Group Ring Theory

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with 𝑆 𝐺 ⊆ 𝑆 being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring 𝑆 𝐺 is a pure subring of S, equivalently, 𝑆 𝐺 is a direct summand of S as an 𝑆 𝐺 -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion 𝑆 𝐺 ⊆ 𝑆 is pure. It turns out that if 𝑆 𝐺 ⊆ 𝑆 is pure, then either the invariant ring 𝑆 𝐺 is regular or the group G is linearly reductive.

The covering number of a nontrivial finite group G, denoted σ(G), is the smallest number of proper subgroups of G whose set-theoretic union equals G. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group G, we define the intersection number of G, denoted ι(G), to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of G. We elucidate some basic properties of this invariant, and give an exact formula for ι(G) when G is a nontrivial finite nilpotent group. In addition, we determine the intersection numbers of a few infinite families of non-nilpotent groups. We conclude by discussing a generalization of the intersection number of a nontrivial finite group and pose some open questions about these invariants.

The covering number of a nontrivial finite group $G$, denoted $\sigma(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $\iota(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $\iota(G)$ when $G$ is a nontrivial finite nilpotent group. In addition, we determine the intersection numbers of a few infinite families of non-nilpotent groups. We conclude by discussing a generalization of the intersection number of a nontrivial finite group and pose some open questions about these invariants.

Let M be a complete n-dimensional Riemannian spin manifold, partitioned by q two-sided hypersurfaces which have a compact transverse intersection N and which in addition satisfy a certain coarse transversality condition. Let E be a Hermitean bundle on M with connection. We define a coarse multi-partitioned index of the spin Dirac operator on M twisted by E. Our main result is the computation of this multi-partitioned index as the Fredholm index of the Dirac operator on the compact manifold N, twisted by the restriction of E to N. We establish the following main application: if the scalar curvature of M is bounded below by a positive constant everywhere (or even if this happens only on one of the quadrants defined by the partitioning hypersurfaces) then the multi-partitioned index vanishes. Consequently, the multi-partitioned index is an obstruction to uniformly positive scalar curvature on M. The proof of the multi-partitioned index theorem proceeds in two steps: first we establish a strong new localization property of the multi-partitioned index which is the main novelty of this note. This we establish even when we twist with an arbitrary Hilbert A-module bundle E (for an auxiliary C*algebra A). This allows to reduce to the case where M is the product of N with Euclidean space of dimension q. For this special case, standard methods for the explicit calculation of the index in this product situation can be adapted to obtain the result.

Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra H N (q = 1). We show in the case of the reflection algebra that there exists a "boundary" finite sub-algebra for some special choice of "boundary" elements of the B-type Hecke algebra B N (q = 1, Q). We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings and also allow the investigation of the symmetries of the double row transfer matrix.

We describe the structure of the collineation groups of Figueroa planes, giving examples and explanations that show why the description in Dempwolff [3] is not completely accurate. We also give criteria for when Figueroa planes are isomorphic, and show that certain subplanes are Figueroa or Pappian planes.

Let RG denote the group ring of the torsion group G over a commutative ring R with identity. In this paper we present proofs of some statements that appear without to be proved in the literature. We establish the valid implications between the ring-theoretic conditions duo, reversible, SI property and symmetric in the setting of group rings. We further show that if the group ring RG possesses any of these properties, then G is a Hamiltonian group and the characteristic of R is either 0 or 2. Moreover, we characterize the same properties in group rings RG in the following cases: (1) RG is a semi-simple group ring and (2) R is a semi-simple ring and G any group.

This is an introduction to quantum algebra, from a geometric perspective. The classical spaces X, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras A, defined over various fields F. These algebras A satisfy a commutativity type condition, and the general idea is that of lifting this condition, and calling quantum spaces the underlying space-like objects X. One problem comes from the fact that different fields F lead, via different algebras A, to different classes of quantum spaces X. Our aim here is to identify and put at the center of the presentation those quantum spaces X which do not depend on the choice of F .

We state the Eilenberg-Zilber type theorem for the product of two small categories and consider the Bredon homology of an equivariant space with local coefficients in the light of homologies of small categories. Then, in particular, an exterior product, Künneth formula and cap-product for the Bredon Homology of equivariant spaces with local coefficients are established. we also use exactly the same methods as above. 2 Observe that explicit formulas for Φ * and Ψ * can also be found in a routine way (see, e.g., Chapter VI] and [10, Chapter VIII]).

