Primitive Central Idempotents Research Articles

The enumeration of equivalent classes of minimal general dihedral group codes

A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.

Structure of group rings and the group of units of integral group rings: an invitation

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $$\mathcal {U}(\mathbb {Z}G)$$ of the integral group ring $$\mathbb {Z}G$$ of a finite group G. These constructions rely on explicit constructions of units in $$\mathbb {Z}G$$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $$\mathbb {Q}G$$ . The latter relies on explicit constructions of primitive central idempotents and the rational representations of G. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.

Normal Supercharacter Theory

There are three main constructions of supercharacter theories for a group G. The first, defined by Diaconis and Isaacs, comes from the action of a group A via automorphisms on our given group G. Another general way to construct a supercharacter theory for G, defined by Diaconis and Isaacs, uses the action of a group A of automor- phisms of the cyclotomic field Q[ζ|G|]. The third, defined by Hendrickson, is combining a supercharacter theories of a normal subgroup N of G with a supercharacter theory of G/N . In this paper we construct a supercharacter theory from an arbitrary set of normal subgroups of G. We show that when we consider the set of all normal subgroups of G, the corresponding supercharacter theory is related to a partition of G given by certain values on the central primitive idempotents. Also, we show the supercharacter theories that we construct can not be obtained via automorphisms or a single normal subgroup.

Структура конечной групповой алгебры одного полупрямого произведения абелевых групп и её приложения

В 1978 году Р. Мак-Элисом построена первая асимметричная кодовая криптосистема, основанная на применении помехоустойчивых кодов Гоппы, при этом эффективные атаки на секретный ключ этой криптосистемы до сих пор не найдены. К настоящему времени известно много криптосистем, основанных на теории помехоустойчивого кодирования. Одним из способов построения таких криптосистем является модификация криптосистемы Мак-Элиса с помощью замены кодов Гоппы на другие классы кодов. Однако, известно что криптографическая стойкость многих таких модификаций уступает стойкости классической криптосистемы Мак-Элиса. В связи с развитием квантовых вычислений кодовые криптосистемы, наряду с криптосистемамми на решётках, рассматриваются как альтернатива теоретико-числовым. Поэтому актуальна задача поиска перспективных классов кодов, применимых в криптографии. Представляется, что для этого можно использовать некоммутативные групповые коды, т.е. левые идеалы в конечных некоммутативных групповых алгебрах.Для исследования некоммутативных групповых кодов полезной является теорема Веддерберна, доказывающая существование изоморфизма групповой алгебры на прямую сумму матричных алгебр. Однако конкретный вид слагаемых и конструкция изоморфизма этой теоремой не определены, и поэтому для каждой группы стоит задача конструктивного описания разложения Веддерберна. Это разложение позволяет легко получить все левые идеалы групповой алгебры, т.е. групповые коды. В работе рассматривается полупрямое произведение $$Q_{m,n} = (\mathbb{Z}_m \times \mathbb{Z}_n) \leftthreetimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$$ абелевых групп и конечная групповая алгебра $$\mathbb{F}_q Q_{m,n}$$ этой группы. Для этой алгебры при условиях $$n \mid q -1$$ и $$\text{НОД}(2mn, q) = 1$$ построено разложение Веддербёрна. В случае поля чётной характеристики, когда эта групповая алгебра не является полупростой, также получена сходная структурная теорема. Описаны все неразложимые центральные идемпотенты этой групповой алгебры. Полученные результаты используются для алгебраического описания всех групповых кодов над $$Q_{m,n}.$$

Endomorphism algebras of 2-row permutation modules in characteristic 3

Given r∈N, let λ be a partition of r with at most two parts. Let F be a field of characteristic 3. Write Mλ for the FSr-permutation module corresponding to the action of the symmetric group Sr on the cosets of the maximal Young subgroup Sλ. We construct a full set of central primitive idempotents in EndFSr(Mλ) in this case. We also determine the Young module corresponding to each primitive idempotent that we construct.

Orthogonal Bases of Involution in Hadamard Algebras

The notion of a Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrices, which correspond to the case of commutative algebras. Algebras admitting a Hadamard decomposition are said to be Hadamard. Images of orthogonal bases of involution in Hadamard algebras under the canonical projections of these algebras onto their simple components are studied. Using a technique related to the study of central primitive idempotents of Hadamard algebras, we obtain a necessary condition for a family of involutory matrices of fixed order to be such an image. It is also shown that this necessary condition is not sufficient. We also present new proofs of results proved earlier.

Varietal Terwilliger algebras arising from wreath products of rank two association schemes

As abstractions of Terwilliger algebras, generalized Terwilliger algebras and varietal Terwilliger algebras have been studied in [5,6,17]. In general, it is difficult to determine the structure of a varietal Terwilliger algebra. As proposed in the concluding remarks in [4], in this paper we study the structures of those varietal Terwilliger algebras arising from the wreath products of rank two association schemes. In particular, we give the explicit formulas for their central primitive idempotents. Applications to Terwilliger algebras of association schemes are also discussed, and an affirmative answer to the Terwilliger conjecture mention in [4] is given.

On central idempotents in the Brauer algebra

We provide a method for constructing central idempotents in the Brauer algebra (using the splitting of short exact sequences of bimodules). From this we determine certain primitive central idempotents. By working over a suitable integral ring we hence demonstrate an efficient method of constructing pieces of the representation theory of the Brauer algebra over Artinian rings from the integral case.

