Richardson Elements for Parabolic Subgroups of Classical Groups in Positive Characteristic

Let G be a simple algebraic group of classical type over an algebraically closed field k. Let P be a parabolic subgroup of G and let be the Lie algebra of P with Levi decomposition , where is the Lie algebra of the unipotent radical of P and ł is a Levi complement. Thanks to a fundamental theorem of Richardson (Bull. London Math. Soc. 6:21–24, 1974), P acts on with an open dense orbit; this orbit is called the Richardson orbit and its elements are called Richardson elements. Recently Baur (J. Algebra 297(1):168–185, 2006), the first author gave constructions of Richardson elements in the case for many parabolic subgroups P of G. In this note, we observe that these constructions remain valid for any algebraically closed field k of characteristic not equal to 2 and we give constructions of Richardson elements for the remaining parabolic subgroups. Presented by Peter Littelmann.     

Torsion and Ortho-Slender Classes

This paper generalizes a number of results obtained by Dimitrić in (Glas. Mat. 21(41):327–329, 1986; Proceedings of Hobart Conference on Rings, Modules and Radicals 1987, 204:41–50, Gordon and Breach, 1989) and Dimitrić and Goldsmith in (Glas. Mat. 23(43):241–246, 1988). The original papers were restricted to the category of Abelian groups and orthogonality was to the group of integers ℤ. Here, we are in a general Abelian category with products and coproducts, with applications to module categories and further to modules over PID’s. Another generalization is in replacing ℤ by an entire class of subobjects of the underlying category. We examine properties of the torsion class , Hom(T,C)=0} in relation to purity, direct summands and indecomposability as well as commutation with direct products, for example. Of special interest are members of this class when is a class of slender objects in the ground category; in this case, members of are called ortho-slender objects. In a sense, ortho-slenderness represents complementary, if not dual, notion to slenderness.      

Range of the Fractional Weak Discrepancy Function

In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset and characterize semiorders in terms of these values. In [6], we defined the fractional weak discrepancy of a poset to be the minimum nonnegative for which there exists a function satisfying (1) if then and (2) if then . This notion builds on previous work on weak discrepancy in [3, 7, 8]. We prove here that the range of values of the function is the set of rational numbers that are either at least one or equal to for some nonnegative integer . Moreover, is a semiorder if and only if , and the range taken over all semiorders is the set of such fractions .The third author's work was supported in part by a Wellesley College Brachman Hoffman Fellowship.     

Duality, Vector Spaces and Absolutely Convex Modules

It is well-known (see Semadeni, Queen Pap. Pure Appl. Math., 33:1–98, 1973 and Pumplün and Röhrl, Commun. Algebra, 12(8):953–1019, 1984, 1985) that the embedding of vector spaces into the category of absolutely convex modules is reflective. As we will show, under a separatedness condition on these modules it is at the same time coreflective. This is a peculiar situation, see Kannan, Math. Ann., 195:168–174, (1972) and Hu $\textrm {\u{s}}It is well-known (see Semadeni, Queen Pap. Pure Appl. Math., 33:1–98, 1973 and Pumplün and R?hrl, Commun. Algebra, 12(8):953–1019, 1984, 1985) that the embedding of vector spaces into the category of absolutely convex modules is reflective. As we will show, under a separatedness condition on these modules it is at the same time coreflective. This is a peculiar situation, see Kannan, Math. Ann., 195:168–174, (1972) and Huek, Reflexive and coreflexive subcategories of unif and top, Seminar Uniform Spaces, Prague, 113–126, (1973), but we do find it also in the embedding (Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, Oxford University Press, London, UK, 1997) and, by extension, in the embedding (see Lowen and Verwulgen, Houst. J. Math, 30(4):1127–1142, 2004, and Sioen and Verwulgen, Appl. Gen. Topol., 4(2):263–279, 2003. We demonstrate that, in this setting, by duality arguments, absolutely convex modules are indeed the numerical counterpart of vector spaces. All these, at first sight unrelated facts, are comprised in the commutative scheme below with natural dualisation functors and their left adjoints.      

