S-module

	S-module In algebraic topology, an \mathbb-object (also called a symmetric sequence) is a sequence \ of objects such that each X(n) comes with an actionAn action of a group ''G'' on an object ''X'' in a category ''C'' is a functor from ''G'' viewed as a category with a single object to ''C'' that maps the single object to ''X''. Note this functor then induces a group homomorphism G \to \operatorname(X); cf. Automorphism group#In category theory. of the symmetric group \mathbb_n. The category of combinatorial species is equivalent to the category of finite \mathbb-sets (roughly because the permutation category is equivalent to the category of finite sets and bijections.) S-module By ''\mathbb-module'', we mean an \mathbb-object in the category \mathsf of finite-dimensional vector spaces over a field ''k'' of characteristic zero (the symmetric groups act from the right by convention). Then each \mathbb-module determines a Schur functor on \mathsf. This definition of \mathbb-module share ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Highly Structured Ring Spectrum In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the .... Background Highly structured ring spectra have better formal properties than multiplicative cohomology theories – a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory. Beside their formal properties, E_\infty-structures are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Algebraic Topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Automorphism Group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Symmetric Group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Combinatorial Species In combinatorics, combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for deriving the generating functions of discrete structures, which allows one to not merely count these structures but give bijective proof, bijective proofs involving them. Examples of combinatorial species are (wiktionary:finite, finite) Graph (discrete mathematics), graphs, permutations, Tree (graph theory), trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size. One goal of species theory is to be able to analyse complicated structures by describing them in terms of transformations and combinations of simpler structures. These operations correspond to equivalent manipulations of generating functions, so producing such functions for complicated structures is much easier than with other methods. The theory was introduced, carefully elaborated and applied by Canadian researchers around André ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]