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29
A FINITE LOOP SPACE NOT RATIONALLY EQUIVALENT TO A COMPACT LIE GROUP
, 2003
"... Abstract. We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we ver ..."
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Abstract. We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5. 1.
Reflection Groups. A Contribution to the Handbook of Algebra
, 2003
"... This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included. ..."
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This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included.
A Cohomology Decomposition Theorem
"... this paper we will prove a parallel algebraic decomposition theorem for certain kinds of unstable algebras over the mod p Steenrod algebra. This algebraic result gives a new proof of the theorem of Jackowski and McClure and has the potential to lead to homotopy decompositon theorems for many spaces ..."
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this paper we will prove a parallel algebraic decomposition theorem for certain kinds of unstable algebras over the mod p Steenrod algebra. This algebraic result gives a new proof of the theorem of Jackowski and McClure and has the potential to lead to homotopy decompositon theorems for many spaces which are not of the form BG (see x6).
Symplectic groups are Ndetermined 2compact groups
"... Abstract. We show that for n ≥ 3 the symplectic group Sp(n) is as a 2compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus. 1. ..."
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Abstract. We show that for n ≥ 3 the symplectic group Sp(n) is as a 2compact group determined up to isomorphism by the isomorphism type of its maximal torus normalizer. This allows us to determine the integral homotopy type of Sp(n) among connected finite loop spaces with maximal torus. 1.
PAdic Lattices Of Pseudo Reflection Groups
"... . Let U be a vector space over the padic rationals, and let W \Gamma! Gl(U) be faithful representation of a finite group such that W is generated by pseudo reflections. For odd primes we study the padic W sublattice of this representation and achieve a complete classification. Examples of suc ..."
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. Let U be a vector space over the padic rationals, and let W \Gamma! Gl(U) be faithful representation of a finite group such that W is generated by pseudo reflections. For odd primes we study the padic W sublattice of this representation and achieve a complete classification. Examples of such situations are given by the Weyl group acting on the 1dimesional homology of the maximal torus of a connected compact Lie group, or of the so called pcompact groups, a homotopy theoretic generalisation of compact Lie groups. The associated lattices are an important algebraic invariant in the study of these geometric object. Introduction. Let U be a finite dimensional vector space over the padic rationals Q p . For a faithful representation ae : W \Gamma! Gl(U) of a group W , an element 1 6= oe 2 W is called a pseudo reflection if oe or ae(oe) has finite order and if the kernel of ae(oe) \Gamma id U has codimension 1. The element oe is called a honest reflection or a reflection if...
Automorphisms of pcompact groups and their root data
"... Abstract. We construct a model for the space of automorphisms of a connected pcompact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a pcompact group can be lifted to ..."
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Abstract. We construct a model for the space of automorphisms of a connected pcompact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a pcompact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper “The classification of 2compact groups”, where we prove the conjectured classification of 2compact groups and determine their automorphism spaces. 1.
Partial mirror symmetry I: reflection monoids
, 2007
"... Abstract. This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their ord ..."
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Abstract. This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.
SEPARATING INVARIANTS AND FINITE REFLECTION GROUPS
, 805
"... Abstract. Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a more geometric notion of separating algebra. This allows us to prove that when there ..."
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Abstract. Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a more geometric notion of separating algebra. This allows us to prove that when there is a polynomial separating algebra, the group is generated by reflections, and when there is a complete intersection separating algebra, the group is generated by bireflections. 1.
AUTOMORPHISMS OF ROOT DATA, MAXIMAL TORUS NORMALIZERS, AND pCOMPACT GROUPS
, 2005
"... Abstract. We describe the outer automorphism group of a compact connected Lie group as a certain subgroup of the outer automorphism group of its maximal torus normalizer, expressed in terms of the associated root datum. The same subgroup can be defined for connected 2compact groups. We use this to ..."
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Abstract. We describe the outer automorphism group of a compact connected Lie group as a certain subgroup of the outer automorphism group of its maximal torus normalizer, expressed in terms of the associated root datum. The same subgroup can be defined for connected 2compact groups. We use this to show that any homomorphism to the outer automorphism group of a pcompact group can be lifted to an action, analogous to a classical theorem of de Siebenthal for compact Lie groups, and we find a candidate formula for the whole space of selfhomotopy equivalences of any connected 2compact group. The results of this paper play a key role in a subsequent paper by the authors where we prove the conjectured classification of 2compact groups and describe their automorphism spaces. 1.
Normalizers of Tori
"... Abstract We determine the groups which can appear as the normalizer of a maximal torus in a connected 2compact group. The technique depends on using ideas of Tits to give a novel description of the normalizer of the torus in a connected compact Lie group, and then showing that this description can ..."
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Abstract We determine the groups which can appear as the normalizer of a maximal torus in a connected 2compact group. The technique depends on using ideas of Tits to give a novel description of the normalizer of the torus in a connected compact Lie group, and then showing that this description can be extended to the 2compact case. AMS Classification