Let I be a graded ideal of a standard graded polynomial ring S with coefficients in a field K. The asymptotic behaviour of the v-number of the powers of I is investigated. Natural lower and upper bounds which are linear functions in k are determined for v(I k ). We call v(I k ) the v-function of I. We prove that v(I k ) is a linear function in k for k large enough, of the form v(I k ) = α(I)k + b, where α(I) is the initial degree of I, and b ∈ Z is a suitable integer. For this aim, we construct new blowup algebras associated to graded ideals. Finally, for a monomial ideal in two variables, we compute explicitly its v-function.

We introduce the notion of a matrad M = {Mn,m} whose submodules M * ,1 and M 1, * are non-Σ operads. We define the free matrad H∞ generated by a singleton θ n m in each bidegree (m, n) and realize H∞ as the cellular chains on a new family of polytopes {KKn,m = KKm,n} , called biassociahedra, of which KK n,1 = KK 1,n is the associahedron Kn. We construct the universal enveloping functor from matrads to PROPs and define an A∞bialgebra as an algebra over H∞.

Here the reader can find some basic definitions and notations in order to better understand the model for social choise described by L. Marengo and S. Settepanella in their paper: Social choice among complex objects. The interested reader can refer to [Bou68], [Massey] and [OT92] to go into more depth.

Consider the ring of smooth function germs at the origin of $\mathbb{R}^n$. The Taylor expansion is the completion map from this ring to the ring of formal power series. Borel's lemma ensures the surjectivity of this map. In this short note we address the surjectivity of the completion map for the rings of smooth functions and rather general filtrations. This generalizes the classical Whitney extension theorem.

We know from the classical sequence spaces theory that there is a useful relationship between continuous and -duals of a scalar-valued FK-space originated by the AK-property. Our main interest in this work is to expose relationships between the operator space and and the generalized -duals of some -valued AK-space where and are Banach spaces and . Further, by these results, we obtain the generalized -duals of some vector-valued Orlicz sequence spaces.

The cyclic, periodic cyclic and negative cyclic homologies of associative algebras are fitted into the context of cotriple homology of Barr and Beck. As applications of these results, an axiomatic description of the cyclic homology theory and the Hopf type formulas in the sense of Brown-Ellis are given.

Object oriented programming and design patterns introduce a high level of abstraction that allows us to implement and work with mathematical abstractions. We analyze and design an object oriented algebraic library, that allows working not only with concrete algebraic structures, but also with abstract algebraic structures. The advantages of this approach mainly result from the usage of creational design patterns, reflection and dynamic loading, and representation independence. These introduce significant flexibility and abstraction. Using this library, we may work with abstract algebraic structures, such as: groups, rings, fields, etc., define new algebraic structures, and operate with them in abstract and concrete mathematical ways.

Teaching about design patterns is not easy, especially for the students that don't have so much experience in OOP. Hence, finding example from well-known arias is very important. Usually, students in Computer Science also learn a lot of Mathematics, and Algebra is a field that is normally very well treated. This article presents an example that deals with algebraic structures and uses creational design patterns. The possibility of constructing a general polynomial built over any kind of algebraic field (K, +, * ) is discussed. In order to work with polynomials that have different coefficient types: real, complex, etc., the type of a polynomial coefficient is consider to be an abstract type -FieldElem. Three creational design patterns are discussed and advantages of using them are also analyzed. Patterns are used for the creation of the special values of the coefficients: zero and one. The example helps the students to learn about creational design patterns, without creating big applications. This leads to an easy and well understanding of the concepts.

The authors examine when a finite group G can have the property that the group of units in the integral group ring ℤG can have a subgroup U of finite index such that U is a non-trivial free product of Abelian groups. The main theorem states that if G is neither Abelian nor a Hamiltonian 2-group, then the existence of such a U implies that the Wedderburn decomposition of the group algebra ℚG over the rationals is of a restricted type. Only for very few groups G this Wedderburn decomposition can occur. Most of the work is to prove the restriction on the isomorphism type of G. This part can be simplified by use of results of B. Huppert and R. Gow [J. Reine Angew. Math. 381, 136-147 (1987; Zbl 0618.20005), ibid. 389, 122-132 (1988; Zbl 0639.20007)].

