Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces

For discrete Hecke pairs (G,H), we introduce a notion of covariant representation which reduces in the case where H is normal to the usual definition of covariance for the action of G/H on c₀(G/H) by right translation; in many cases where G is a semidirect product, it can also be expressed in terms of covariance for a semigroup action. We use this covariance to characterise the representations of c₀(G/H) which are multiples of the multiplication representation on ?²(G/H), and more generally, we prove an imprimitivity theorem for regular representations of certain crossed products by coactions of homogeneous spaces. We thus obtain new criteria for extending unitary representations from H to G.     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P^′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P^′ the supremum T^∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T^∨⊆T#.The authors conjecture for abelian G∈P^′ that T^∨=T#. That equality is established here for abelian G∈P^′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c²; (c) r₀(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r₀(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r₀(G)=|G|. Furthermore, the product of finitely many abelian G∈P^′, each with the property T^∨(G)=T#(G), has the same property.     

On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ₀?m?|G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erd?s-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits ²|G|-many μ-nonmeasurable dense subgroups.     

Reynolds operator on functors

Let be an affine R-monoid scheme. We prove that the category of dual functors (over the category of commutative R-algebras) of G-modules is equivalent to the category of dual functors of A^∗-modules. We prove that G is invariant exact if and only if A^∗=R×B^∗ as R-algebras and the first projection A^∗→R is the unit of A. If M is a dual functor of G-modules and w_G?(1,0)∈R×B^∗=A^∗, we prove that M^G=w_G⋅M and M=w_G⋅M⊕(1−w_G)⋅M; hence, the Reynolds operator can be defined on M.     

Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξ in semiparametric models of the form Y=G(ξ^TX,ε) where G(⋅) is an unknown link function, ε is an independent error, and ξ is a p_n×1 vector. To recover the direction of ξ, we first propose an inverse regression approach regardless of the link function G(⋅); to construct a data-driven confidence region for the direction of ξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G(⋅) or its derivative. When p_n remains fixed, the empirical likelihood ratio without bias correlation can be asymptotically standard chi-square. Moreover, the asymptotic normality of the empirical likelihood ratio holds true even when the dimension p_n follows the rate of p_n=o(n^1/4) where n is the sample size. Simulation studies are carried out to assess the performance of our proposal, and a real data set is analyzed for further illustration.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

Distance distributions for graphs modeling computer networks

The Wiener polynomial of a graph G is a generating function for the distance distribution dd(G)=(D₁,D₂,…,D_t), where D_i is the number of unordered pairs of distinct vertices at distance i from one another and t is the diameter of G. We use the Wiener polynomial and several related generating functions to obtain generating functions for distance distributions of unweighted and weighted graphs that model certain large classes of computer networks. These provide a straightforward means of computing distance and timing statistics when designing new networks or enlarging existing networks.     

A completion theorem for pro-G-spectra

We give a very general completion theorem for pro-spectra. We show that, if G is a compact Lie group, M[∗] is a pro-G-spectrum, and F is a family of (closed) subgroups of G, then the mapping pro-spectrum F(EF₊,M[∗]) is the F-adic completion of M[∗], in the sense that the map M[∗]→F(EF₊,M[∗]) is the universal map into an algebraically F-adically complete pro-spectrum. Here, F(EF₊,M[∗]) denotes the pro-G-spectrum , where runs over the finite subcomplexes of EF₊.     

Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.     

Free generating sets of lattice-ordered abelian groups

We prove that the generators g₁,…,g_n of a lattice-ordered abelian group G form a free generating set iff each ?-ideal generated by any n−1 linear combinations of the g_i is strictly contained in some maximal ?-ideal of G.     

Note on the 3-graph counting lemma

Szemerédi's regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a k-partite graph with k-partition V₁∪?∪V_k, |V₁|=?=|V_k|=n, where all induced bipartite graphs G[V_i,V_j] are (d,ε)-regular, then the number of k-cliques K_k in G is . Frankl and Rödl extended Szemerédi's regularity lemma to 3-graphs and Nagle and Rödl established an accompanying 3-graph counting lemma analogous to the graph counting lemma above. In this paper, we provide a new proof of the 3-graph counting lemma.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Nilmanifolds with a calibrated G2-structure

We introduce obstructions to the existence of a calibrated G₂-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G₂-structure.