LIFTING HAMILTONIAN LOOPS TO ISOTOPIES IN FIBRATIONS

Let G be a Lie group, H a closed subgroup and M the homogeneous space G/H. Each representation Ψ of H determines a G-equivariant principal bundle [Formula: see text] on M endowed with a G-invariant connection. We consider subgroups [Formula: see text] of the diffeomorphism group Diff (M), such that, each vector field [Formula: see text] admits a lift to a preserving connection vector field on [Formula: see text]. We prove that [Formula: see text]. This relation is applicable to subgroups [Formula: see text] of the Hamiltonian groups of the flag varieties of a semisimple group G. Let MΔ be the toric manifold determined by the Delzant polytope Δ. We put φb for the loop in the Hamiltonian group of MΔ defined by the lattice vector b. We give a sufficient condition, in terms of the mass center of Δ, for the loops φb and [Formula: see text] to be homotopically inequivalent.

Integral Cayley Multigraphs over Abelian and Hamiltonian Groups

It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$. The converse holds in the case when $G$ is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set.

Quantum solution to the hidden subgroup problem for Poly-Near-Hamiltonian groups

The Hidden Subgroup Problem (HSP) has been widely studied in the context of quantum computing and is known to be efficiently solvable for Abelian groups, yet appears to be difficult for many non-Abelian ones. An efficient algorithm for the HSP over a group \f G\ runs in time polynomial in \f{n\deq\log|G|.} For any subgroup \f H\ of \f G, let \f{N(H)} denote the normalizer of \f H. Let \MG\ denote the intersection of all normalizers in \f G (i.e., \f{\MG=\cap_{H\leq G}N(H)}). \MG\ is always a subgroup of \f G and the index \f{[G:\MG]} can be taken as a measure of ``how non-Abelian'' \f G is (\f{[G:\MG] = 1} for Abelian groups). This measure was considered by Grigni, Schulman, Vazirani and Vazirani, who showed that whenever \f{[G:\MG]\in\exp(O(\log^{1/2}n))} the corresponding HSP can be solved efficiently (under certain assumptions). We show that whenever \f{[G:\MG]\in\poly(n)} the corresponding HSP can be solved efficiently, under the same assumptions (actually, we solve a slightly more general case of the HSP and also show that some assumptions may be relaxed).