In mathematics, the Bruhat order (also called the strong order, strong Bruhat order, Chevalley order, 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.
mathematics, the Bruhatorder (also called the strong order, strong Bruhatorder, Chevalley order, Bruhat–Chevalley order, or Chevalley–Bruhatorder) is a partial...
source of problems involving Young tableaux, plactic monoids, and the Bruhatorder. Subgroups of symmetric groups are called permutation groups and are...
weak Bruhatorder on S ~ n / S n {\displaystyle {\widetilde {S}}_{n}/S_{n}} is isomorphic to a certain subposet of Young's lattice. The Bruhatorder on...
bigrassmannian}}\}} where ≤ {\displaystyle \leq } denotes Bruhatorder in the symmetric groups. This graded partial order often appears in the context of Coxeter groups...
those are the only relations in W {\displaystyle W} . Weyl groups have a Bruhatorder and length function in terms of this presentation: the length of a Weyl...
Monomial order Weak order of permutations Bruhatorder on a Coxeter group Incidence algebra Monotonic Pointwise order of functions Galois connection Order embedding...
(right) weak order, the absolute order and the Bruhatorder (named for François Bruhat). An element v exceeds an element u in the Bruhatorder if some (or...
the word metric. These are also two resulting orders, the weak order and the Bruhatorder, and corresponding Hasse diagrams. An important example is in...
true of the Bruhatorder: if u and w are elements of W I {\displaystyle W_{I}} , then u ≤ w {\displaystyle u\leq w} in the Bruhatorder on W I {\displaystyle...
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{\displaystyle (T_{w})_{w\in W}} , the group is partially ordered by the Bruhatorder which is interval finite and has a dualization operation defined by T...
unique maximal element with respect to the Bruhatorder. The longest element is an involution (has order 2: w 0 − 1 = w 0 {\displaystyle w_{0}^{-1}=w_{0}}...
(product of reflections) and a longest element of a Coxeter group in the Bruhatorder; this is an analogy. It corresponds to and generalizes a volume form...
set, or equivalently its Cayley graph, ordered by the weak or strong Bruhatorder, and ranked by word length (length of shortest reduced word) In particular...
every w in W, M contains a unique minimal element with respect to the w-Bruhatorder. Suppose that the Coxeter group W is the symmetric group Sn and P is...
W_{\lambda }} if and only if w ≤ w ′ {\displaystyle w\leq w'} in the Bruhatordering of the Weyl group. Let 0 ⊂ A ⊂ B ⊂ W λ {\displaystyle 0\subset A\subset...
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inclusion of their closures, is Eulerian. Let W be a Coxeter group with Bruhatorder. Then (W,≤) is an Eulerian poset. The defining condition of an Eulerian...
using the inner product α(v) = (α,v). The choice of Δ gives rise to a Bruhatorder on the Weyl group determined by the ways of writing elements minimally...
=w\cdot {\tilde {\lambda }}} and w ≤ w ′ {\displaystyle w\leq w'} in the Bruhatordering (otherwise, there is no homomorphism of (true) Verma modules M μ →...
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