|
Longest Element Of A Coxeter Group
In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by ''w''0. See and . Properties * A Coxeter group has a longest element if and only if it is finite; "only if" is because the size of the group is bounded by the number of words of length less than or equal to the maximum. * The longest element of a Coxeter group is the unique maximal element with respect to the Bruhat order. * The longest element is an involution (has order 2: w_0^ = w_0), by uniqueness of maximal length (the inverse of an element has the same length as the element). * For any w \in W, the length satisfies \ell(w_0w) = \ell(w_0) - \ell(w). * A reduced expression for the longest element is not in general unique. * In a reduced expression for the longest element, every simple reflection must occur at least once. * If the Coxeter group is finite t ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Length Function
In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group. Definition A length function ''L'' : ''G'' → R+ on a group ''G'' is a function satisfying: :\beginL(e) &= 0,\\ L(g^) &= L(g)\\ L(g_1 g_2) &\leq L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G. \end Compare with the axioms for a metric and a filtered algebra. Word metric An important example of a length is the word metric: given a presentation of a group by generators and relations, the length of an element is the length of the shortest word expressing it. Coxeter groups (including the symmetric group) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: length of a Weyl group element. A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections). Prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite 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. Standard re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
