Derived Categories of Schur Algebras

Abstract

In this paper we classify the derived tame Schur and infinitesimal Schur algebras and describe indecomposable objects in their derived categories.     

The Flagged Double Schur Function

The double Schur function is a natural generalization of the factorial Schur function introduced by Biedenharn and Louck. It also arises as the symmetric double Schubert polynomial corresponding to a class of permutations called Grassmannian permutations introduced by A. Lascoux. We present a lattice path interpretation of the double Schur function based on a flagged determinantal definition, which readily leads to a tableau interpretation similar to the original tableau definition of the factorial Schur function. The main result of this paper is a combinatorial treatment of the flagged double Schur function in terms of the lattice path interpretations of divided difference operators. Finally, we find lattice path representations of formulas for the symplectic and orthogonal characters for sp(2n) and so(2n + 1) based on the tableau representations due to King and El-Shakaway, and Sundaram. Based on the lattice path interpretations, we obtain flagged determinantal formulas for these characters.     

On <Emphasis Type="Italic">p</Emphasis>-convergent Operators on Banach Lattices

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sánchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.     

Plücker Relations on Schur Functions

We present a set of algebraic relations among Schur functions which are a multi-time generalization of the discrete Hirota relations known to hold among the Schur functions of rectangular partitions. We prove the relations as an application of a technique for turning Plücker relations into statements about Schur functions and other objects with similar definitions as determinants. We also give a quantum analogue of the relations which incorporates spectral parameters. Our proofs are mostly algebraic, but the relations have a clear combinatorial side, which we discuss.     

On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?₃ × ?_{3 ^k} or ?₃ × ?₃ × ? _p where k ≥ 1 and p is a prime. In addition, we prove that ?₂ × ?₂ × ? _p is a Schur group for every prime p.     

On Schur Superfunctors

We introduce super-analogues of the Schur functors defined by Akin, Buchsbaum and Weyman. These Schur superfunctors may be viewed as characteristic-free analogues of finite dimensional irreducible polynomial representations of the Lie superalgebra ????(m|n) studied by Berele and Regev. Our construction realizes Schur superfunctors as objects of a certain category of strict polynomial superfunctors. We show that Schur superfunctors are indecomposable objects of this category. In characteristic zero, these correspond to the set of all simple supermodules for the Schur superalgebra, S(m|n, d), for any m, n, d ? 0. We also provide decompositions of Schur bisuperfunctors in terms of tensor products of skew Schur superfunctors.     

Pluriassociative algebras I: The pluriassociative operad

Diassociative algebras form a category of algebras recently introduced by Loday. A diassociative algebra is a vector space endowed with two associative binary operations satisfying some very natural relations. Any diassociative algebra is an algebra over the diassociative operad, and, among its most notable properties, this operad is the Koszul dual of the dendriform operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of diassociative algebras, called γ-pluriassociative algebras, so that 1-pluriassociative algebras are diassociative algebras. Pluriassociative algebras are vector spaces endowed with 2γ associative binary operations satisfying some relations. We provide a complete study of the γ-pluriassociative operads, the underlying operads of the category of γ-pluriassociative algebras. We exhibit a realization of these operads, establish several presentations by generators and relations, compute their Hilbert series, show that they are Koszul, and construct the free objects in the corresponding categories. We also study several notions of units in γ-pluriassociative algebras and propose a general way to construct such algebras. This paper ends with the introduction of an analogous generalization of the triassociative operad of Loday and Ronco.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.     

EMBEDDINGS INTO THE CATEGORY OF SUPER UNIFORM SPACES

Abstract

The category US of uniform spaces has been generalised in various ways. The category FUS, of fuzzy uniform spaces and the category GUS, of generalised uniform spaces have both been shown to be good extensions in the sense that US can be embedded into them. We show here that the category SUS, of super uniform spaces also enjoys this property and furthermore, the categories FUS and GUS can be embedded into SUS.     

A property of Dunford-Pettis type in topological groups

The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.

In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.

For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.     

Multi-graded nilpotent tuples

We discuss multi-graded nilpotent tuples of multi-graded vector spaces which are a generalization of graded nilpotent pairs. The multi-grading yields a natural notion of a shape of such tuple and our main interest is to answer the question “Is the number of multi-graded nilpotent tuples of a fixed shape, up to base change in the homogeneous components, finite?” Our methods make use of a translation to the class of so-called “Multi-staircase algebras” and we classify their representation types.     

On a new geometric property for Banach spaces

In this paper we study a geometric property for Banach spaces called condition (*), introduced by de Reynaet al in [3], A Banach space has this property if for any weakly null sequencex _n of unit vectors inX, ifx _* ⁿ is any sequence of unit vectors inX ^* that attain their norm at x_n’s, then . We show that a Banach space satisfies condition (*) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.     

两类对称函数的Schur凸性

本文用一种新方法研究两类对称函数的Schur凸性.首先,对x=(x1,...,xn)∈(-∞,1)n∪(1,+∞)n和r∈{1,2,...,n},讨论Guan(2007)定义的对称函数Fn(x,r)=Fn(x1,x2,...,xn;r)=∑1≤i1≤i2≤···≤ir≤n r∏j=1xij/(1-xij)的Schur凸性,其中i1,i2,...,in为正整数;推广褚玉明等人(2009)的主要结果,因而用新方法推广并解决Guan(2007)提出的一个公开问题.然后,对x=(x1,...,xn)∈(-∞,1)n∪(1,+∞)n和r∈{1,2,...,n},研究本文定义的对称函数Gn(x,r)=Gn(x1,x2,...,xn;r)=∑1≤i1≤i2≤···≤ir≤n(r∏j=1xij/(1-xij))1/r的Schur凸性、Schur乘性凸性和Schur调和凸性,其中i1,i2,...,in为正整数.作为应用,用Schur凸函数自变量的双射变换得到其他几类对称函数的Schur凸性,用控制理论建立一些不等式,特别地,由此给出Sharpiro不等式和Ky Fan不等式一个共同的推广,导出Safta猜想在高维空间的推广.     

Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.

     

Trace theorems for vector fields

In the setting of Carnot-Carathéodory spaces we prove some trace theorems for Sobolev functions. We consider the trace on a non characteristic surface for H?rmander vector fields of step and the trace on the boundary of a class of domains in the Grushin plane. Received: 27 January 2000 / in final form: 7 May 2001 / Published online: 1 February 2002     

Porosity of Mutually Nearest and Mutually Furthest Points Problems in Geodesic Space

The notion of superframe in general Hilbert spaces was introduced in the context of multiplexing, which has been widely used in mobile communication network, satellite communication network and computer area network. The notion of oblique dual frame is a generalization of conventional dual frame. It has provided us with a frame-like expansion. Using oblique dual frames one can extend frame expansions to include redundant expansions in which the analysis and synthesis frames lie in different spaces. Given positive integers L, M and N, an N?-periodic set 𝕊 in ?, let 𝒢(g, N, M) be a frame for l ²(𝕊, ? ^L ), and let 𝒢(h, N, M) be a frame for ?(h, N, M) (generated by 𝒢(h, N, M)). This article addresses super Gabor duals of g in ?(h, N, M). We obtain a necessary and sufficient condition on h admitting super oblique Gabor duals of g, and present a parametrization expression of all super oblique Gabor duals and all oblique canonical Gabor duals of g. We also characterize the uniqueness of super oblique Gabor dual and oblique canonical Gabor dual of g. Some examples are also provided.     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.