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Topic: Coxeter matrix

	PlanetMath: Coxeter group
		Alternative methods to study Coxeter groups is through the use of a length measurement on elements in the group.
		Coxeter groups arrise as the Weyl groups of Lie algebra, Lie groups, and groups of with a BN-pair.
		This is version 9 of Coxeter group, born on 2006-01-21, modified 2007-04-12.
	www.planetmath.org /encyclopedia/CoxeterGroup.html (504 words)

	Dictionary of Combinatorics -- C
		A diagram used to visualize a Coxeter group, it is a labeled graph with nodes indexed by the generators of a Coxeter group and (P
		is an entry in the Coxeter matrix corresponding to the given Coxeter group.
		A Coxeter matrix of rank n is an n x n matrix M with M
	www.southernct.edu /~fields/comb_dic/C.html (420 words)

	Finite and Affine Coxeter Groups
		A Coxeter group is called affine if it is infinite and it has a representation as a discrete, properly acting, affine reflection group (see [Bou68] for more details on discreteness and proper action).
		Print the Dynkin diagramof a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name given by the string N. If the corresponding group is neither affine nor crystallographic, an error is flagged.
		Print the Coxeter diagramof a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name given by the string N. If the corresponding group is not affine or is not crystallographic, an error is flagged.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1023.htm (1261 words)

	Weyl Groups
		A Cartan matrix is: 2 0 -1 0 2 -1 -1 -1 2 The Coxeter-Dynkin diagram is
		A Cartan matrix is: 2 0 -1 0 0 0 2 -1 0 0 -1 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 The Coxeter-Dynkin diagram is
		A Cartan matrix is: 2 0 -1 0 0 0 0 2 0 -1 0 0 -1 0 2 -1 0 0 0 -1 -1 2 -1 0 0 0 0 -1 2 -1 0 0 0 0 -1 2 The Coxeter-Dynkin diagram is
	www.valdostamuseum.org /hamsmith/Weyl.html (5287 words)

	List of matrices - Wikipedia, the free encyclopedia
		Companion matrix — a matrix whose eigenvalues are equal to the roots of the polynomial.
		Permutation matrix — a matrix representation of a permutation.
		Toeplitz matrix — a matrix with constant diagonals.
	en.wikipedia.org /wiki/List_of_matrices (2368 words)

	Root Systems
		This notation was suggested by Coxeter and used by Cohen in his thesis on the classification of the groups.
		The (i, j)-th entry of the root system matrix for the roots a_1, a_2,..., a_k is delta_(ij) + (alpha_j - 1)(a_i, a_j), where alpha_j is an m-th root of unity, for some m.
		The k-th reflection matrix is obtained from the identity matrix by replacing its k-th column with the k-th column of the root system matrix.
	www.umich.edu /~gpcc/scs/magma/text514.htm (828 words)

	Weyl Groups
		A Cartan matrix is: 2 The Coxeter-Dynkin diagram is
		A Cartan matrix is: 2 -1 -1 2 The Coxeter-Dynkin diagram is
		The polytope corresponding to the A2 Lie algebra by The McKay Correspondence is a triangle
	valdostamuseum.org /hamsmith/Weyl.html (5287 words)

	Finitely presented Coxeter groups
		Coxeter groups can be described in a standard manner as finitely presented groups.
		A reduced word for w in the FP Coxeter group F. The word w may be given either as an element of F or as a sequence of integers.
		The braid group B of the FP Coxeter group F as a finitely presented group, together with the projection B to F. The braid group is the group with the same generators and braid relations as F, but with no order relations.
	www.math.niu.edu /help/math/magmahelp/text477.html (715 words)

	Finite and Affine Coxeter Groups
		A Coxeter group is finite if, and only if, all its irreducible components are finite; a Coxeter group is affine if, and only if all its irreducible components are finite or affine, and at least one component is affine.
		Print the Dynkin diagram of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name N. If the corresponding group is not affine or is not crystallographic, an error is flagged.
		Print the Coxeter diagram of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name N. If the corresponding group is not affine or is not crystallographic, an error is flagged.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1013.htm (1093 words)

	[No title]
		The 1-by-1 matrix that you get at the end is the answer (that is, the determinant of the original matrix).
		This is not a polynomial in the entries of the matrix, but rather a Laurent polynomial, where the coefficients are polynomials in lambda.
		Here's a kludge we can use: replace the matrix by a t^0 t^1 t^3 t^6 t^0 b t^0 t^1 t^3 t^1 t^0 c t^0 t^1 t^3 t^1 t^0 d t^0 t^6 t^3 t^1 t^0 e (where the exponents of t are successive triangular numbers).
	www.math.wisc.edu /~propp/somos/dodgson (1213 words)

