Quadratic forms in characteristic 2 and a Wedderburn decomposition over rationals

2019 ◽  
Vol 18 (01) ◽  
pp. 1950010
Author(s):  
Dilpreet Kaur ◽  
Amit Kulshrestha
Keyword(s):  
Group Algebra ◽  
Quadratic Forms ◽  
Rational Group ◽  
Interesting Application ◽  
Special Formula ◽  
Wedderburn Decomposition ◽  
Characteristic 2

Let [Formula: see text] be a field of characteristic 2. In this paper, we provide an interesting application of quadratic forms over [Formula: see text] in determination of the Wedderburn decomposition of the rational group algebra [Formula: see text], where [Formula: see text] is a real special [Formula: see text]-group. We further apply these computations to exhibit two non-isomorphic real special [Formula: see text]-groups with isomorphic rational group algebra.

THE RATIONAL GROUP ALGEBRA OF A FINITE GROUP

2012 ◽  
Vol 12 (03) ◽  
pp. 1250168 ◽  
Author(s):  
GURMEET K. BAKSHI ◽  
RAVINDRA S. KULKARNI ◽  
INDER BIR S. PASSI
Keyword(s):  
Finite Group ◽  
Explicit Expression ◽  
Group Algebra ◽  
Irreducible Character ◽  
Metabelian Group ◽  
Central Idempotent ◽  
Rational Group ◽  
Wedderburn Decomposition ◽  
Complete Set ◽  
A Finite Group

An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] of a finite group G associated with any complex irreducible character of G is obtained. A complete set of primitive central idempotents and the Wedderburn decomposition of the rational group algebra of a finite metabelian group is also computed.

UNITS OF THE GROUP ALGEBRA 𝔽2κ(C2 × D8)

2011 ◽  
Vol 10 (04) ◽  
pp. 643-647 ◽  
Author(s):  
JOE GILDEA
Keyword(s):  
Group Algebra ◽  
Unit Group ◽  
Cyclic Groups ◽  
Split Extensions ◽  
Characteristic 2

The structure of the unit group of the group algebra of the group C2 × D8 over any field of characteristic 2 is established in terms of split extensions of cyclic groups.

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

On the Garsia Lie Idempotent

2005 ◽  
Vol 48 (3) ◽  
pp. 445-454 ◽  
Author(s):  
Frédéric Patras ◽  
Christophe Reutenauer ◽  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Group Algebra ◽  
Orthogonal Projection ◽  
Gram Matrix ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Rational Group ◽  
Lie Idempotent

AbstractThe orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group Sn, in each homogenous degree n. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of Sn−1. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in Sn.

Central Idempotents and Units in Rational Group Algebras of Alternating Groups

1998 ◽  
Vol 08 (04) ◽  
pp. 467-477 ◽  
Author(s):  
A. Giambruno ◽  
E. Jespers
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Explicit Description ◽  
Alternating Group ◽  
Group Algebras ◽  
Young Tableaux ◽  
Rational Group ◽  
Alternating Groups ◽  
Group Of Units

Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.

A family of crystallographic groups with 2-torsion in K0 of the rational group algebra

1991 ◽  
Vol 34 (2) ◽  
pp. 325-331 ◽  
Author(s):  
P. H. Kropholler ◽  
B. Moselle
Keyword(s):  
Group Algebra ◽  
Euler Class ◽  
Crystallographic Group ◽  
Rational Group ◽  
Crystallographic Groups

We calculate K0 of the rational group algebra of a certain crystallographic group, showing that it contains an element of order 2. We show that this element is the Euler class, and use our calculation to produce a whole family of groups with Euler class of order 2.

CONSTRUCTION OF SELF-DUAL RADICAL 2-CODES OF GIVEN DISTANCE

2012 ◽  
Vol 04 (04) ◽  
pp. 1250052
Author(s):  
CAROLIN HANNUSCH ◽  
PIROSKA LAKATOS
Keyword(s):  
Abelian Group ◽  
Group Algebra ◽  
Abelian Groups ◽  
Dual Code ◽  
Decomposition Formula ◽  
Construction Of Self ◽  
Characteristic 2

We prove that for arbitrary n ∈ ℕ and [Formula: see text] and for a field K of characteristic 2 there exists an abelian group G of order 2n such that one of the powers of the radical of the group algebra K[G] is a (2n, 2n-1, 2d)-self-dual code. These codes are constructed for abelian groups G with decomposition [Formula: see text] where a1 ≥ 3 and si ≥ 0(1 ≤ i ≤ 3).

Invertible powers of ideals over orders in commutative separable alǵebras

1970 ◽  
Vol 67 (2) ◽  
pp. 237-242 ◽  
Author(s):  
Michael Singer
Keyword(s):  
Group Algebra ◽  
Sylow Subgroup ◽  
Basic Result ◽  
Rational Group ◽  
Powers Of Ideals ◽  
Dedekind Domains ◽  
Result Theorem ◽  
Quantitative Results ◽  
Separable Algebras ◽  
Special Case

The purpose of this paper is to obtain quantitative results on invertible powers of (fractional) ideals in commutative separable algebras over dedekind domains. This is connected with the work of Dade, Taussky and Zassenhaus(2) on ideals in noetherian domains. We do not, however, make use of their paper, but rather draw on the general theorems on ideals in commutative separable algebras established by Fröhlich (3), in particular his qualitative result that some power of any given ideal is invertible. Our basic result (Theorem 1) concerns the invertibility of powers of a particular type of ideal, the componentwise dedekind ideals defined below. From this we deduce a general result (Theorem 2), which includes as a special case Theorem C of (2) for the case of separable field extensions; specifically, the (n – 1)th power of any ideal is invertible, where n is the dimension of the algebra. Although, as we show, it is possible to deduce Theorem 2 from (2), we have here an independent proof of one of the main results of (2) based entirely on the results in (3). As a further application of Theorem 1 we obtain a new result on ideals over the group ring of an abelian group over the ring of rational integers; the (t – 1)th power of such an ideal is invertible, where t is the maximum number of simple components of the rational group algebra of any Sylow subgroup. We also show that this is the best possible result when some Sylow subgroup whose rational group algebra has t components is cyclic.

Sign in / Sign up

Export Citation Format

Share Document