Idempotents and codes in some non semi-simple metacyclic group algebras

We consider group algebra of a metacyclic group of order pnq over a field of characteristic p, where p and q are distinct prime numbers. We determine the terms of the primary decomposition of this algebra. The central primitive idempotents are also calculated. In the end we give the characteristics (dimension and minimal weight) of codes generated by these idempotents.

Character triples and Shoda pairs

In this paper, a construction of Shoda pairs using character triples is given for a large class of monomial groups including abelian-by-supersolvable and subnormally monomial groups. The computation of primitive central idempotents and the structure of simple components of the rational group algebra for groups in this class are also discussed. The theory is illustrated with examples.

Pseudo-direct sums and wreath products of loose-coherent algebras with applications to coherent configurations

We introduce the notion of a loose-coherent algebra, which is a special semisimple subalgebra of the matrix algebra, and define two operations to obtain new loose-coherent algebras from the old ones: the pseudo-direct sum and the wreath product. For two arbitrary coherent configurations C, D and their wreath product C≀D, it is difficult to express the Terwilliger algebra T(x,y)(C≀D) in terms of the Terwilliger algebras Tx(C) and Ty(D). By using the concept and operations of loose-coherent algebras, we find a very simple such expression. As a direct consequence of this expression, we obtain the central primitive idempotents of T(x,y)(C≀D) in terms of the central primitive idempotents of Tx(C) and Ty(D). Many results in [4,6,10,12] are special cases of the results in this paper.

Finite Metabelian Group Algebras

Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of order p2, p odd prime, the objective of this paper is to obtain a complete algebraic structure of semisimple group algebra Fq[G] in terms of primitive central idempotents, Wedderburn decomposition and the automorphism group.

Normal supercharacter theories and their supercharacters

In order to find a tractable theory to substitute for the wild character theory of the group of n×n unipotent upper-triangular matrices over a finite field Fq, André and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set S of normal subgroups of G. We call such supercharacter theory the normal supercharacter theory generated by S. It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of G are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, we argue that normal supercharacter theories cannot be obtained by previously known supercharacter theory constructions.

Group algebras of metacyclic groups over finite fields

In this paper, we consider semisimple group algebras \({\mathbb {F}}_{q}G\) of non abelian split metacyclic groups over a finite field. We give conditions for them to have a minimal number of simple components and find the primitive central idempotents of \({\mathbb {F}}_{q}G\) in the case when the order G is equals \(p^{m}\ell ^{n}\), where p and \(\ell \) are different prime numbers.

Center of the Schur algebra

We describe a basis of the center of the Schur algebra that comes from conjugacy classes in the symmetric group via Schur–Weyl duality. We give a combinatorial description of expansions of these basis elements in terms of the basis originally used by Schur. The primitive central idempotents of the Schur algebra can be written down using this basis and the character table of the symmetric group in the semisimple case. Along the way we prove a result on the non-singularity of the submatrix of the character table matrix of a symmetric group obtained by taking rows and columns indexed by partitions with at most [Formula: see text] parts for any [Formula: see text].

Terwilliger algebras of wreath products of association schemes

The Terwilliger algebra of an association scheme of order n introduced in [13] is a subalgebra of the matrix algebra of all n×n matrices. Terwilliger algebras of wreath products of special association schemes are studied in several papers. In this paper we study the Terwilliger algebra of the wreath product T≀S of two arbitrary association schemes S and T. We will express the Terwilliger algebra of T≀S and its primitive central idempotents in terms of the Terwilliger algebras of S and T and their primitive central idempotents. The known results of Hanaki, Kim, etc. (cf. [7,10]) are special cases of our results.

Extremely strong Shoda pairs with GAP

We provide algorithms to compute a complete irredundant set of extremely strong Shoda pairs of a finite group G and the set of primitive central idempotents of the rational group algebra Q[G] realized by them. These algorithms are also extended to write new algorithms for computing a complete irredundant set of strong Shoda pairs of G and the set of primitive central idempotents of Q[G] realized by them. Another algorithm to check whether a finite group G is normally monomial or not is also described.

On Idempotents and the Number of Simple Components of Semisimple Group Algebras

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly monomial groups. Further, for this class of group algebras, we describe when the number of simple components agrees with the number of simple components of the rational group algebra. Finally, we give a formula for the rank of the central units of the group ring over the ring of integers of a number field for a strongly monomial group.

The Algebraic Structure of Finite Metabelian Group Algebras

An algorithm for the explicit computation of a complete set of primitive central idempotents, Wedderburn decomposition, and the automorphism group of the semisimple group algebra of a finite metabelian group is developed. The algorithm is illustrated with its application to the semisimple group algebra of an arbitrary metacyclic group, and certain indecomposable groups whose central quotient is the Klein four-group.

Equivalences between blocks of cohomological Mackey algebras

Let \(G\) be a finite group and \((K,\mathcal {O},k)\) be a \(p\)-modular system which is large enough. Let \(R=\mathcal {O}\) or \(k\). There is a bijection between the blocks of the group algebra \(RG\) and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra \(co\mu _{R}(G)\). Here, we introduce the notion of permeable derived equivalence and we prove that a permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Broué’s abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.