ''w''. The pairs (''i'', ''j'') appearing in the sum are exactly those such that ''i'' ≤ ''r'' < ''j'', ''w''i < ''w''j, and there is no ''i'' < ''k'' < ''j'' with ''w''i < ''w''k < ''w''j; each ''wt''ij is a cover of ''w'' in Bruhat order In mathematics, ...
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