1,949 research outputs found
Exponential formulas for models of complex reflection groups
In this paper we find some exponential formulas for the Betti numbers of the
De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the
complex reflection groups G(r,p,n). Our formulas are different from the ones
already known in the literature: they are obtained by a new combinatorial
encoding of the elements of a basis of the cohomology by means of set
partitions with weights and exponents.
We also point out that a similar combinatorial encoding can be used to
describe the faces of the real spherical wonderful models of type
A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=G(2,2,n). This provides exponential
formulas for the f-vectors of the associated nestohedra: the Stasheff's
associahedra (in this case closed formulas are well known) and the graph
associahedra of type D_n.Comment: with respect to v.1: misprint corrected in Example 3.
Markov random fields and iterated toric fibre products
We prove that iterated toric fibre products from a finite collection of toric
varieties are defined by binomials of uniformly bounded degree. This implies
that Markov random fields built up from a finite collection of finite graphs
have uniformly bounded Markov degree.Comment: several improvements, final versio
MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence
Motivated by string-theoretic arguments Manschot, Pioline and Sen discovered
a new remarkable formula for the Poincare polynomial of a smooth compact moduli
space of stable quiver representations which effectively reduces to the abelian
case (i.e. thin dimension vectors). We first prove a motivic generalization of
this formula, valid for arbitrary quivers, dimension vectors and stabilities.
In the case of complete bipartite quivers we use the refined GW/Kronecker
correspondence between Euler characteristics of quiver moduli and Gromov-Witten
invariants to identify the MPS formula for Euler characteristics with a
standard degeneration formula in Gromov-Witten theory. Finally we combine the
MPS formula with localization techniques, obtaining a new formula for quiver
Euler characteristics as a sum over trees, and constructing many examples of
explicit correspondences between quiver representations and tropical curves.Comment: 31 page
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