Twisted homology of symmetric groups

We study the homology of symmetric groups with coefficients coming from the functor . We are primarily interested in the limit where . Our main goal is to compare the described above situation with the case of general linear groups.

     

Twisted products in pseudo-Riemannian geometry

Twisted products are generalizations of warped products, namely the warping function may depend on the points of both factors. The two canonical foliations of a twisted product are mutually perpendicular and their leaves are totally geodesic, resp. totally umbilic. The main result is a decomposition theorem of de Rham type: If on a simply connected, geodesically complete pseudo-Riemannian manifoldM two foliations with the above properties are given, thenM is a twisted product.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Characterizations of products of symmetric matrices

Characterizations are obtained for matrices C of the form C = AΣ, where A, Σ are n×n matrices over the real field such that A is symmetric and C is nonnegative definite. Among others, a proof of recent generalization of Cochran's theorem is given.     

Twisted Yangians for symmetric pairs of types B,C, D

We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan’s classification. These algebras can be regarded as coideal subalgebras of the Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincaré–Birkoff–Witt Theorem, determine their centres and study also extended reflection algebras.     

Finite graphs have unique symmetric products

Let X be a metric continuum and let Fn(X) be the nth symmetric product of X (Fn(X) is the hyperspace of nonempty subsets of X with at most n points). In this paper we prove that if Fn(X) is homeomorphic to Fn(Y), where X is a finite graph and Y is a continuum, then X is homeomorphic to Y.     

Fixed point property on symmetric products

For a metric continuum X, let Fn(X)={A⊂X:A is nonempty and has at most n points}. In this paper we show a continuum X such that F₂(X) has the fixed point property while X does not have it.     

Truncated symmetric products and configuration spaces

Research partially supported by the NSF and the Sonderforschungsbereich 170, G?ttingen     

Fixed point index in symmetric products

Let be an open subset of a locally compact metric ANR and let be a continuous map. In this paper we study the fixed point index of the map that induces in the -symmetric product of , . This index can detect the existence of periodic orbits of period of , and it can be used to obtain the Euler characteristic of the -symmetric product of a manifold , . We compute for all orientable compact surfaces without boundary.

     

Effective scalar products of D-finite symmetric functions

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel's work by providing algorithms that compute differential equations, these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel's class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself.     

Tautological integrals on symmetric products of curves

We propose a conjecture on the generating series of Chern numbers of tautological bundles on symmetric products of curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn's conjecture for surfaces.     

On the symmetric products of a curve

Both authors are supported by MURST and CNR of Italy.