It is a classical result in matrix algebra that any square matrix over a field can be conjugated to its transpose by a symmetric matrix. For
F
a non-Archimedean local field, Tupan used this to give ...an elementary proof that transpose inverse takes each irreducible smooth representation of
GL
n
(
F
)
to its dual. We re-prove the matrix result and related observations using module-theoretic arguments. In addition, we write down a generalization that applies to central simple algebras with an involution of the first kind. We use this generalization to extend Tupan’s method of argument to
GL
n
(
D
)
for
D
a quaternion division algebra over
F
.
Let
RG
be the group ring of a finite group
G
over a commutative ring
R
with 1. An element
x
in
RG
is said to be skew-symmetric with respect to an involution
σ
of
RG
if
σ
(
x
)
=
-
x
.
A structure ...theorem for the Lie algebra of skew-symmetric elements of
FG
is given where
F
is an algebraic extension of
Q
which generalizes some previously known results in this direction.
Let k be a field, K/k be a quadratic separable field extension, and a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent ...henselian local domain with an algebraically closed residue field of characteristic zero and the degree of is odd, we prove that all K/k-involutions on are cyclic.
In this article, the first general constructions of fast-decodable, more specifically (conditionally) g-group decodable, space-time block codes for the nonorthogonal amplify and forward (NAF) ...multiple-input multiple-output (MIMO) relay channel under the half-duplex constraint are proposed. In this scenario, the source and the intermediate relays used for data amplification are allowed to employ multiple antennas for data transmission and reception. The worst-case decoding complexity of the obtained codes is reduced by up to 75%. In addition to being fast-decodable, the proposed codes achieve full-diversity and have nonvanishing determinants, which has been shown to be useful for achieving the optimal diversity-multiplexing tradeoff (DMT) of the NAF channel. Furthermore, it is shown that the same techniques as in the cooperative scenario can be utilized to achieve fast-decodability for K-user MIMO multiple-access channel (MAC) space-time block codes. The resulting codes in addition exhibit the conditional nonvanishing determinant property which, for its part, has been shown to be useful for achieving the optimal MAC-DMT.
We prove that if two division p-algebras of prime degree share an
inseparable field extension of the center then they also share a
cyclic separable one. We show that the converse is in general not
...true. We also point out that sharing all the inseparable field
extensions of the center does not imply sharing all the cyclic
separable ones.
In this paper we show that a split central simple algebra with quadratic pair which decomposes into a tensor product of quaternion algebras with involution and a quaternion algebra with quadratic ...pair is adjoint to a quadratic Pfister form. This result is new in characteristic two, otherwise it is equivalent to the Pfister Factor Conjecture proven in Becher (Invent Math 173:1–6,
2008
).
This article combines two independent theories: firstly, the algorithm of Faddeev–Leverrier which calculates characteristic polynomials of matrices; secondly, the Descent Theory which, in particular, ...lets many properties of matrix algebras descend down to Azumaya algebras, especially the characteristic polynomials. The algorithm of Faddeev–Leverrier is completely revisited. The details of the descent are explained as far as they are needed for Clifford algebras over fields.
A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders ...representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.
In this article, we present a decomposition theorem for unitary spaces over a central simple division algebra with involution in characteristic 2. This is a generalization of a decomposition result ...for quadratic forms in characteristic 2 from
4
and extends a generalization of the Witt decomposition theorem for nonsingular spaces in this setting to cover spaces that may be singular.
Soient F un corps de caractéristique 2 et T(F) l'ensemble des F-formes quadratiques totalement singulières (à isométrie près). Le uˆ-invariant de F est la dimension maximale d'une F-forme quadratique ...anisotrope. Dans cet article, on introduit de nouveaux invariants de F liés au uˆ-invariant. Le premier d'entre eux, appelé u˜-invariant, est la dimension maximale d'une F-forme quadratique anisotrope non totalement singulière. La motivation d'introduire cet invariant est de savoir comment les valeurs du uˆ-invariant peuvent être données sans l'utilisation des formes de T(F), ceci puisque la majorité des valeurs connues du uˆ-invariant est réalisée par les formes de T(F) (voir 12). Aussi, pour r,s≥1 des entiers, on introduit le ur-invariant (resp. le u˜s-invariant) qui est la dimension maximale d'une F-forme quadratique anisotrope de partie régulière de dimension 2r (resp. la dimension maximale d'une F-forme quadratique anisotrope non totalement singulière de partie quasi-linéaire de dimension s). On fera une comparaison entre ces nouveaux invariants, puis on donnera une liste de valeurs qu'ils peuvent prendre, en faisant le lien avec le u-invariant et le uˆ-invariant. Aussi, on discutera la question classique, due à Baeza 3, sur l'universalité de certaines F-formes quadratiques non singulières lorsque F est de u-invariant fini.
Let F be a field of characteristic 2, and T(F) the set of totally singular F-quadratic forms up to isometry. The uˆ-invariant of F is the maximal dimension of an anisotropic F-quadratic form. In this paper we introduce new invariants of F related to the uˆ-invariant. The first one, called u˜-invariant, is the maximal dimension of an anisotropic not totally singular F-quadratic form. The motivation of introducing this invariant is to see how the values of the uˆ-invariant can be given without the use of forms of T(F), this is because most of the known values of the uˆ-invariant are realized by the forms of T(F) (see 12). Also, for r,s≥1 integers, we introduce the ur-invariant (resp. the u˜s-invariant) which is the maximal dimension of an anisotropic F-quadratic form having a regular part of dimension 2r (resp. the maximal dimension of an anisotropic not totally singular F-quadratic form having a quasilinear part of dimension s). We make a comparison between these new invariants by relating them to the u-invariant and the uˆ-invariant, and we give a list of values that they can take. We also discuss the classical question, due to Baeza 3, on the universality of some nonsingular F-quadratic forms when F has finite u-invariant.
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