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Length Of A Weyl Group Element
In mathematics, the length of an element ''w'' in a Weyl group ''W'', denoted by ''l''(''w''), is the smallest number ''k'' so that ''w'' is a product of ''k'' reflections by simple roots. (So, the notion depends on the choice of a positive Weyl chamber.) In particular, a simple reflection has length one. The function ''l'' is then an integer-valued function of ''W''; it is a 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 satisfy ... of ''W''. It follows immediately from the definition that ''l''(''w''−1) = ''l''(''w'') and that ''l''(''ww'''−1) ≤ ''l''(''w'') + ''l''(''w' ''). References * {{cite book, last1=Kac, first1=Victor G., title=Infinite dimensional Lie algebras, date=1994, publisher=Cambridge University Press, location=Cambridge, isbn=9780521466936, ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl group W of \Phi is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Weyl Chamber
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the Root system of a semi-simple Lie algebra, root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner pro ... [...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). Properties A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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