Natural Action Of SL Research Articles

The pentagram map on Grassmannians

In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n,mn). We define invariants of Grassmannian twisted polygons under the natural action of SL(nm), invariants that define coordinates in the moduli space of twisted polygons. We then prove that when written in terms of the moduli space coordinates, the pentagram map is preserved by a certain scaling. The scaling is then used to construct a Lax representation for the map that can be used for integration.

Vector spaces of skew-symmetric matrices of constant rank

We study the orbits of vector spaces of skew-symmetric matrices of constant rank 2 r and type ( N + 1 ) × ( N + 1 ) under the natural action of SL ( N + 1 ) , over an algebraically closed field of characteristic zero. We give a complete description of the orbits for vector spaces of dimension 2, relating them to some 1 -generic matrices of linear forms. We also show that, for each rank two vector bundle on P 2 defining a triple Veronese embedding of P 2 in G ( 1 , 7 ) , there exists a vector space of 8 × 8 skew-symmetric matrices of constant rank 6 whose kernel bundle is the dual of the given rank two vector bundle.

Unipotent flows on the space of branched covers of Veech surfaces

There is a natural action of SL $(2,\mathbb{R})$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \big\{\big(\begin{smallmatrix}1 & * 0 & 1\end{smallmatrix}\big)\big\}$ . We classify the U -invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL $(2,\mathbb{R})$ -orbit of the set of branched covers of a fixed Veech surface.) For the U -action on these submanifolds, this is an analogue of Ratner's theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is $2 \pi/n$ , with $n \ge 5$ and n odd.

SL(2,Bbb Z) action on three-dimensional CFTs and holography

We show that there is a natural action of SL(2,Z) on the two-point functions of the energy momentum tensor and of higher-spin conserved currents in three-dimensional CFTs. The dynamics behind the S-operation of SL(2,Z) is that of an irrelevant current-current deformation and we point out its similarity to the dynamics of a wide class of three-dimensional CFTs. The holographic interpretation of our results raises the possibility that many three-dimensional CFTs have duals on AdS4 with SL(2,Z) duality properties at the linearized level.

Relatively prime numbers and invariant measures under the natural action of SL (n, \mathbb{Z}) on \mathbb{R}^{n}

Relatively prime numbers and invariant measures under the natural action of SL (n, \mathbb{Z}) on \mathbb{R}^{n}

Orbit distribution on [formula omitted]2 under the natural action of SL(2, [formula omitted

Let x be a point in R 2. In the present paper, we give an explicit description of the set { Ax: A ϵ SL(2, Z )}, the orbit of x, under the natural action of SL(2, Z ) on R 2. Our approach allows us to analyze the distribution of an orbit in the real plane. As an application we evaluate, for r, t > 0, the asymptotic behaviour of the set {A∈SL(2,Z): Ax 0∈[0,r] 2, ‖A‖≦t} , where x 0=(( 5 −1) 2 ,1) and ‖A‖=max |a ij|, for A=(a ij) .

Invariants of pencils of binary cubics

Let U denote the variety of pencils of binary cubics without base point defined over an algebraically closed field k whose characteristic is not equal to 2 or 3. (For full details of the terminology, see § 2.) There is a natural action of SL(2) on U; our main object is to show that U possesses a good quotient for this action in the sense of ((7), Definition 1·5) or ((4), p. 70), and to identify this quotient with the affine line over k. In fact, we proveTheorem 1. There exists a morphism φ: such that (, φ) is a good quotient of U by SL(2). Moreover, all but one of the fibres of φ are orbits; the exceptional fibre consists of two orbits corresponding to pencils with one or two triple points.

Asymptotically invariant sequences and an action of SL (2,Z) on the 2-sphere

LetG be a countable group which acts non-singularly and ergodically on a Lebesgue space (X, ȑ, μ). A sequence (Bn) in ℒ is calledasymptotically invariant in limn μ (BnΔgBn)=0 for everygeG. In this paper we show that the existence of such sequences can be characterized by certain simple assumptions on the cohomology of the action ofG onX. As an explicit example we prove that a natural action of SL (2,Z) on the 2-sphere has no asymptotically invariant sequences. The last section deals with a particular cocycle for this action which has an interpretation as a random walk on the integers with “time” in SL (2,Z).