Monk's Formula

	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, ... [...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]
	Schubert Polynomial In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by and are named after Hermann Schubert. Background described the history of Schubert polynomials. The Schubert polynomials \mathfrak_w are polynomials in the variables x_1,x_2,\ldots depending on an element w of the infinite symmetric group S_\infty of all permutations of \N fixing all but a finite number of elements. They form a basis for the polynomial ring \Z _1,x_2,\ldots/math> in infinitely many variables. The cohomology of the flag manifold \text(m) is \Z _1, x_2,\ldots, x_mI, where I is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial \mathfrak_w is the unique homogeneous polynomial of degree \ell(w) representing the Schubert cycle of w in the cohomology of the flag manifold \text(m) for all sufficiently large m. Properties If w_0 is the permutation ... [...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]
picture info	Cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Flag Manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Transposition (mathematics) In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of ''X''. If ''S'' has ''k'' elements, the cycle is called a ''k''-cycle. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For example, given ''X'' = , the permutation (1, 3, 2, 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 (so ''S'' = ''X'') is a 4-cycle, and the permutation (1, 3, 2) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 (so ''S'' = and 4 is a fixed element) is a 3-cycle. On the other hand, the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs and . The set ''S'' is called the orbit of the cycle. Every permutation on finitely many eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object. Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length square ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Bruhat Order In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties. History The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by , and the analogue for more general semisimple algebraic groups was studied by . started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat. The left and right weak Bruhat orderings were studied by . Definition If (''W'', ''S'') is a Coxeter system with generators ''S'', then the Bruhat order is a partial order on the group ''W''. Recall that a reduced word for an element ''w'' of ''W'' is a minimal length expression of ''w'' as a product of elements of ''S'', and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]