61 research outputs found
Steinberg lattice of the general linear group and its modular reduction
We study the Steinberg lattice of the general linear group when reduced
modulo a prime different from the defining characteristic.Comment: 6th version, as accepted in journa
Clifford theory for infinite dimensional modules
Clifford theory of possibly infinite dimensional modules is studie
Modular reduction of the Steinberg lattice of the general linear group
The modular reduction of the Steinberg lattice of the general linear group is
studie
Modular representations of Heisenberg algebras
Let be be an arbitrary field and let be the Heisenberg algebra of
dimension over . It was shown by Burde that if has characteristic
0 then the minimum dimension of a faithful -module is . We show here
that his result remains valid in prime characteristic , as long as
.
We construct, as well, various families of faithful irreducible
-modules if has prime characteristic, and classify these when is
algebraically closed. Applications to matrix theory are given
Equivalence and congruence of matrices under the action of standard parabolic subgroups
We find necessary and sufficient conditions for -equivalence of arbitrary
matrices and -congruence of symmetric and alternating matrices, where is
standard parabolic subgroup of and is an arbitrary field
Groups having a faithful irreducible representation
We address the problem of finding necessary and sufficient conditions for an
arbitrary group, not necessarily finite, to admit a faithful irreducible
representation over an arbitrary field
Equivalence and normal forms of bilinear forms
We present an alternative account of the problem of classifying and finding
normal forms for arbitrary bilinear forms. Beginning from basic results
developed by Riehm, our solution to this problem hinges on the classification
of indecomposable forms and in how uniquely they fit together to produce all
other forms. We emphasize the use of split forms, i.e., those bilinear forms
such that the minimal polynomial of the asymmetry of their non-degenerate part
splits over ground field, rather than restricting the field to be algebraically
closed. In order to obtain the most explicit results, without resorting to the
classification of hermitian, symmetric and quadratic forms, we merely require
that the underlying field be quadratically closed
Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings
Let be a ring with , not necessarily finite, endowed with an
involution~, that is, an anti-automorphism of order . Let
be the additive group of all hermitian matrices over relative
to . Let be the subgroup of of all
upper triangular matrices with 1's along the main diagonal. Let
, where acts on
by -congruence transformations. We may view as a unipotent subgroup of
either a symplectic group , if (in which case
is commutative), or a unitary group if . In
this paper we construct and classify a family of irreducible representations of
over a field that is essentially arbitrary. In particular, when is
finite and we obtain irreducible representations of of the
highest possible degree
On the splitting ring of a polynomial
Let be a monic
polynomial with coefficients in a ring~ with identity, not necessarily
commutative. We study the ideal of generated by
, where are the
elementary symmetric polynomials, as well as the quotient ring
The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring
We find all irreducible constituents of the Weil representation of a unitary
group of rank associated to a ramified quadratic extension of
a finite, commutative, local and principal ring of odd characteristic. We
show that this Weil representation is multiplicity free with monomial
irreducible constituents. We also find the number of these constituents and
describe them in terms of Clifford theory with respect to a congruence
subgroup. We find all character degrees in the special case when is a
field
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