	Help Texts for coxeter and weyl, v2.4
		If R is a Coxeter matrix, then the assigned permutation (if requested) indicates a re-ordering of the rows and columns of R that agrees with cox_matrix(N), where N = name_of(R).
		FUNCTION: stab_chain - stabilizer chain for a Coxeter group CALLING SEQUENCE: stab_chain(pg); PARAMETERS: pg = a permutation representation of a Coxeter group SYNOPSIS: For a finite permutation group W, a stabilizer chain is a sequence of subgroups W = W_0 > W_1 > W_2 >...
		Coxeter matrices: A root system can also be (imperfectly) specified by a Coxeter matrix.
	www.math.lsa.umich.edu /~jrs/software/coxeterhelp.html (8531 words)

	Coxeter3 (Site not responding. Last check: 2007-09-23)
		Coxeter is a computer program for the study of combinatorial aspects of Coxeter group theory, particularly those related to the Bruhat ordering and Kazhdan-Lusztig polynomials.
		The program now works for essentially any Coxeter group (with some mild restrictions on the rank and the coefficients of the Coxeter matrix); in contrast, version 1 only worked for finite groups.
		The author would like to stress that Coxeter was written primarily with his own research interests in mind, and that it is made available without any promise of technical or other assistance, nor of further development.
	math.univ-lyon1.fr /~ducloux/coxeter/coxeter3/english/coxeter3_e.html (293 words)

	Weyl Groups
		A Cartan matrix, which determines the commutation relations of the Lie algebra, is determined by the D2 root vector space is determined by ratios of the inner products of the positive roots.
		A Cartan matrix is: 2 0 0 2 Since 4-1 = 3 is the dimension of the imaginary quaternions, Spin(0,4) has quaternionic structure and in fact generates two copies of the Lie group S3, one for the group of Lorentz rotations and another for the group of Lorentz boosts.
		According to Fulton and Harris, Representation Theory, Springer-Verlag 1991 and Bourbaki, Groupes et Algebres de Lie, Chapitres 4, 5, et 6, for all Lie algebras the weight lattice /\w of irreducible finite-dimensional representations includes as a sublattice the root lattice /\r generated by the root vectors of the Lie algebra.
	www.valdostamuseum.org /hamsmith/WeyLie.html (3488 words)

	Homogeous Coordinates: Methods
		This follows from the definition of matrix multiplication, and is useful in constructing or verifying matrix representations of transformations.
		We can choose fixed representations for the axis and corresponding center: say for the axis an n x r independent hyperplane matrix h, and for the center an r x n point matrix C. If the common eigenvalue of h and C is nonzero, we presume it is 1 by scaling T.
		Theorem 7 states that f has a matrix representation of the form T = I + hC, where C is an r x n independent point matrix representing the center of f corresponding to axis h.
	www.silcom.com /~barnowl/homogcoords.htm (3952 words)

	Base and Strong Generator Functions
		Construct a presentation for the matrix group G on a set of strong generators and return the presentation in the form of a finitely presented group F that is isomorphic to G. In Magma, the Todd-Coxeter Schreier algorithm is used to construct the presentation.
		The lengths of the basic orbits as defined by the current base for the matrix group G. This function assumes that a BSGS is known for G. The lengths are returned as a sequence of integers.
		Given a matrix group G for which a base and strong generating set are known, and an integer i, where 1 <= i <= k with k the length of the base, return the subgroup of G which fixes the first i - 1 points of the base.
	www.math.uiuc.edu /Software/magma/text269.html (3325 words)

	GAP Manual: 75 Root systems and finite Coxeter groups
		The matrix {m(i,j)}_{i,j} is called the Coxeter matrix; the set of Coxeter matrices such that the defined group is finite have been completely classified.
		A Coxeter group has a natural representation on a real vector space V of dimension the number of generators, its reflection representation, where the s_i act as reflections (a reflection in a vector space V is an element of text{GL}(V) of order 2 which leaves fixed a hyperplane).
		For the study of Coxeter groups it would be sufficient to consider root systems as certain subsets of Euclidean spaces which contain a basis of that space.
	www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C075S000.htm (1814 words)

	[No title]
		Here a reflection is a matrix which is conjugate in GL (r, Q) to the diagonal matrix diag(-1, 1,.
		Let (mi,j) be the Coxeter matrix (2.14) of (L, W), so that W is generated by the simple reflections {s1,.
		Since (L^, W) is rationally of Coxeter type, there is a lattice L0 and an action of W on L0 such that (Q ^L, W) is isomorphic to (Q2 L0, W); we use this isomorphism to identify Q ^Lwith Q2 L0.
	hopf.math.purdue.edu /Dwyer-Wilkerson/normaltorus/NT.txt (14335 words)