Restriction matrices for numerically exploiting symmetry

In this paper we develop a technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group of congruences of . This technique will be based upon suitable restriction matrices strictly related to a system of irreducible matrix representation of . Both Abelian and non-Abelian finite groups are considered. In the framework of symmetric Galerkin boundary element method (SGBEM), where the discretization matrices are typically full, to increase the computational gain we couple Panel Clustering Method [30] and Adaptive Cross Approximation algorithm [13] with restriction matrices introduced in this paper, showing some numerical examples. Applications of restriction matrices to SGBEM under the weaker assumption of partial geometrical symmetry, where the boundary has disconnected components, one of which is invariant, are proposed. The paper concludes with several numerical tests to demonstrate the effectiveness of the introduced technique in the numerical resolution of Dirichlet or Neumann invariant BVPs, in their differential or integral formulation.      

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Decomposing triples of \mathbb{Z}_{p^n } and \mathbb{Z}_{3p^n } into cyclic designs

In this paper, we give some decompositions of triples of Zp^n or Z3p^n into cyclic triple systems. New constructions of difference families are given. Some infinite classes of simple cyclic triple systems are obtained from these decompositions.     

Plurisubharmonic Functions on the Octonionic Plane and <Emphasis Type="Italic">Spin</Emphasis>(9)-Invariant Valuations on Convex Sets

A new class of plurisubharmonic functions on the octonionic plane is introduced. An octonionic version of theorems of A.D. Aleksandrov (Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13(1):5–24, 1958) and Chern-Levine-Nirenberg (Global Analysis, pp. 119–139, 1969), and Błocki (Proc. Am. Math. Soc. 128(12):3595–3599, 2000) are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of . In particular, a new example of Spin(9)-invariant valuation on ℝ¹⁶ is given. Partially supported by ISF grant 1369/04.     

Distances Between <Emphasis Type="Italic">σ</Emphasis>-Fields on a Probability Space

For a probability space (Ω,ℱ,P) and two sub-σ-fields we consider two natural distances: and . We investigate basic properties of these distances. In particular we show that if a distance (ρ or ) from ℬ to is small then there exists Z∈ℱ with small P(Z), such that for every B∈ℬ there exists such that B∖Z and A∖Z differ by a set of probability zero. This improves results of Neveu (Ann. Math. Stat. 43(4):1369–1371, [1972]), Jajte and Paszkiewicz (Probab. Math. Stat. 19(1):181–201, [1999]).      

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.     

Chern Forms on Mapping Spaces

We state a Chern–Weil type theorem which is a generalization of a Chern–Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1, 19, 20]. We specialize to the case to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group . In this last example, we precise Freed's construction [5] on the loop group: The cohomology of the first Chern form of any holomorphic connection in the class considered is given by the Kähler form.     

Global error bounds for the extended vertical LCP

A new necessary and sufficient condition for the row -property is given. By using this new condition and a special row rearrangement, we provide two global error bounds for the extended vertical linear complementarity problem under the row -property, which extend the error bounds given in Chen and Xiang (Math. Program. 106:513–525, 2006) and Mathias and Pang (Linear Algebra Appl. 132:123–136, 1990) for the P-matrix linear complementarity problem, respectively. We show that one of the new error bounds is sharper than the other, and it can be computed easily for some special class of the row -property block matrix. Numerical examples are given to illustrate the error bounds. The work was in part supported by a Grant-in-Aid from Japan Society for the Promotion of Science, and the National Natural Science Foundation of China (10671010).     

