3,132 research outputs found
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page
Refined matrix models from BPS counting
We construct a free fermion and matrix model representation of refined BPS
generating functions of D2 and D0 branes bound to a single D6 brane, in a class
of toric manifolds without compact four-cycles. In appropriate limit we obtain
a matrix model representation of refined topological string amplitudes. We
consider a few explicit examples which include a matrix model for the refined
resolved conifold, or equivalently five-dimensional U(1) gauge theory, as well
as a matrix representation of the refined MacMahon function. Matrix models
which we construct have ordinary unitary measure, while their potentials are
modified to incorporate the effect of the refinement.Comment: 27 pages, 4 figures, published versio
Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory
We study the partition function of the compactified 5D U(1) gauge theory (in
the Omega-background) with a single adjoint hypermultiplet, calculated using
the refined topological vertex. We show that this partition function is an
example a periodic Schur process and is a refinement of the generating function
of cylindric plane partitions. The size of the cylinder is given by the mass of
adjoint hypermultiplet and the parameters of the Omega-background. We also show
that this partition function can be written as a trace of operators which are
generalizations of vertex operators studied by Carlsson and Okounkov. In the
last part of the paper we describe a way to obtain (q,t) identities using the
refined topological vertex.Comment: 40 Page
Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures
We prove two identities of Hall-Littlewood polynomials, which appeared
recently in a paper by two of the authors. We also conjecture, and in some
cases prove, new identities which relate infinite sums of symmetric polynomials
and partition functions associated with symmetry classes of alternating sign
matrices. These identities generalize those already found in our earlier paper,
via the introduction of additional parameters. The left hand side of each of
our identities is a simple refinement of a relevant Cauchy or Littlewood
identity. The right hand side of each identity is (one of the two factors
present in) the partition function of the six-vertex model on a relevant
domain.Comment: 34 pages, 14 figure
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