Students would be able to: CO1 Apply group theoretic reasoning to group actions. CO2 Learn properties and analysis of solvable & nilpotent groups, Noetherian & Artinian modules and rings. CO3 Apply Sylow's theorems to describe the structure of some finite groups and use the concepts of isomorphism and homomorphism for groups and rings. CO4 Use various canonical types of groups and rings-cyclic groups and groups of permutations, polynomial rings and modular rings. CO5 Analyze and illustrate examples of composition series, normal series, subnormal series. Conjugates and centralizers in Sn, p-groups, Group actions, Counting orbits. Sylow subgroups, Sylow theorems, Applications of Sylow theorems, Description of group of order p2 and pq, Survey of groups upto order 15. Normal and subnormal series, Solvable series, Derived series, Solvable groups, Solvability of Sn-the symmetric group of degree n ≥ 2, Central series, Nilpotent groups and their properties, Equivalent conditions for a finite group to be nilpotent, Upperandlower central series. Composition series, Zassenhaus lemma, Jordan-Holder theorem. Modules, Cyclic modules, Simple and semi-simple modules, Schur lemma, Free modules, Torsion modules, Torsion free modules, Torsion part of a module, Modules over principal ideal domain and its applications to finitely generated abelian groups. Noetherian and Artinian modules, Modules of finite length, Noetherian and Artinian rings, Hilbert basis theorem. HomR(R,R), Opposite rings, Wedderburn -Artin theorem, Maschk theorem, Equivalent statement for left Artinian rings having non-zero nilpotent ideals. Radicals: Jacobson radical, Radical of an Artinian ring. Note :The question paper of each course will consist of five Sections. Each of the sections I to IV will contain two questions and the students shall be asked to attempt one question from each. Section-V shall be compulsory and will contain eight short answer type questions without any internal choice covering the entire syllabus.

In this note we construct five new symmetric 2-(61,16,4) designs invariant under the dihedral group of order 10. As a by-product we obtain 25 new residual 2-(45,12,4) designs. The automorphism groups of all new designs are computed. 1. Introduction. Let V = {0, 1, . . . , v -1} be a finite set of elements called points, and let B = {B 0 , B 1 , . . . , B b-1 } be a collection of k-element subsets of V called blocks. The incidence structure D = (V, B) is a 2 -(v, k, λ) design (or BIB (v, k, λ) design) if every unordered pair of points is contained in exactly λ blocks of B. Each point from the point set of a 2-design is contained in a constant number of blocks. This number is usually denoted by r. Obviously, λ(v -1) = r(k -1), λv(v -1) = bk(k -1).

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra QG for G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in QG, we describe a subgroup of finite index in the group of units U (ZG) of the integral group ring ZG that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.

Let G be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of G. This resolution is used to define representations of groups which act compatibly on G, generalizing classical constructions of Magnus, Burau, and Gassner. Our construction also yields algorithms for computing the homology of the Milnor fiber of a fibertype hyperplane arrangement, and more generally, the homology of the complement of such an arrangement with coefficients in an arbitrary local system.

This paper suggests the method to investigate the properties of semifield projective planes. The coordinatizing set of such plane is a semifield, or division ring. The features of coordinatizing set allow to consider semifield plane as a subclass of translation planes, and so allow to use a vector space of even dimension and the certain family of linear transformations (regular set, spread set) to construct semifield planes [1, 2]. A number of papers ([3, 4] and others), devoted to construction and investigation of translation planes of rank 2, was published in 1985–1995 years. Translation plane of rank 2 can be determined by four-dimension vector space and 2 × 2-matrices over the finite field of order p. However the functions which define the spread set are the polinoms of p-th degree (in a common case) and therefore the construction of such a plane of great order is so complicated even for small rank. The method that is presented by this paper is based on a consideration of a spre...

I introduce φHash, a novel one-way hash function defined entirely in the categorical framework of Alpay Algebra. In this setting, each input (finite or infinite sequence) is encoded as an object in a small cartesian-closed category A with a distinguished initial state I, and a transfinite evolution functor φ : A → A. Repeated application of φ with inputdependent updates yields a unique fixed-point object φ ∞ (I) that encapsulates the entire input data. The hash digest is then obtained by a universal φ-algebra "fold" (projection) onto an n-bit object in A. I formalize this construction axiomatically, prove determinism and one-wayness, and show that φHash resists both classical and quantum brute-force attacks. As illustrations, I symbolically compute φHash-256, φHash-512, and φHash-1024 of the word "alpay" within Alpay Algebra, verifying repeatability and irreversibility. Finally, I explain how φHash extends to arbitrary output lengths (including φHash-∞) via universal fold operators. Throughout, I rely solely on Faruk Alpay's foundational axioms and the SHA-256 standard for contrast.

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm.

We analyze the structure of finite commutative rings with respect to its idempotent and nilpotent elements. Based on this analysis we provide a quantum-classical IND-CCA attack for ring homomorphic encryption schemes. Moreover, when the plaintext space is a finite reduced ring, i.e. a product of finite fields, we present a key-recovery attack based on representation problem in black-box finite fields. In particular, if the ciphertext space has smooth characteristic the key-recovery attack is effectively computable. We also extend the work of Maurer and Raub on representation problem in black-box finite fields to the case of a black-box product of finite fields of equal characteristic.