	[No title]
		The Coxeter diagram is the graph with vertex set S and one edge joining each pair {si, sj}, i 6= j, with label mij, and the conventions: if mij = 2 we omit the edge; if mij = 3 we omit the label.
		For n -1 define n as the union of Coxeter cells corresponding to finite WT with rank(T) n.
		If W is a right-angled Coxeter group, the equivariant K-homo- logy of E_W coincides with its Bredon homology at degree 0 and 1 respectively (given in the previous theorem).
	hopf.math.purdue.edu /SanchezGarcia/coxeter.txt (5782 words)

	Finite and Affine Coxeter Groups
		The Cartan name of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, or Dynkin digraph D. If the corresponding Coxeter group is neither finite nor affine, an error is flagged.
		Print the Dynkin diagramof a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name N. If the corresponding group is neither affine nor crystallographic, an error is flagged.
		Print the Coxeter diagramof a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name N. If the corresponding group is not affine or is not crystallographic, an error is flagged.
	www.math.lsu.edu /magma/text982.htm (1101 words)

	GAP Manual: 75.5 CoxeterGroup
		When the rank is equal to the semisimple rank (we then say that the Coxeter datum is semisimple), this can be given as a permutation group (on the roots).
		This is used to return the Coxeter group of objects derived from Coxeter groups, such as Coxeter cosets, Hecke algebras and braid elements.
		We document the following entries in a Coxeter datum record which are guaranteed to remain present in future versions of the package.
	www-groups.dcs.st-and.ac.uk /~gap/Gap3/Manual3/C075S005.htm (501 words)

	Coxeter Groups I
		This is not the case if the Coxeter group is the Weyl group of a Kac-Moody Lie algebra, since in that case the roots themselves are part of the structure of the Lie algebra.
		Coxeter groups which occur as the Weyl groups of Kac-Moody algebras are called
		dimensions is a Coxeter group, and that the fundamental domain of the action is a slice through a fundamental domain in a realization of the group.
	www.math.ubc.ca /~cass/coxeter/crm1.html (3666 words)

	GAP Manual: 76 Coxeter groups (Site not responding. Last check: 2007-09-23)
		Because of the easy solution of the word problem in Coxeter groups, a convenient way to represent their elements is as words in the Coxeter generators.
		As a rule of thumb one should keep in mind that for a Coxeter group which is a permutation group, if in some application one has to do a lot of computations with Coxeter group elements then using the low level GAP functions for permutations is usually much faster than manipulating lists of reduced expressions.
		The group is constructed as a matrix group, using the standard reflection representation for Coxeter groups.
	www.math.jussieu.fr /~jmichel/htm/CHAP076.htm (2676 words)

	Client Programs
		Rows of the incidence matrix correspond to the vectors of the first set, columns correspond to the second set.
		The transformation matrix, when applied to the points of the polytope, will put them on the projection facet, the latter will stay fixed.
		If the objective function is linear and the corresponding LP problem is unbounded, then the affine vertices that would become optimal after the removal of the rays are painted pale.
	cgm.cs.mcgill.ca /labdocs/polymake-1.5.1/apps/polytope/clients.html (3903 words)

	Seminar "Group theory and topology"
		The pair (W,S) is a Coxeter system if W is a group with Coxeter presentation , where R consists of the relations (st)^{m(s,t)} where m(s,t)=m(t,s), m(s,t)=1 iff s=t (i.e each generator is order 2) and m(s,t) in {1, 2, 3, …, \infty} (here m(s,t)=\infty, simply means st has infinite order).
		These centralizers and centers are shown to be even Coxeter groups that are convex in (W,S).
		We also show that for an arbitrary Coxeter system (W,S) and single generator s in S, the centralizer of s is convex in (W,S).
	www.math.vanderbilt.edu /~msapir/altop.html (1495 words)

	Cartan Matrices
		Then the group generated by s_1,..., s_n is a Coxeter group with Coxeter matrix M. In other words, a Cartan matrix specifies a representation of the Coxeter group as a real reflection group.
		A Cartan matrix corresponding to the Coxeter matrix M or Coxeter graph G. Note that the Cartan matrix of a Coxeter system is not unique.
		If the Coxeter matrix is reducible, this function also returns a nontrivial subset I of {1,..., n} such that m_(ij)=2 (i.e.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1030.htm (692 words)

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