Harmonic Analysis on SO (n, ℂ)/SO (n−1, ℂ), n≥3

In this article we compute the Plancherel measure for SO(n, ℂ)/SO(n − 1, ℂ) following the approach of Van den Ban. This result is required in order to calculate the explicit decomposition of the oscillator representation wn for the dual pair G = SL(2, ℂ) × SO(n, ℂ) and to prove that every wn(G)-invariant Hilbert subspace of the space of tempered distributions decomposes multiplicity free.     

The geometry of SO(n)\SO0(n, 1)

Consider the Lie group SO₀(n, 1) with the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left and right. This paper studies the geometry of the quotient space of the homogeneous submersion SO₀(n, 1) → SO(n)\SO₀(n, 1). It is a cohomogeneity one manifold, which can be expressed as a warped product. Its group of isometries, geodesics, and sectional curvatures are calculated.     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

On Pairs of Commuting Involutions in Sym(n) and Alt(n)

The number of pairs of commuting involutions in Sym(n) and Alt(n) is determined up to isomorphism. It is also proven that, up to isomorphism and duality, there are exactly two abstract regular polyhedra on which the group Sym(6) acts as a regular automorphism group.     

Solvable Subgroups of Out(F n ) are Virtually Abelian

Let F _n be the free group of rank n, let Aut(F _n ) be its automorphism group and let Out(F _n ) be its outer automorphism group. We show that every solvable subgroup of Out(F _n ) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(F _n ) has a finite index subgroup that lifts to Aut(F _n ).     

Singular Manakov flows and geodesic flows on homogeneous spaces of SO(N)

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces SO(n)/SO(k₁) ×⋯× SO(k _r ), for any choice of k ₁,…,k _r , k ₁ + ⋯ + k _r ⩽ n. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented. Also, the proof of integrability of the SO(n)-invariant Einstein metrics on SO(k ₁ + k ₂ + k ₃)/SO(k ₁) × SO(k ₂) × SO(k ₃) and on the Stiefel manifolds V (n, k) = SO(n)/SO(k) is given.     

A theorem of the Wiener—Tauberian type forL 1(H n)

The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hⁿ and the unitary group U(n), acts on Hⁿ in a natural way. Here we prove a Wiener-Tauberian theorem for L¹ (Hⁿ) with this HM(n)-action on Hⁿ i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L¹ (Hⁿ) in order that the linear span of^gf : g∈HM(n) is dense in L¹(Hⁿ), where^gf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hⁿ.     

Embedding the Outer Automorphism Group Out(F n ) of a Free Group of Rank n in the Group Out(F m ) for m>n

It is proved that for every n 1, the group Out(F _n )is embedded in the group Out(F _m ) with m=1+(n-1)k ⁿ , where k is an arbitrary natural number coprime to n-1.     

Homology of the double loop space of the homogeneous space SU(n)/SO(n)

We study the mod 2 homology of the double loop space of SU(n)/SO(n) using the Serre spectral sequence along with the Eilenberg-Moore spectral sequence. Then we also get the homology of the double loop space of the set of all Lagrangian subspaces of the symplectic vector space R²ⁿ.     

Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a H?lder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order O(n^-a+e)\mathcal{O}(n^{-\alpha+\epsilon}) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

Braid group action via GL n (q) and U n (q), and Galois realizations

We determine the braid group action on generating systems of a group that is the semi-direct product of a finite vector space with a group of scalars. This leads to Galois realizations of certain groups GL _n (q) and PU _n (q). Dedicated to Prof. J. G. Thompson Partially supported by an NSA grant. This work was done while the author was a fellow of the Institute for Advanced Studies in Jerusalem.     

Dade's Invariant Conjecture for the Symplectic Group Sp4(2 n ) and the Special Unitary Group SU4(22n ) in Defining Characteristic

In this article, we verify Dade's projective invariant conjecture for the symplectic group Sp₄(2 ⁿ ) and the special unitary group SU₄(2²ⁿ ) in the defining characteristic, that is, in characteristic 2. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for Sp₄(2 ⁿ ) and SU₄(2²ⁿ ) in the defining characteristic, that is, Sp₄(2 ⁿ ) and SU₄(2²ⁿ ) are good for the prime 2 in the sense of Isaacs, Malle, and Navarro.     

On the general theory of (m,n) rings

In this paper, the lattice of congruences of an (m, n) ring is determined, a generalization of the Wedderburn theorem for finite division rings is considered, all (2,n) fields, (2,n) rings of prime order, and all (3,n) rings of prime order are determined. A special class of (2,n) fields, called super-simple (2,n) fields, is characterized. Presented by J. D. Monk.     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

Generating Sets of Finite Transformation Semigroups PK(n,r) and K (n,r)

Let P _n and T _n be the partial transformation and the full transformation semigroups on the set {1,…, n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of height r in the subsemigroup PK(n, r) = {α ∈P _n : |im (α)| ≤r} of P _n to be a (minimal) generating set of PK(n, r); and similarly, for any set of full transformations of height r in the subsemigroup K(n, r) = {α ∈T _n : |im (α)| ≤r} of T _n to be a (minimal) generating set of K(n, r) for 2 ≤ r ≤ n ? 1.     

Stochastic flows on the boundaries of SL(n,R)

Summary We study the asymptotic stability of the stochastic flows on a class of compact spaces induced by a diffusion process in SL(n, R) or GL(n, R). These compact spaces are called boundaries of SL(n, R), which include SO(n), the flag manifold, the sphereS ⁿ–1 and the Grassmannians. The one point motions of these flows are Brownian motions. For almost every, , we determine the set of stable points. This is a random open set whose complement has zero Lebesgue measure. The distance between any two points in the same component of this set tends to zero exponentially fast under the flow. The Lyapunov exponents at stable points are computed explicitly. We apply our results to a stochastic flow onS ⁿ–2 generated by a stochastic differential equation which exhibits some nice symmetry.Research supported in part by Hou Yin Dong Education Foundation of China On leave from Nankai University, Tianjin, China     

Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.