709 research outputs found
Deformed Kazhdan-Lusztig elements and Macdonald polynomials
We introduce deformations of Kazhdan-Lusztig elements and specialised
nonsymmetric Macdonald polynomials, both of which form a distinguished basis of
the polynomial representation of a maximal parabolic subalgebra of the Hecke
algebra. We give explicit integral formula for these polynomials, and
explicitly describe the transition matrices between classes of polynomials. We
further develop a combinatorial interpretation of homogeneous evaluations using
an expansion in terms of Schubert polynomials in the deformation parameters.Comment: major revision, 29 pages, 22 eps figure
Cluster algebras of type D: pseudotriangulations approach
We present a combinatorial model for cluster algebras of type in terms
of centrally symmetric pseudotriangulations of a regular -gon with a small
disk in the centre. This model provides convenient and uniform interpretations
for clusters, cluster variables and their exchange relations, as well as for
quivers and their mutations. We also present a new combinatorial interpretation
of cluster variables in terms of perfect matchings of a graph after deleting
two of its vertices. This interpretation differs from known interpretations in
the literature. Its main feature, in contrast with other interpretations, is
that for a fixed initial cluster seed, one or two graphs serve for the
computation of all cluster variables. Finally, we discuss applications of our
model to polytopal realizations of type associahedra and connections to
subword complexes and -cluster complexes.Comment: 21 pages, 21 figure
Non-commutative Donaldson-Thomas theory and the conifold
Given a quiver algebra A with relations defined by a superpotential, this
paper defines a set of invariants of A counting framed cyclic A-modules,
analogous to rank-1 Donaldson-Thomas invariants of Calabi-Yau threefolds. For
the special case when A is the non-commutative crepant resolution of the
threefold ordinary double point, it is proved using torus localization that the
invariants count certain pyramid-shaped partition-like configurations, or
equivalently infinite dimer configurations in the square dimer model with a
fixed boundary condition. The resulting partition function admits an infinite
product expansion, which factorizes into the rank-1 Donaldson-Thomas partition
functions of the commutative crepant resolution of the singularity and its
flop. The different partition functions are speculatively interpreted as
counting stable objects in the derived category of A-modules under different
stability conditions; their relationship should then be an instance of wall
crossing in the space of stability conditions on this triangulated category.Comment: Infinite product form, conjectured in v1, now a theorem of Ben Young.
Additional discussion of small-volume expansion related to Eisenstein-like
serie
Multispecies Weighted Hurwitz Numbers
The construction of hypergeometric Toda -functions as generating
functions for weighted Hurwitz numbers is extended to multispecies families.
Both the enumerative geometrical significance of multispecies weighted Hurwitz
numbers, as weighted enumerations of branched coverings of the Riemann sphere,
and their combinatorial significance in terms of weighted paths in the Cayley
graph of are derived. The particular case of multispecies quantum
weighted Hurwitz numbers is studied in detail.Comment: this is substantially enhanced version of arXiv:1410.881
The number of rhombus tilings of a "punctured" hexagon and the minor summation formula
We compute the number of all rhombus tilings of a hexagon with sides
, of which the central triangle is removed, provided
have the same parity. The result is a product of four numbers, each of which
counts the number of plane partitions inside a given box. The proof uses
nonintersecting lattice paths and a new identity for Schur functions, which is
proved by means of the minor summation formula of Ishikawa and Wakayama. A
symmetric generalization of this identity is stated as a conjecture.Comment: 21 pages, AmS-TeX, uses TeXDra
Chow rings of toric varieties defined by atomic lattices
We study a graded algebra D=D(L,G) defined by a finite lattice L and a subset
G in L, a so-called building set. This algebra is a generalization of the
cohomology algebras of hyperplane arrangement compactifications found in work
of De Concini and Procesi. Our main result is a representation of D, for an
arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we
construct from L and G. We describe this variety both by its fan and
geometrically by a series of blowups and orbit removal. Also we find a Groebner
basis of the relation ideal of D and a monomial basis of D over Z.Comment: 23 pages, 7 figures, final revision with minor changes, to appear in
Invent. Mat
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