Multivalued Exponentiation Analysis

We continue with the exponentiation analysis of multivalued maps defined on Banach spaces. In Part I of this work we have explored the Maclaurin exponentiation technique which is based on the use of a suitable power series. Now we focus the attention on the so-called recursive exponentiation method. Recursive exponentials are specially useful when it comes to study the reachable set associated to a differential inclusion of the form . The definition of the recursive exponential of uses as ingredient the set of trajectories associated to the discrete time system . Although we are taking inspiration from a recent paper by Alvarez et al. [1] on the relation between continuous and discrete time evolution systems, our analysis and results go far beyond the particular context of convex processes considered by these authors.     

Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Let be a finite-dimensional complex reductive Lie algebra and S() its symmetric algebra. The nilpotent bicone of is the subset of elements (x, y) of whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of generated by the 2-order polarizations of invariants of . The main result of this paper is that the nilpotent bicone is a complete intersection of dimension , where and are the dimensions of Borel subalgebras and the rank of , respectively. This affirmatively answers a conjecture of Kraft and Wallach concerning the nullcone [KrW2]. In addition, we introduce and study in this paper the characteristic submodule of . The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by Mustaţǎ in [Mu]. Finally, we give applications of our results to invariant theory.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Hermite interpolation by Pythagorean hodograph quintics in Minkowski space

Curves in the Minkowski space are very well suited to describe the medial axis transform (MAT) of planar domains. Among them, Minkowski Pythagorean hodograph (MPH) curves correspond to domains where both the boundaries and their offsets admit rational parameterizations (Choi et al., Comput Aided Design 31:59–72, 1999; Moon, Comput Aided Geom Design 16:739–753; 1999). We construct MPH quintics which interpolate two points with associated first derivative vectors and analyze the properties of the system of solutions, including the approximation order of the ‘best’ interpolant.      

Sur les idéaux stables dans certains anneaux différentiels de formes quasi-modulaires de Hilbert

Nesterenko (Sb. Math. 187:1319–1348, [1996]) proved, among other results, the algebraic independence over ℚ of the numbers π and e ^π . A very important feature of his proof is a multiplicity estimate for quasi-modular forms associated to SL ₂(ℤ) which involves differential properties of certain non-linear differential systems. The aim of this article is to begin the study of the corresponding properties for Hilbert modular and quasi-modular forms, especially those which are associated with the number field . We show that the differential structure of these functions has several analogies with the differential structure of the quasi-modular forms associated to SL ₂(ℤ).      

Real Commutative Division Algebras

The category of all two-dimensional real commutative division algebras is shown to split into two full subcategories. One of them is equivalent to the category of the natural action of the cyclic group of order 2 on the open right half plane . The other one is equivalent to the category of the natural action of the dihedral group of order 6 on the set of all ellipses in which are centered at the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are easily described. Together with they classify all real commutative division algebras up to isomorphism. Moreover we describe all morphisms between the objects in this classifying set, thus obtaining a complete picture of the category of all real commutative division algebras, up to equivalence. This supplements earlier contributions of Kantor and Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Nauka, Moscow, 1973; Benkart et al., Hadronic J., 4: 497–529, 1981; and Althoen and Kugler, Amer. Math. Monthly, 90: 625–635, 1983, who achieved partial results on the classification of the real commutative division algebras. Dedicated to Claus Michael Ringel on the occasion of his 60th birthday.     

The Orthogonal Subcategory Problem and the Small Object Argument

A classical result of P. Freyd and M. Kelly states that in “good” categories, the Orthogonal Subcategory Problem has a positive solution for all classes of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on , the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular -injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal λ every member of the class is either λ-presentable or an epimorphism. Both the Orthogonal Subcategory Problem and the Small Object Argument are valid for quasi-presentable classes. Surprisingly, in locally ranked categories (used previously to generalize Quillen’s result), this is no longer true: we present a class of morphisms, all but one being epimorphisms, such that the orthogonality subcategory is not reflective and the injectivity subcategory Inj is not weakly reflective. We also prove that in locally presentable categories, the injectivity logic and the Orthogonality Logic are complete for all quasi-presentable classes. Financial support by Centre for Mathematics of University of Coimbra and by School of Technology of Viseu is acknowledged by the third author.