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Descent Algebra
In algebra, Solomon's descent algebra of a Coxeter group is a subalgebra of the integral group ring of the Coxeter group, introduced by . The descent algebra of the symmetric group In the special case of the symmetric group ''S''''n'', the descent algebra is given by the elements of the group ring such that permutations with the same descent set have the same coefficients. (The descent set of a permutation σ consists of the indices ''i'' such that σ(''i'') > σ(''i''+1).) The descent algebra of the symmetric group ''S''''n'' has dimension 2''n-1''. It contains the peak algebra In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group ''S'n'', studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutati ... as a left ideal. References * Reflection groups {{group-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of '' ... [...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 represen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peak Algebra
In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group ''S''''n'', studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on is an index ''i'' such that σ(''i''–1)σ(''i''+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all ''n'' has a natural structure of a Hopf algebra Hopf is a German surname. Notable people with the surname include: * Eberhard Hopf (1902–1983), Austrian mathematician * Hans Hopf (1916–1993), German tenor * Heinz Hopf (1894–1971), German mathematician * Heinz Hopf (actor) (1934–2001), Sw .... References *{{citation, mr=2001673 , last=Nyman, first= Kathryn L. , title=The peak algebra of the symmetric group , journal=J. Algebraic Combin., volume= 17 , year=2003, issue= 3, pages= 309–322 , doi=10.1023/A:1025000905826, doi-access= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left Ideal
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