In this paper we consider a partial action α of a polycyclic by finite group G on a ring R. We prove that if R is right noetherian, then the partial skew group ring R α G is also right noetherian. Extending the methods of Passman in Passman (Trans Am Math Soc 301:737-759, 1987), we obtain a description of the prime spectrum of R α G. The results obtained are applied to get bounds for the Krull dimension and the classical Krull dimension of R α G. Partial actions • Partial skew group rings • Noetherian rings • Prime ideals Mathematics Subject Classifications (2000) 16D25 • 16P40 • 16S35 • 16W22 Presented by Donald Passman.

This paper deals with Moufang-Klingenberg planes M (A) defined over a local alternative ring A of dual numbers. The definition of cross-ratio is extended to M (A). Also, some properties of cross-ratios and 6-figures that are well-known for Desarguesian planes are investigated in M (A); so we obtain relations between algebraic properties of A and geometric properties of M (A). In particular, we show that pairwise non-neighbour four points of the line g are in harmonic position if and only if they are harmonic, and that µ is Menelaus or Ceva 6-figure if and only if r (µ) = -1 or r (µ) = 1, respectively.

Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let S X be the group of permutations of X. The group S X acts on R in a natural way, and this in turn gives R the structure of a left module over the group ring R[S X ]. We prove that all ideals of R invariant under the action of S X are finitely generated as R[S X ]-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let S X be the group of permutations of X. The group S X acts on R in a natural way, and this in turn gives R the structure of a left module over the group ring R[S X ]. We prove that all ideals of R invariant under the action of S X are finitely generated as R[S X ]-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

𝑝 𝑘 ,+𝑝 𝑗 2 ,𝑝 𝑘 𝑝 𝑗 , where 𝑝 𝑘 is the SLPF and 𝑝 𝑗 ≥𝑝 𝑘 , we generate a family of smooth curves. Each curve corresponds to a unique SLPF and encodes all semiprimes sharing that factor. This visualization allows for the enumeration of Goldbach partitions for any even number 2𝑚 , since the number of semiprimes with arithmetic mean 𝑚 directly equates to the number of such partitions. This method offers a continuous, structural approach to understanding Goldbach pairs and highlights deeper relationships between arithmetic means and semiprime structures.

The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. As an application, for each cyclic cocycle of at most exponential growth, we prove a formal higher Atiyah-Patodi-Singer index theorem on manifolds with boundary, under the condition that the operator on the boundary has a sufficiently large spectral gap at zero. Our second main result is to obtain an explicit formula of the delocalized Connes-Chern character of all C * -algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between C * -algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the C * -algebraic secondary invariant is a K-theoretic higher rho invariant of an invertible differential operator, we show this pairing is precisely the higher analogue of Lott's delocalized eta invariant alluded to above. Our work uses Puschnigg's smooth dense subalgebra for word hyperbolic groups in an essential way. We emphasize that our construction of the delocalized Connes-Chern character is at C * -algebra K-theory level. This is of essential importance for applications to geometry and topology. As a consequence, we compute the paring between delocalized cyclic cocycles and C * -algebraic Atiyah-Patodi-Singer index classes for manifolds with boundary, when the fundamental group of the given manifold is hyperbolic. In particular, this improves the formal delocalized higher Atiyah-Patodi-Singer theorem from above and removes the condition that the spectral gap of the operator on the boundary is sufficiently large.

The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. Our second main result is to obtain an explicit formula of the delocalized Connes-Chern character of all $C^\ast$-algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between $C^\ast$-algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the $C^\ast$-algebraic secondary invariant is a $K$-theoretic higher rho invariant of an invertible differential operator, we show this pairing is precisely the higher analogue of Lott's delocalized eta invariant alluded to above. Our work uses Pusch...

A module M is said to be modest if the injectivity domain of M is the class of all crumbling modules. In this paper, we investigate the basic properties of modest modules. We provide characterizations of some classes of rings using modest modules. In particular, we show that a ring having the class of crumbling modules as the only right middle class of injectivity domains is either a right V -ring or right Noetherian; and a commutative ring with this property is regular. We also give criteria for a ring having the class of crumbling modules as the only right middle class of injectivity domains.

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring ZG of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center Z(U (ZG)) of the unit group U (ZG) in case G is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in Z(U (ZG)) for all finite strongly monomial groups G. We call these units generalized Bass units. Finally, we show that the commutator group U (ZG)/U (ZG) ′ and Z(U (ZG)) have the same rank if G is a finite group such that QG has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra, or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of Q. This allows us to prove that in this case the natural images of the Bass units of ZG generate a subgroup of finite index in U (ZG)/U (ZG) ′ .

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups J 1 , J 2 and J 3 is the same as that of the normalized unit group of their respective integral group ring.