Edwin C. Ihrig

Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W. We use this correspondence to find all W for which L is supersolvable. In particular, we show that the only infinite Coxeter group for which L is supersolvable is the infinite dihedral group. Also, we show how this isomorphism gives an embedding of L into the partition lattice whenever W is of type An, Bn or Dn. In addition, we give several results concerning non-broken circuit bases (NBC bases) when W is finite. We show that L is supersolvable if and only if all NBC bases are obtainable by a certain specific combinatorial procedure, and we use the lattice of parabolic subgroups to identify a natural subcollection of the collection of all NBC bases.

A q-umbral calculus☆

Abstract An algebraic setting for the Roman-Rota umbral calculus is introduced. It is shown how many of the umbral calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map. The same approach is used to construct a q-umbral calculus. Our umbral calculus yields some of Andrews recent results on Eulerian families of polynomials as corollaries. The homogeneous Eulerian families are studied. Operator and functional expansions are also included.

Homogeneous spacetimes of zero curvature

In the following we show the only possible flat, connected, incomplete homogeneous spacetimes are H/A where H = {v E Rn1g(v,N) > 0}, N is a null vector, and A is a discrete subgroup of translations. In [WI] Wolf classifies all complete flat homogeneous spacetimes. He does this by observing that the universal covering of such a spacetime must be Minkowski space, reducing the problem to finding all discrete subgroups of the Poincar& group which act properly discontinuously and whose normalizer acts transitively on Minkowski space. Classification of complete compact flat spacetimes in dimension 4 is given by Fried in [F]. In [W2] Wolf remarks that there are homogeneous, flat, incomplete spacetimes, and he gives an example. In this paper we will classify all incomplete, flat, homogeneous spacetimes. The first step in such a classification is to find all the possible universal covers of these spacetimes. Let N be a null vector in n dimensional Minkowski space. We show that the subspace H of n dimensional Minkowski space defined as {v: g(v , N) > 0} is the unique (to within isometry), incomplete, homogeneous, simply connected, flat n dimensional spacetime (3.1). Next we find all discrete subgroups A of the isometry group of H which have transitive normalizers. Then every connected flat incomplete homogeneous spacetime is isometric to H/A. This result is given in Theorem 3.7. Besides finishing the classification of all homogeneous flat spaces, this result relates to some other questions. Auslander ([Au, p. 809]) conjectured that every nilpotent, simply transitive group of affine transformations must contain a one parameter subgroup of central translations. This result was shown by Scheuneman [S, p. 226]. Auslander gives an example ([Au, p. 810]) of a three dimensional solvable group that acts affinely and simply transitively on R3 and contains no translations. A corollary of our result is that every subgroup of the Poincare group having an open orbit in Minkowski space must contain a one Received by the editors July 5, 1988 and, in revised form, December 8, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C50, 53C30; Secondary 22E43. ? 1989 American Mathematical Society 0002-9939/89

Symmetry groups related to the construction of perfect one factorizations of K 2n

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Existence of Hartree--Fock solutions

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Modular elements in the lattice L(A) when A is a real reflection arrangement

Abstract We classify the elements of order two contained in the symmetry groups of perfect one factorizations of K 2 n . Using these results, we show that if one assumes the symmetry group of a perfect one factorization contains a noncentral element of order two with at least one fixed point, then the one factorization must be one of the known perfect one factorizations on K p + 1 or K 2 p , where p is prime. Moreover, the assumption of the existence of such a symmetry element gives rise to an explicit construction of these one factorizations. A number of other results are also shown. The only perfect one factorizations with doubly transitive symmetry groups are the known perfect one factorizations on K p + 1 . The symmetry group of a perfect one factorization generated by the even starter construction is the starter group if it is not the known K p + 1 one factorization. The order of the symmetry group of a perfect one factorization of K 2 n generated by the starter construction must be odd and divide (2 n −1)( n −1) if n is not prime and the perfect one factorization is not the known K p + 1 one factorization.

The structure of the symmetry groups of perfect 1-factorizations of K 2 n

For a finite‐dimensional space with only a mild restriction on the Hamiltonian, it is shown that there exist at least as many Hartree–Fock states as the dimension of the many‐fermion space. The index of the random phase approximation matrix is determined for these HF states and the relationship between that index and the number of real and complex excitation energies established.

Cyclic Perfect One Factorizations of K2n

Abstract Let W be a real reflection group, and let L W denote the lattice consisting of all possible intersections of reflecting hyperplanes of reflections in W . Let p W ( t ) be the characteristic polynomial of L W . To every element X of L W there corresponds a parabolic subgroup of W denoted by Gal( X ). If W is irreducible, we show that an element X of L W is modular if and only if p Gal( X ) ( t ) divides p W ( t ). This characterization is not true if W is not irreducible. Also, we show that if W is neither A n nor B n , then the only modular elements are 0 , 1 and the atoms of L W .

Flat Pseudo-Reimannian manifolds with a nilpotent transitive group of isometries

We give a characterization of the structure of the symmetry groups of perfect 1-factorizations of the complete graph on 2n vertices. This characterization is strong for symmetry groups which have no trivial stability subgroups. In this case if n is not prime, we show that the full group of symmetries must be a semi-direct product of a nilpotent group which acts strictly transitively on 2n−1 vertices with a subgroup of its automorphism group consisting of fixed-point-free automorphisms. As a consquence there are certain values of 2n (for example 16, 36, 40, 52, 56, 64, 66, 70, 76, 88, 92, and 98) for which every symmetry group must have at least one trivial stability subgroup, and thus its order must be less than 2n. We also show that the order of any symmetry group must divide one of the numbers 2n(2n−1), 2n(n−1), or (2n−1)(2n−2). Moreover if the order of the symmetry group is (2n−1)(n−1) or greater, then it must act 2-homogeneously on the 1-factors, and it must have (Zp)m as the nilponent group mentioned above where p is prime.

Three-dimensional Schwarzschild spacetimes

Abstract Recently Hartman and Rosa characterized those n for which K 2n admits a cyclic one factorization. We show that K 2n admits a cyclic perfect one factorization if and only if n is prime. We also show that if n is prime there is a one to one, onto correspondence between cyclic perfect one factorizations on K 2n and starter induced perfect one factorizations on K n+1 . Moreover the full symmetry group of the cyclic perfect one factorization is that of the corresponding starter induced perfect one factorization direct sum Z 2