Complete Set Of Primitive Idempotents Research Articles

A decomposition of the group algebraof a hyperoctahedral group

The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafe asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ⊆ J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.

Primitive idempotents in group algebras and minimal abelian codes

In this paper, the minimal abelian codes for several classes of non-cyclic abelian groups are constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Furthermore, we acquire the minimum Hamming distances and the dimensions of these minimal abelian codes in the group algebra \({\mathbb {F}}_q(G_{\ell _1} \, \times \, G_{\ell _2})\), where \(G_{\ell _1} \, \times \, G_{\ell _2}\) is the direct product of two cyclic groups and \(\ell _1, \, \ell _2\) are prime divisors of \(q-1\).

Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.

SEMICENTRAL IDEMPOTENTS IN A RING

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and <TEX>$S_{\ell}$</TEX>(R) (resp. <TEX>$S_r$</TEX>(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) <TEX>$e{\in}S_{\ell}(R)$</TEX> (resp. <TEX>$e{\in}S_r(R)$</TEX>) if and only if re=ere (resp. er=ere) for all nilpotent elements <TEX>$r{\in}R$</TEX> if and only if <TEX>$fe{\in}I(R)$</TEX> (resp. <TEX>$ef{\in}I(R)$</TEX>) for all <TEX>$f{\in}I(R)$</TEX> if and only if fe=efe (resp. ef=efe) for all <TEX>$f{\in}I(R)$</TEX> if and only if fe=efe (resp. ef=efe) for all <TEX>$f{\in}I(R)$</TEX> which are isomorphic to e if and only if <TEX>$(fe)^n=(efe)^n$</TEX> (resp. <TEX>$(ef)^n=(efe)^n$</TEX>) for all <TEX>$f{\in}I(R)$</TEX> which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any <TEX>$0{\neq}e{\in}S_{\ell}(R)$</TEX> (resp. 0<TEX>$0{\neq}e{\in}S_r(R)$</TEX>).

The Coarse Structure of the Representation Algebra of a Finite Monoid

Let M be a monoid, and let L be a commutative idempotent submonoid. We show that we can find a complete set of orthogonal idempotents L^0 of the monoid algebra A of M such that there is a basis of A adapted to this set of idempotents which is in one-to-one correspondence with elements of the monoid. The basis graph describing the Peirce decomposition with respect to L^0 gives a coarse structure of the algebra, of which any complete set of primitive idempotents gives a refinement, and we give some criterion for this coarse structure to actually be a fine structure, which means that the nonzero elements of the monoid are in one-to-one correspondence with the vertices and arrows of the basis graph with respect to a set of primitive idempotents, with this basis graph being a canonical object.

Explicit determination of certain minimal constabelian codes

Constabelian codes can be viewed as ideals in twisted group algebras over finite fields. In this paper we study decomposition of semisimple twisted group algebras of finite abelian groups and prove results regarding complete determination of a full set of primitive orthogonal idempotents in such algebras. We also explicitly determine complete sets of primitive orthogonal idempotents of twisted group algebras of finite cyclic and abelian p-groups. We also describe methods of determining complete set of primitive idempotents of abelian groups whose orders are divisible by more than one prime and give concrete (numerical) examples of minimal constabelian codes, illustrating the above mentioned results.

Additive Set of Idempotents in Rings

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f ∈ I(R) (e ≠ f), e + f ∈ I(R), and M(R) is additive in I(R) if for all e, f ∈ M(R)(e ≠ f), e + f ∈ I(R). In this article, the following points are shown: (1) I(R) is additive if and only if I(R) is multiplicative and the characteristic of R is 2; M(R) is additive in I(R) if and only if M(R) is orthogonal. If 0 ≠ ef ∈ I(R) for some e ∈ M(R) and f ∈ I(R), then ef ∈ M(R), (2) If R has a complete set of primitive idempotents, then R is a finite product of connected rings if and only if I(R) is multiplicative if and only if M(R) is additive in I(R).

EXPLICIT DETERMINATION OF CERTAIN MINIMAL ABELIAN CODES AND THEIR MINIMUM DISTANCES

In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.

Matrix units and generic degrees for the Ariki–Koike algebras

This paper uses the Murphy basis of the Ariki–Koike algebras to explicitly construct a complete set of primitive idempotents when these algebras are semisimple and q ≠ 1 . As a consequence, we obtain an explicit Wedderburn basis for the Ariki–Koike algebras. Finally, we use these idempotents to compute the generic degrees of the Ariki–Koike algebras.

On the structure of nonsemisimple Hopf algebras

The left module structure of finite-dimensional quantum algebras is analysed using the theory of primitive idempotents. In particular, a complete structural result in terms of principal (projective) indecomposable modules (p.i.m.) is given in the case of (q a root of 1) by finding a complete set of primitive idempotents. The structure of p.i.m. is analysed in detail. The Jacobson radical of the algebra is investigated and its significance in the study of nonsemisimple symmetries in physical systems is discussed.

String-like lattice models and Hecke algebras

The author discusses lattice statistical mechanical models with n-dimensional (static endpoint) string-like degrees of freedom. In the infinite string tension limit these models give representations of Hecke algebras Hk(q), quotient algebras of which may be used to construct Virasoro algebras in the limit k to infinity and when q=4 cos2( pi /r) (r epsilon z). They give the regular representation and well defined representations associated with and containing each irreducible representation, including non-unitarisable cases. They use these models to obtain a complete set of primitive idempotents, and hence the central idempotents, of the generic algebra.