Inclusion matrix of k vs. / affine subspaces and a permutation module of the general affine group

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

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Recurrence on affine Grassmannians

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.18 ◽

2018 ◽

Vol 39 (12) ◽

pp. 3207-3223

Author(s):

YVES BENOIST ◽

CAROLINE BRUÈRE

Keyword(s):

Lyapunov Exponent ◽

Probability Measure ◽

Affine Group ◽

Stationary Measure ◽

Compactly Supported ◽

Affine Subspaces ◽

Affine Grassmannians

We study the action of the affine group $G$ of $\mathbb{R}^{d}$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly supported Zariski dense probability measure $\unicode[STIX]{x1D707}$ on $G$, we show that $X_{k,d}$ supports a $\unicode[STIX]{x1D707}$-stationary measure $\unicode[STIX]{x1D708}$ if and only if the $(k+1)\text{th}$ Lyapunov exponent of $\unicode[STIX]{x1D707}$ is strictly negative. In particular, when $\unicode[STIX]{x1D707}$ is symmetric, $\unicode[STIX]{x1D708}$ exists if and only if $2k\geq d$.

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Affine Subspaces of Curvature Functions from Closed Planar Curves

Results in Mathematics ◽

10.1007/s00025-021-01378-6 ◽

2021 ◽

Vol 76 (2) ◽

Author(s):

Leonardo Alese

Keyword(s):

Arc Length ◽

Planar Curves ◽

Affine Subspaces ◽

Planar Curve ◽

Real Functions ◽

Discrete Counterpart ◽

Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

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Solitons of the midpoint mapping and affine curvature

Journal of Geometry ◽

10.1007/s00022-020-00567-y ◽

2021 ◽

Vol 112 (1) ◽

Author(s):

Christine Rademacher ◽

Hans-Bert Rademacher

Keyword(s):

Differential Equation ◽

Large Class ◽

Affine Group ◽

Arc Length ◽

Smooth Curves ◽

Affine Map ◽

Affine Curvature

AbstractFor a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

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Polycyclic-by-Finite Affine Group Schemes

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-52.3.495 ◽

1986 ◽

Vol s3-52 (3) ◽

pp. 495-516 ◽

Cited By ~ 1

Author(s):

W. W. Crawley-Boevey

Keyword(s):

Affine Group ◽

Group Schemes ◽

Finite Affine

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IMBEDDING OF TRANSFORMATION GROUPS IN AFFINE ONES AND RENORMALIZATION OF TWO-DIMENSIONAL HOMOGENEOUS σ-MODELS

Modern Physics Letters A ◽

10.1142/s0217732393003354 ◽

1993 ◽

Vol 08 (31) ◽

pp. 2937-2942

Author(s):

A. V. BRATCHIKOV

Keyword(s):

Homogeneous Space ◽

Transformation Group ◽

Affine Group ◽

Coupling Constants ◽

Transformation Groups ◽

Two Dimensional

The BLZ method for the analysis of renormalizability of the O(N)/O(N − 1) model is extended to the σ-model built on an arbitrary homogeneous space G/H and in arbitrary coordinates. For deriving Ward-Takahashi (WT) identities an imbedding of the transformation group G in an affine group is used. The structure of the renormalized action is found. All the infinities can be absorbed in a coupling constants renormalization and in a renormalization of auxiliary constants which are related to the imbedding.

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Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.103 ◽

2014 ◽

Vol 36 (4) ◽

pp. 1156-1166 ◽

Cited By ~ 4

Author(s):

IGORS GORBOVICKIS

Keyword(s):

Periodic Orbits ◽

Moduli Space ◽

Algebraic Independence ◽

Complex Polynomials ◽

Generic Point ◽

Algebraic Functions ◽

Polynomial Maps ◽

Affine Subspaces

We consider the space of complex polynomials of degree $n\geq 3$ with $n-1$ distinct marked periodic orbits of given periods. We prove that this space is irreducible and the multipliers of the marked periodic orbits, considered as algebraic functions on that space, are algebraically independent over $\mathbb{C}$. Equivalently, this means that at its generic point the moduli space of degree-$n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. We also prove a similar result for a certain class of affine subspaces of the space of complex polynomials of degree $n$.

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