Schubert Class and Cyclotomic NilHecke Algebras

Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

Distributed rrays: an algebra for generic distributed query processing

We propose a simple model for distributed query processing based on the concept of a distributed array. Such an array has fields of some data type whose values can be stored on different machines. It offers operations to manipulate all fields in parallel within the distributed algebra. The arrays considered are one-dimensional and just serve to model a partitioned and distributed data set. Distributed arrays rest on a given set of data types and operations called the basic algebra implemented by some piece of software called the basic engine. It provides a complete environment for query processing on a single machine. We assume this environment is extensible by types and operations. Operations on distributed arrays are implemented by one basic engine called the master which controls a set of basic engines called the workers. It maps operations on distributed arrays to the respective operations on their fields executed by workers. The distributed algebra is completely generic: any type or operation added in the extensible basic engine will be immediately available for distributed query processing. To demonstrate the use of the distributed algebra as a language for distributed query processing, we describe a fairly complex algorithm for distributed density-based similarity clustering. The algorithm is a novel contribution by itself. Its complete implementation is shown in terms of the distributed algebra and the basic algebra. As a basic engine the Secondo system is used, a rich environment for extensible query processing, providing useful tools such as main memory M-trees, graphs, or a DBScan implementation.

Interval Sheffer Stroke Basic Algebras and Yang-Baxter Equation

In this study, we give basic definitions and notions about Sheffer stroke operation and Sheffer stroke basic algebra. After presenting Sheffer stroke basic algebra on a given interval, named interval Sheffer stroke basic algebra, we give some features of an interval Sheffer stroke basic algebra. Then we investigate solutions to the set-theoretical Yang-Baxter equation in this algebraic structure by using its features.