Set Of Orthogonal Primitive Idempotents Research Articles

Generalized Burnside algebra of type B_{n}

In this paper, we firstly give an alternative method to determine the size of $C(S_{n})$ which is the set of elements of type $S_{n}$ in a finite Coxeter system $(W_{n},S_{n})$ of type $B_{n}$. We also show that all cuspidal classes of $W_{n}$ are actually the conjugate classes $\mathcal{K}_{\lambda}$ for every $\lambda \in \mathcal{DP}^{+}(n)$. We then define the generalized Burnside algebra $HB(W_{n})$ for $W_{n}$ and construct a surjective algebra morphism between $HB(W_{n})$ and Mantaci-Reutenauer algebra $\mathcal{MR}(W_{n})$. We obtain a set of orthogonal primitive idempotents $e_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $HB(W_{n})$, that is, all the characteristic class functions of $W_{n}$. Finally, we give an effective formula to compute the number of elements of all the conjugate classes $\mathcal{K}_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $W_{n}$.

Annihilator Conditions related to the quasi-Baer Condition

We call a ring R an EGE-ring if for each $I \leq R$, which is generated by a subset of right semicentral idempotents there exists an idempotent $e$ such that $r(I) = eR$. The class EGE includes quasi-Baer, semiperfect rings (hence all local rings) and rings with a complete set of orthogonal primitive idempotents (hence all Noetherian rings) and is closed under direct product, full and upper triangular matrix rings, polynomial extensions (including formal power series, Laurent polynomials, and Laurent series) and is Morita invariant. Also we call $R$ an AE-ring if for each $I \unlhd R$, there exists a subset $S \subseteq S_{r}(R)$ such that $r(I) = r(RSR)$. The class AE includes the principally quasi-Baer ring and is closed under direct products, full and upper triangular matrix rings and is Morita invariant. For a semiprime ring $R$, it is shown that $R$ is an EGE (resp., AE)-ring if and only if the closure of any union of clopen subsets of $Spec(R)$ is open (resp., $Spec(R)$ is an EZ-space).

Reduced representations of rooted trees

Let k be a field and Q a quiver. The category of finite dimensional representations of Q will be denoted by repk(Q). We can define the pointwise tensor product on the category repk(Q), i.e. for two representations N, M the representation N⊗M is defined at a vertex x∈Q0 as (N⊗M)x=Nx⊗Mx and for an arrow α∈Q1 we define (N⊗M)α=Nα⊗Mα. This tensor product has been investigated in recent work of Herschend and Kinser [1–3]. In [3] Kinser investigated the representation ring of rooted trees. A rooted tree (Q,σ) is a quiver Q, whose underlying graph is tree and which has a unique sink σ. The representation ring of Q is the free abelian group generated by the isomorphism classes of indecomposable objects in repk(Q). It is endowed with a multiplication induced by the pointwise tensor product. See [3] for details. For these rooted trees, Kinser constructed reduced representations, which are defined inductively. These reduced representations then lead to a set of orthogonal primitive idempotents in the representation ring of Q. On the other hand, there is an inductive construction of so-called radiation modules for a tree with arbitrary orientation in [5]. In the special case of a rooted tree quiver, this construction results in the reduced representations introduced by Kinser. The idea used in the construction of the radiation modules then allows us to prove a new characterization of reduced representations which isTheoremLet Q be a rooted tree quiver. An indecomposable representationV∈repk(Q)is reduced if and only if V is a direct summand ofV⊗V.This theorem then allows us to prove further results for the special case, where Q is a rooted tree quiver, whose underlying graph is of Dynkin type. In that case V being reduced is also equivalent to the condition that the vector space of V at the sink σ is one dimensional.

Construction of minimal non-abelian left group codes

Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra \({\mathbb F}G\) for a large class of groups \(G\). As an illustration of our methods, alternative constructions to some best linear codes over \({\mathbb F}_2\) and \({\mathbb F}_3\) are given. Furthermore, we give constructions of non-abelian left group codes.

Group rings of finite strongly monomial groups: Central units and primitive idempotents

We compute the rank of the group of central units in the integral group ring ZG of a finite strongly monomial group G. The formula obtained is in terms of the strong Shoda pairs of G. Next we construct a virtual basis of the group of central units of ZG for a class of groups G properly contained in the finite strongly monomial groups. Furthermore, for another class of groups G inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of QG.Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZG, this for metacyclic groups G of the form G=Cqm⋊Cpn with p and q different primes and the cyclic group Cpn of order pn acting faithfully on the cyclic group Cqm of order qm.

Rational Group Algebras of Finite Groups: From Idempotents to Units of Integral Group Rings

We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring ${\mathbb Z} G$ of a finite nilpotent group G has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups (H, K), called strong Shoda pairs, and explicit constructed central elements e(G, H, K) play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such e(G, H, K), with (H, K) strong Shoda pairs in subgroups of G.

Finite group algebras of nilpotent groups: A complete set of orthogonal primitive idempotents

We provide an explicit construction for a complete set of orthogonal primitive idempotents of finite group algebras over nilpotent groups. Furthermore, we give a complete set of matrix units in each simple epimorphic image of a finite group algebra of a nilpotent group.

Endomorphism rings of permutation modules over maximal Young subgroups

Let K be a field of characteristic two, and let λ be a two-part partition of some natural number r. Denote the permutation module corresponding to the (maximal) Young subgroup Σ λ in Σ r by M λ . We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra S K ( λ ) = 1 λ S K ( 2 , r ) 1 λ = End K Σ r ( M λ ) of the Schur algebra S K ( 2 , r ) . These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.

Efficient Decomposition of Associative Algebras over Finite Fields

We present new, efficient algorithms for some fundamental computations with finite-dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra A we show how to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of A. If A is given by a generating set of matrices inFm×m, then our algorithm requires aboutO (m3) operations inF, in addition to the cost of factoring a polynomial inF[ x ] of degree O(m), and the cost of generating a small number of random elements from A. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a finite field in this same time.