Dimension 2p Research Articles

The p-primary subgroup of the cohomology of BPUn in dimension 2p + 6

Let PUn denote the projective unitary group of rank n, and let BPUn be its classifying space. We show that the p-primary subgroup of H2p+6(BPUn;Z) is trivial, where p is an odd prime.

Non-normal purely log terminal centres in characteristic p geqslant 3

We show, building on a recent work of Totaro (The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen–Macaulay, 2017. arXiv:1710.04364v1), that for every prime number p geqslant 3 there exists a purely log terminal pair (Z, S) of dimension 2p+2 whose plt centre S is not normal.

Non-avoided Crossings for n-Body Balanced Configurations in $${\mathbb {R}}^3$$ R 3 Near a Central Configuration

The balanced configurations are those n-body configurations which admit a relative equilibrium motion in a Euclidean space E of high enough dimension 2p. They are characterized by the commutation of two symmetric endomorphisms of the $$(n-1)$$ -dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism B which encodes the shape and the Wintner–Conley endomorphism A which encodes the forces. In general, p is the dimension d of the configuration, which is also the rank of B. Lowering to $$2(d-1)$$ the dimension of E occurs when the restriction of A to the (invariant) image of B possesses a double eigenvalue. It is shown that, while in the space of all $$d\times d$$ symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition (H) is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if $$d=n-1$$ ), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of four bodies with no three of the masses equal, of exactly three families of balanced configurations which admit relative equilibrium motion in a four dimensional space.

Uniqueness of tangent cones to positive-(p,p) integral cycles

Let $(M, \om)$ be a symplectic manifold, endowed with a compatible almost complex structure J and the associated metric g . For any p \in {1, 2, ... (dim M)/2} the form $\Om := \frac{\om^p}{p!}$ is a calibration. More generally, dropping the closedness assumption on $\om$, we get an almost hermitian manifold $(M, \om, J, g)$ and then $\Om$ is a so-called semi-calibration. We prove that integral cycles of dimension 2p (semi-)calibrated by $\Om$ possess at every point a unique tangent cone. The argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting.

Extremal sequences of polynomial complexity

AbstractThe joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer p ≥ 1, there exists a pair of square matrices of dimension 2p(2p+1 − 1) for which every extremal sequence has subword complexity at least 2−p2np.

MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

Kazhdan, Kostant, Binegar–Zierau and Kobayashi–Ørsted constructed a distinguished infinite-dimensional irreducible unitary representation π of the indefinite orthogonal group G = O(2p, 2q) for p, q ≥ 1 with p + q > 2, which has the smallest Gelfand–Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G′ of G, the restriction π|G′ is multiplicity-free. We prove that the restriction of π to any subgroup containing the direct product group U(p1) × U(p2) × U(q) for p1, p2 ≥ 1 with p1 + p2 = p is multiplicity-free, whereas the restriction to U(p1) × U(p2) × U(q1) × U(q2) for q1, q2 ≥ 1 with q1 + q2 = q has infinite multiplicities.

Realizations of the Lie superalgebraq(2) and applications

The Lie superalgebra q(2) and its class of irreducible representations V_p of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of V_p is given explicitly, and from here two realizations of q(2) are determined. The q(2) generators are realized as differential operators in one variable x, and the basis vectors of V_p as 2-arrays of polynomials in x. Following such realizations, it is observed that the Hamiltonian of certain physical models can be written in terms of the q(2) generators. In particular, the models given here as an example are the sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each of these, it is shown how the q(2) realization of the Hamiltonian is helpful in determining the spectrum.

THE RAMOND–RAMOND SELF-DUAL FIVE-FORM'S PARTITION FUNCTION ON T10

In view of the recent interest in formulating a quantum theory of Ramond–Ramond p-forms, we exhibit an [Formula: see text] invariant partition function for the chiral four-form of Type IIB string theory on the ten-torus. We follow the strategy used to derive a modular invariant partition function for the chiral two-form of the M-theory five-brane. We also generalize the calculation to self-dual quantum fields in space–time dimension 2p = 2 + 4k, and display the [Formula: see text] automorphic forms for odd p > 1. We relate our explicit calculation to a computation of the B-cycle periods, which are discussed in the work of Witten.

Reduction in codimension of mixed foliate CR-submanifolds of a Kaehler manifold

Starting with a proper mixed foliate Ci?-submanifold M of dimension 2p-\-q in a hyperbolic complex space form M(—4) of dimension m(rn^2p+2q), it has been proved that under a suitable condition on second fundamental form there exists a (2p+2q) -dimensional totally geodesic submanifold M' of M such that Mis a mixed foliate Ctf-submanifold of M'.