On multiplicities for SL(n)

We use the strong Artin conjecture for Galois extensions of Heisenberg type to show that a cuspidal automorphic representation of SL(N)/F, forF a number field andN>2, can occur with multiplicity greater than one. We also exhibit two cuspidalL-packets (forF=Q andN prime) which are locally isomorphic for primesp different fromN, but which are disjoint atN, i.e. thatL-packets are not rigid.     

An example of an irregular random measure

Summary We consider the set of random measures which consists of measurable maps from [0, 1] to the set of measures on . As it is the dual space ofL ₁ ([0, 1];C()), we can equip this space with the weak^* topology. We construct a special random measure , which appears as the weak^* limit of a sequence of Dirac random measures , where (X _n ) _n is a bounded sequence inL _p [0, 1], (1p<2). The special form of this random measure, which oscillates randomly between twoq-stable standard measures on with different normalizations (p<q<2) allows us to prove two properties of (X _n ) _n is equivalent to the unit vector basis ofl _q and has no almost symmetric subsequence.     

2-Fusion Systems with Components of Type SL2(q)

We study a problem in fusion systems designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. A 2-fusion system is a 2-group S with a family of injective homomorphisms on subgroups of S. Fusion systems arise in the study of modular representations and classifying spaces, and our results have potential ramifications beyond finite group theory. One problem is to determine S or, whenever possible, the entire 2-fusion system from the knowledge of certain subgroups and maps. We consider the case where S contains a subgroup and maps that arise in the Classification with standard component of type SL₂(q).     

An Asymptotic Order of Fourier Transform on SL(2, R)

Let SL(2, R) be the group of all real matrices of order 2 with determinant 1. In this paper, weuse G to denote both SL(2, R) and the lie group SU(1,1) = {(α/ββ/α):|α|2-|β|2=1,α,β∈C}because they are isomorphic to each other[1].     

An example of an atomic pullback without the ACCP

We find necessary and sufficient conditions on a pullback diagram in order that every nonzero nonunit in its pullback ring admits a finite factorization into irreducible elements. As a result, we can describe a method of easily producing atomic domains that do not satisfy the ascending chain condition on principal ideals.     

Mean dimension and an embedding problem: An example

For any positive integer D, we construct a minimal dynamical system with mean dimension equal to D/2 that cannot be embedded into (([0, 1] ^D )^?, shift).     

A fake projective plane constructed from an elliptic surface with multiplicities (2,4)

Given any (2,4)-elliptic surface with nine smooth rational curves,eight (2)-curves and one (3)-curve,forming a Dynkin diagram of type [2,2][2,2][2,2][2,2,3],we show that a fake projective plane can be constructed from it by taking a degree 3 cover and then a degree 7 cover.We also determine the types of singular fibres of such a (2,4)-elliptic surface.     

The Structure of an SL2‐module of finite Morley rank

We consider a universe of finite Morley rank and the following definable objects: a field , a non‐trivial action of a group on a connected abelian group V , and a torus T of G such that . We prove that every T‐minimal subgroup of V has Morley rank . Moreover V is a direct sum of ‐minimal subgroups of the form , where W is T‐minimal and ζ is an element of G of order 4 inverting T .     

On multiplicities of cocharacters for algebras with superinvolution

In this paper we deal with finitely generated superalgebras with superinvolution, satisfying a non-trivial identity, whose multiplicities of the cocharacters are bounded by a constant. Along the way, we prove that the codimension sequence of such algebras is polynomially bounded if and only if their colength sequence is bounded by a constant.     

SL2-modules of small homological dimension

Let V _n be the SL₂-module of binary forms of degree n and let V = V_n1 ???V_np V = {V_{{n_1}}} \oplus \cdots \oplus {V_{{n_p}}} . We consider the algebra R = O(V)^{\textS\textL₂} R = \mathcal{O}{(V)^{{\text{S}}{{\text{L}}_2}}} of polynomial functions on V invariant under the action of SL₂. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hd R. Popov gave in 1983 a classification of the cases in which hd R ≤ 10 for a single binary form (p = 1) or hd R ≤ 3 for a system of two or more binary forms (p > 1). We extend Popov’s result and determine for p = 1 the cases with hd R ≤ 100, and for p > 1 those with hd R ≤ 15. In these cases we give a set of homogeneous parameters and a set of generators for the algebra R.     

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 (B _n) in ℒ is calledasymptotically invariant in lim _n μ (B _nΔgB _n)=0 for everygεG. 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).