601,439 research outputs found
On the Generic Type of the Free Group
We answer a question raised by Pillay, that is whether the infinite weight of
the generic type of the free group is witnessed in . We also prove
that the set of primitive elements in finite rank free groups is not uniformly
definable. As a corollary, we observe that the generic type over the empty set
is not isolated. Finally, we show that uncountable free groups are not
-homogeneous.Comment: To appear in J. of Symbolic Logi
q-Krawtchouk polynomials as spherical functions on the Hecke algebra of type B
The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group
of type B, contains the generic Hecke algebra for the symmetric group, i.e. the
Weyl group of type A, as a subalgebra. Inducing the index representation of the
subalgebra gives a Hecke algebra module, which splits multiplicity free. The
corresponding zonal spherical functions are calculated in terms of q-Krawtchouk
polynomials. The result covers a number of previously established
interpretations of (q-)Krawtchouk polynomials on the hyperoctahedral group,
finite groups of Lie type, hypergroups and the quantum SU(2) group. Jimbo's
analogue of the Frobenius-Schur-Weyl duality is a key ingredient in the proof.Comment: AMS-TeX v. 2.1, 30 page
Generic Stationary Measures and Actions
Let be a countably infinite group, and let be a generating
probability measure on . We study the space of -stationary Borel
probability measures on a topological space, and in particular on ,
where is any perfect Polish space. We also study the space of
-stationary, measurable -actions on a standard, nonatomic probability
space.
Equip the space of stationary measures with the weak* topology. When
has finite entropy, we show that a generic measure is an essentially free
extension of the Poisson boundary of . When is compact, this
implies that the simplex of -stationary measures on is a Poulsen
simplex. We show that this is also the case for the simplex of stationary
measures on .
We furthermore show that if the action of on its Poisson boundary is
essentially free then a generic measure is isomorphic to the Poisson boundary.
Next, we consider the space of stationary actions, equipped with a standard
topology known as the weak topology. Here we show that when has property
(T), the ergodic actions are meager. We also construct a group without
property (T) such that the ergodic actions are not dense, for some .
Finally, for a weaker topology on the set of actions, which we call the very
weak topology, we show that a dynamical property (e.g., ergodicity) is
topologically generic if and only if it is generic in the space of measures.
There we also show a Glasner-King type 0-1 law stating that every dynamical
property is either meager or residual.Comment: To appear in the Transactions of the AMS, 49 page
Three-point Functions in Sine-Liouville Theory
We calculate the three-point functions in the sine-Liouville theory
explicitly. The same calculation was done in the (unpublished) work of Fateev,
Zamolodchikov and Zamolodchikov to check the conjectured duality between the
sine-Liouville and the SL(2,R)/U(1) coset CFTs. The evaluation of correlators
boils down to that of a free-field theory with a certain number of insertion of
screening operators. We prove that the winding number conservation is violated
up to (+-)1 in three-point functions, which is in agreement with the result of
FZZ that in generic N-point correlators the winding number conservation is
violated up to N-2 units. A new integral formula of Dotsenko-Fateev type is
derived, using which we write down the generic three-point functions of
tachyons explicitly. When the winding number is conserved, the resultant
expression is shown to reproduce the correlators in the coset model correctly,
including the group-theoretical factor. As an application, we also study the
superstring theory on linear dilaton background which is described by
super-Liouville theory. We obtain the three-point amplitude of tachyons in
which the winding number conservation is violated.Comment: 26 pages, 5 ps figures, v2:minor corrections(references to the work
of FZZ are made more precise
Torsion-free and distal dp-minimal groups
Let be a dp-minimal group. We discuss two different hypotheses on ;
first we show that, if is torsion-free, then it is abelian. Then we
investigate the structure of when it admits a distal f-generic type,
showing in particular that the quotient of by its FC-center can then be
naturally equipped with the structure of a valued group. As an application of
this, we show that, in this case, is virtually nilpotent
A class of Calogero type reductions of free motion on a simple Lie group
The reductions of the free geodesic motion on a non-compact simple Lie group
G based on the symmetry given by left- and right
multiplications for a maximal compact subgroup are
investigated. At generic values of the momentum map this leads to (new) spin
Calogero type models. At some special values the `spin' degrees of freedom are
absent and we obtain the standard Sutherland model with three
independent coupling constants from SU(n+1,n) and from SU(n,n). This
generalization of the Olshanetsky-Perelomov derivation of the model with
two independent coupling constants from the geodesics on with
G=SU(n+1,n) relies on fixing the right-handed momentum to a non-zero character
of . The reductions considered permit further generalizations and work at
the quantized level, too, for non-compact as well as for compact G.Comment: shortened to 13 pages in v2 on request of Lett. Math. Phys. and
corrected some spelling error
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