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Topic: Borel subgroup

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	Armand Borel - Wikipedia, the free encyclopedia
		Armand Borel (21 May 1923 - 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study, Princeton from 1957 to 1993.
		In an algebraic group G a Borel subgroup H is one such that the homogeneous space G/H is a projective variety, and as small as possible.
		In this case it turns out that H is a maximal solvable subgroup, and that the parabolic subgroups P between H and G have a combinatorial structure (in this case the G/P are the various flag manifolds).
	en.wikipedia.org /wiki/Armand_Borel (291 words)

	Borel subgroup -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-07)
		Subgroups between a Borel subgroup B and G (inclusive) are called parabolic subgroups.
		Another characterisation of a parabolic subgroup P is that G/P is a (Click link for more info and facts about complete variety) complete variety (a posteriori, a (Click link for more info and facts about projective variety) projective variety).
		Therefore another way to distinguish the Borel subgroups is as minimal parabolic subgroups: B is a Borel subgroup precisely when G/B is a (Click link for more info and facts about homogeneous space) homogeneous space for G and a projective variety, and "as large as possible".
	www.absoluteastronomy.com /encyclopedia/b/bo/borel_subgroup.htm (253 words)

	[No title]
		In a geometry with symmetry group G, different types of figure correspond to different *subgroups* of G. The idea is that for each type of figure, there is a space X of all figures of that type, upon which G acts.
		First, every complex simple Lie group G has a bunch of maximal compact subgroups, all of which are isomorphic via conjugation inside G. People often pick one, call it "the" maximal compact subgroup, and denote it by K.
		The smallest parabolic subgroup is B itself, and G/B is the space of "maximal flags".
	math.ucr.edu /home/baez/twf_ascii/week178 (3254 words)

	[No title]
		The normalizer sharpness result improves a result of Webb [* 67] that the collection of p-radical subgroups* is normalizer sharp and verifies a suspic* ion of Smith- Yoshiara [59] that the collection of p-radical and p-centric subgroups* should b* *e normalizer sharp.
		Choose a maximal subgroup H* * in C on which F is nonzero, and let F 0denote the subfunctor of F obtained by setting F* * 0(H0) = 0 if H0 is conjugate to H and F 0(H0) = F (H0) otherwise.
		Hence Sp(G) is subgroup acyclic with respect to H0(-; M) iff it is ce* ntralizer acyclic iff it is normalizer acyclic, iff Hom G(St(G); M) is an acyclic cochai* *n complex.
	hopf.math.purdue.edu /Grodal/limsub.txt (11451 words)

	Open-problems
		Is there a Borel bijective map from the set of countable reduced 2-groups to the set of
		When does such a Borel group have a Borel generating tree (or a generating tree at all)?
		Does every Borel subgroup of a rank 1 Borel 2-group with a Borel basis have a Borel basis?
	www.grossmont.net /carylee/open-pro.htm (225 words)

	Abstracts (Representations of Finite Groups and Related Algebras)
		In this talk, new results are presented about one of these ambiguities, which have been obtained using techniques similar to those introduced by T. Okuyama and K. Waki in their determination of the decomposition numbers of the symplectic groups Sp(4,q) in 1996 and 1998.
		The Hecke algebra H of a Coxeter group W has certain ideals associated with the longest elements of the parabolic subgroups of W which allow a straightforward analysis of the corresponding cell representations.
		We consider a conjecture on the alternating sum of the numbers of complex irreducible characters in certain blocks of chain normalizers having fixed defect and ± fixed residue mod p.
	www.stats.bris.ac.uk /~majcr/durham/abstracts.html (1561 words)

