Brigadier General William D Razz Waff (Commanding General, 99th RSC), speaks after he was promoted to a two star Major General, during Promotion Ceremony at Major John P Pryor Army Reserve Center, at Joint Base McGuire-Dix-Lakehurst.
Brigadier General William D Razz Waff (Commanding General, 99th RSC), speaks after he was promoted to a two star Major General, during Promotion Ceremony at Major John P Pryor Army Reserve Center, at Joint Base McGuire-Dix-Lakehurst.
WAFF-TV: News, Weather and Sports for Huntsville, AL.
WAFF -TV: News, Weather and Sports for Huntsville, ALMarch 16, 2011.
WAFF -TV, Raycom's NBC affiliate serving Huntsville and Decatur will launch local news in high-def come October, according to the station.
Vanessa Oubre, WAFF vice president and general manager said the station is the first in the market to do the news in HD.
WAFF -TV: News, Weather and Sports for Huntsville, ALHow finding the right challenge can keep you focused on your weight loss.
WAFF-TV reports the 6-month-old black Lab, Mitzy, was shot Tuesday night and burned in a pile of debris at a Limestone County home.
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We have the semi-direct product Waff = W (cid:110) ˇQ acting on t, with ˇQ acting as translations and the Weyl group as point symmetries.
Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
Waff is called the aﬃne Weyl group. A fundamental region F for t under the action of Waff can be given as follows.
Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
There is a similar length function for Waff The action of Waff on F tiles the entire space t with copies of F, and in this way F serves as a fundamental domain for it.
Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
In particular, all x ∈ F ◦ have trivial stabilizer in Waff .
Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
Although each element of W and each element of Waff is a product of reﬂections, it is not uniquely so.
Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
Note that none of the reﬂecting hyperplanes (5) or (6), nor indeed any reﬂecting hyperplane of Waff, meets the interior F ◦ of F and so Sρ (x) is never 0 in F ◦ .
Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups
Lemma 3.1. [Hum] The aﬃne Weyl group Waff is the semidirect product W ⋉ L∨ of the usual Weyl group W and the coroot lattice L∨ .
Alcoved Polytopes II
Lemma 3.2. [Hum] The aﬃne Weyl group Waff acts simply transitively on the col lection A of al l alcoves.
Alcoved Polytopes II
Lemma 3.2 implies that all alcoves are obtained from A◦ by the action of Waff .
Alcoved Polytopes II
The closure of the fundamental alcove A◦ is a fundamental domain of Waff .
Alcoved Polytopes II
Condition 2 follows immediately from translating the action of the simple generators of Waff on alcoves.
Alcoved Polytopes II
Let us deﬁne the equivalence relation “∼” on the aﬃne Weyl group Waff by u ∼ w if and only if u(A◦ ) = w(A◦ ) + λ, for some λ ∈ Λ∨, where u, w ∈ Waff .
Alcoved Polytopes II
Actually, C is a subgroup in W and Caff is a subgroup in Waff .
Alcoved Polytopes II
Equivalence classes of elements of the Weyl group (respectively, the aﬃne Weyl group) with respect to the relation “∼” are exactly cosets in W/C (respectively, Waff /Caff ).
Alcoved Polytopes II
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