Let
be a discrete valuation ring with maximal ideal
and with finite residue field
, the field with q elements where q is a power of a prime p. For
, we write
for the reduction of
modulo the ideal
. ...An irreducible ordinary representation of the finite group
is called stable if its restriction to the principal congruence kernel
, where
, consists of irreducible representations whose stabilizers modulo
, where
, are centralizers of certain matrices in
, called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group
for
.
Communicated by Scott Chapman
Let be a discrete valuation ring with maximal ideal and with finite residue field , the field with q elements where q is a power of a prime p. For , we write for the reduction of modulo the ideal . ...An irreducible ordinary representation of the finite group is called stable if its restriction to the principal congruence kernel , where , consists of irreducible representations whose stabilizers modulo , where , are centralizers of certain matrices in , called stable matrices. The study of stable representations is motivated by constructions of strongly semisimple representations, introduced by Hill, which is a special case of stable representations. In this paper, we explore the construction of stable irreducible representations of the finite group for .Communicated by Scott Chapman
An accurate characterization of pose uncertainty is essential for safe autonomous navigation. Early pose uncertainty characterization methods proposed by Smith, Self, and Cheeseman (SCC) used ...coordinate-based first-order methods to propagate uncertainty through nonlinear functions such as pose composition (head-to-tail), pose inversion, and relative pose extraction (tail-to-tail). Characterizing uncertainty in the Lie algebra of the special Euclidean group results in better uncertainty estimates. However, existing Lie-group-based uncertainty propagation techniques assume that individual poses are independent. After solving a pose graph, however, the entire trajectory is jointly distributed as factors induce correlation. Hence, the independence assumption does not capture reality. In addition, prior work has focused primarily on the pose composition operation. This article develops a framework for modeling the uncertainty of jointly distributed poses and describes how to perform the equivalent of the SSC pose operations while characterizing uncertainty in the Lie algebra. Evaluation on simulated and open-source datasets shows that the proposed methods result in more accurate uncertainty estimates and thus more accurate filtering of potential loop closures. An accompanying C++ library implementation is also released.
Let G be an irreducible imprimitive subgroup of GLn(F), where F is a field. Any system of imprimitivity for G can be refined to a nonrefinable system of imprimitivity, and we consider the question of ...when such a refinement is unique. Examples show that G can have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where G is the wreath product of an irreducible primitive H≤GLd(F) and transitive K≤Sk, where n=dk. We show that G has a unique nonrefinable system of imprimitivity, except in the following special case: d=1, n=k is even, |H|=2, and K is a subgroup of C2≀Sn/2. As a simple application, we prove results about inclusions between wreath product subgroups.
The limiting spectral distribution of matrix Formula: see text is considered in this paper. Existing results always focus on the condition of modifying Tn, but for Xn, it is usually assumed to be a ...matrix composed of n × N independent identically distributed elements. Here we specify the joint distribution of column vectors of Xn. In particular, entries on the same column of Xn are correlated, in contrast with more common independence assumptions. Assuming that the columns of Xn are random vectors following the isotropic log-concave distribution, and under some additional regularity conditions, we prove that the empirical spectral distribution Formula: see text of matrix Bn converges to a deterministic probability distribution F almost surely. Moreover, the Stieltjes transformation m = m(z) of F satisfies a deterministic form of equation, and for any Formula: see text, it is the unique solution of the equation.
This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target ...matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group.
We consider what some authors call “parabolic Möbius subgroups” of matrices over Z, Q, and R and focus on the membership problem in these subgroups and complexity of relevant algorithms.
In this contribution, a two-player constant-sum 2-tuple linguistic matrix game is described, and a linguistic linear programming (LLP) approach is proposed to solve this class of games. The proposed ...approach can be perceived as a unified mechanism in the sense that it can be adopted to solve linguistic matrix game problems, LLP problems, and linguistic multi-attribute decision-making (MADM) problems. The latter is exhibited by presenting examples of linguistic MADM problems modeled as two-player constant-sum linguistic matrix games with Nature as the second player.
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