	Elementary Invariants (Site not responding. Last check: 2007-09-07)
		the subgroup of the Borel subgroup of C that fixes all elements of the geometry C. Kernels(C) : CosetGeom -> SeqEnum
		Given a coset geometry C, return a sequence containing the i-kernel K_i of each maximal parabolic subgroup G_i of C. The i-kernel of the subgroup G_i is the subgroup consisting of all the elements of G_i that fix all the elements of the residue of G_i.
		Given a coset geometry C = (G; (G_i)_(i in I)) and a permutation group K, return the coset geometry (G/K; (G_i/K)_(i in I)) provided that K is a normal subgroup of G and of all the maximal parabolic subgroups of C. [Next][Prev] [Right] [Left] [Up] [Index] [Root]
	www.math.lsu.edu /magma/text1465.htm (338 words)

	On Orbit Closures of Symmetric Subgroups in Flag Varieties - Brion, Helminck (ResearchIndex) (Site not responding. Last check: 2007-09-07)
		Abstract: Introduction Let G be a connected reductive group over an algebraically closed field k; let B ` G be a Borel subgroup and K ` G a closed subgroup.
		Assume that K is a spherical subgroup of G, that is, the number of K-orbits in the flag variety G=B is finite; equivalently, the set KnG=B of (K; B)-double cosets in G is finite.
		Brion and A. Helminck, On orbit closures of symmetric subgroups in flag varieties, Canad.
	citeseer.ist.psu.edu /211113.html (431 words)

	week178
		Third, G always has a bunch of maximal solvable subgroups, which again are all isomorphic by conjugation inside G. In case you forgot: a group B is "solvable" if when you take the subgroup B1 generated by commutators
		A maximal solvable subgroup of G is also called a "Borel" subgroup, and it's denoted B. When G = SL(n,C), an obvious choice for B is the group of upper triangular matrices with determinant 1:
		As you can see, each subgroup preserves some sort of "flag": a something on a something on a something, etc. The smaller the subgroup, the bigger the flag.
	math.ucr.edu /home/baez/week178.html (3470 words)

	AGT 4 (2004) Paper 32 (Abstract)
		We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold.
		A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.
		Borel subgroup, cusp cross-section, hyperbolic space, nil manifold, subgroup separability.
	www.maths.warwick.ac.uk /agt/AGTVol4/agt-4-32.abs.html (128 words)

	On the Set of Orbits for a Borel Subgroup (Site not responding. Last check: 2007-09-07)
		On the Set of Orbits for a Borel Subgroup
		Let G be a connected reductive group with Borel subgroup B and let X be a G-variety.
		We study the set of B-orbits in X. In case X=G/H is homogeneous, this is the same as to study H-orbits in the flag variety G/B. Assume for simplicity that B has an open orbit in X. First, we give a new proof that then X contains in fact only finitely many B-orbits.
	www.math.rutgers.edu /~knop/papers/BS.html (134 words)

	The Behaviour At Infinity Of The Bruhat Decomposition - Brion (ResearchIndex) (Site not responding. Last check: 2007-09-07)
		For a connected reductive group G and a Borel subgroup B, we study the closures of double classes BgB in a (G \Theta G)-equivariant "regular" compactification of G. We show that these closures BgB intersect properly all (G \Theta G)-orbits, with multiplicity one, and we describe the intersections.
		Moreover, we show that almost all BgB are singular in codimension two exactly.
		0.8: On Orbit Closures of Borel Subgroups in Spherical Varieties - Brion
	citeseer.ist.psu.edu /brion98behaviour.html (573 words)

	Amazon.com: Books: Lie Theory and Geometry : In Honor of Bertram Kostant (Progress in Mathematics) (Site not responding. Last check: 2007-09-07)
		by Jean-Luc Brylinski (Editor), Ranee Brylinski (Editor), Victor Guillemin (Editor), Victor Kac (Editor) "Let G be a connected, complex reductive group and B, a Borel subgroup..." (more)
		Let G be a connected, complex reductive group and B, a Borel subgroup.
		Algebraic Groups and Lie Groups : A Volume of Papers in Honour of the Late R. Richardson (Australian Mathematical Society Lecture Series) by G. Lehrer on page 218, and page 295
	www.amazon.com /exec/obidos/tg/detail/-/0817637613?v=glance (464 words)

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