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.

On the norm-preservation of squares in real algebra representation

One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality $$\Vert a^{2}\Vert \le \Vert a^{2}+b^{2}\Vert $$ ‖ a 2 ‖ ≤ ‖ a 2 + b 2 ‖ for $$a,b\in {\mathcal {A}}$$ a , b ∈ A is sufficient for a commutative real Banach algebra $${\mathcal {A}}$$ A with a unit to be isomorphic to the space $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) of continuous real-valued functions on a compact Hausdorff space $${\mathcal {K}}$$ K . Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of $${\mathcal {A}}$$ A onto $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces $${\mathcal {C}}_{{\mathbb {R}}}({\mathcal {K}})$$ C R ( K ) which are $$(1+\epsilon )$$ ( 1 + ϵ ) -equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras $${\mathcal {A}}$$ A where $$\Vert a^2\Vert \le k \Vert a^2+b^2\Vert $$ ‖ a 2 ‖ ≤ k ‖ a 2 + b 2 ‖ for all $$a,b\in {\mathcal {A}}$$ a , b ∈ A , for some $$k>1$$ k > 1 , but the inequality fails for $$k=1$$ k = 1 .

Algebraic Characteristics of the Matrix Conversions Class and Its Hardware Implementation

This paper derives the algebraic characteristic of the matrix transformations class by the method of isomorphic mappings on the algebraic characteristic of the class of vector transformations using the primitive program algebras. The paper also describes the hardware implementation of the matrix operations accelerator based on the obtained results. The urgency of the work is caused by the fact that today there is a rapid integration of computer technology in all spheres of society and, as a consequence, the amount of data that needs to be processed per unit time is constantly increasing. Many problems involving large amounts of complex computation are solved by methods based on matrix operations. Therefore, the study of matrix calculations and their acceleration is a very important task. In this paper, as a contribution in this direction, we propose a study of the matrix transformations class using signature operations of primitive program algebra such as multi place superposition, branching, cycling, which are refinements of the most common control structures in most high-level programming languages, and also isomorphic mapping. Signature operations of primitive program algebra in combination with basic partial-recursive matrix functions and predicates allow to realize the set of all partial-recursive matrix functions and predicates. Obtained the result on the basis of matrix primitive program algebra. Isomorphism provides the reproduction of partially recursive functions and predicates for matrix transformations as a map of partially recursive vector functions and predicates. The completeness of the algebraic system of matrix transformations is ensured due to the available results on the derivation of the algebraic system completeness for vector transformations. A name model of matrix data has been created and optimized for the development of hardware implementation. The hardware implementation provides support for signature operations of primitive software algebra and for isomorphic mapping. Hardware support for the functions of sum, multiplication and transposition of matrices, as well as the predicate of equality of two matrices is implemented. Support for signature operations of primitive software algebra is provided by the design of the control part of the matrix computer based on the RISC architecture. The hardware support of isomorphism is based on counters, they allow to intuitively implement cycling in the functions of isomorphic mappings. Fast execution of vector operations is provided by the principle of computer calculations SIMD.

The isomorphism problem for group algebras: A criterion

Let R be a finite unital commutative ring. We introduce a new class of finite groups, which we call hereditary groups over R. Our main result states that if G is a hereditary group over R, then a unital algebra isomorphism between group algebras {RG\cong RH} implies a group isomorphism {G\cong H} for every finite group H. As application, we study the modular isomorphism problem, which is the isomorphism problem for finite p-groups over {R=\mathbb{F}_{p}}, where {\mathbb{F}_{p}} is the field of p elements. We prove that a finite p-group G is a hereditary group over {\mathbb{F}_{p}} provided G is abelian, G is of class two and exponent p, or G is of class two and exponent four. These yield new proofs for the theorems by Deskins and Passi–Sehgal.

Circle Actions

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

Isomorphism classes of commutative algebras generated by idempotents

We study commutative algebras generated by idempotents with particular emphasis on the number of primitive idempotents. Let [Formula: see text] be an integral domain with the field of fractions [Formula: see text] and let [Formula: see text] be an [Formula: see text]-algebra which is torsion-free as an [Formula: see text]-module. We show that if [Formula: see text] satisfies the three conditions: [Formula: see text] is generated by idempotents over [Formula: see text]; [Formula: see text] is countably infinite dimensional over [Formula: see text]; [Formula: see text] has [Formula: see text] primitive idempotents for a nonnegative integer [Formula: see text], then [Formula: see text] is uniquely determined up to [Formula: see text]-algebra isomorphism. We also consider the case where [Formula: see text] has countably many primitive idempotents.

A minimal ring extension of a large finite local prime ring is probably ramified

Given any minimal ring extension [Formula: see text] of finite fields, several families of examples are constructed of a finite local (commutative unital) ring [Formula: see text] which is not a field, with a (necessarily finite) inert (minimal ring) extension [Formula: see text] (so that [Formula: see text] is a separable [Formula: see text]-algebra), such that [Formula: see text] is not a Galois extension and the residue field of [Formula: see text] (respectively, [Formula: see text]) is [Formula: see text] (respectively, [Formula: see text]). These results refute an assertion of G. Ganske and McDonald stating that if [Formula: see text] are finite local rings such that [Formula: see text] is a separable [Formula: see text]-algebra, then [Formula: see text] is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let [Formula: see text] be a finite special principal ideal ring (SPIR), but not a field, such that [Formula: see text] has index of nilpotency [Formula: see text] ([Formula: see text]). Impose the uniform distribution on the (finite) set of ([Formula: see text]-algebra) isomorphism classes of the minimal ring extensions of [Formula: see text]. If [Formula: see text] (for instance, if [Formula: see text]), the probability that a random isomorphism class consists of ramified extensions of [Formula: see text] is at least [Formula: see text]; if [Formula: see text] (for instance, if [Formula: see text] for some odd prime [Formula: see text]), the corresponding probability is at least [Formula: see text]. Additional applications, examples and historical remarks are given.

Neutrosophic Quotient Algebra

The algebraic properties of neutrosphic ideals over algebra, isomorphism properties of neutrosophic ideal and neutrosophic modules over algebra are discussed in this paper. Some of the charactrisations of Neutrosophic quotient algebra are derived and the role of algebraic structures is studied in the context of neutrosophic set. This paper expands the definition of quotient algebra within the context of neutrosophical set.

Multipliers of Uniform Topological Algebras

Let E be a complete uniform topological algebra with Arens-Michael normed factors within an algebra isomorphism ϕ. If each factor Eα is complete, then every multiplier of E is continuous and ϕ is a topological algebra isomorphism where M(E) is endowed with its seminorm topology.

REPRESENTING FILIFORM LIE ALGEBRAS MINIMALLY AND FAITHFULLY BY STRICTLY UPPER-TRIANGULAR MATRICES

In this paper, we compute minimal faithful representations of filiform Lie algebras by means of strictly upper-triangular matrices. To obtain such representations, we use nilpotent Lie algebras [Formula: see text]n, of n × n strictly upper-triangular matrices, because any given (filiform) nilpotent Lie algebra [Formula: see text] admits a Lie-algebra isomorphism with a subalgebra of [Formula: see text]n for some n ∈ ℕ\{1}. In this sense, we search for the lowest natural integer n such that the Lie algebra [Formula: see text]n contains the filiform Lie algebra [Formula: see text] as a subalgebra. Additionally, we give a representative of each representation.