2,868 research outputs found
A symplectic refinement of shifted Hecke insertion
Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to
formulate a combinatorial rule for the expansion of the stable Grothendieck
polynomials indexed by permutations in the basis of stable Grothendieck
polynomials indexed by partitions. Patrias and Pylyavskyy
introduced a shifted analogue of Hecke insertion whose natural domain is the
set of maximal chains in a weak order on orbit closures of the orthogonal group
acting on the complete flag variety. We construct a generalization of shifted
Hecke insertion for maximal chains in an analogous weak order on orbit closures
of the symplectic group. As an application, we identify a combinatorial rule
for the expansion of "orthogonal" and "symplectic" shifted analogues of
in Ikeda and Naruse's basis of -theoretic Schur -functions.Comment: 40 pages; v2: fixed several errors, minor reorganization; v3: further
corrections, condensed expositio
Strings on Bubbling Geometries
We study gauge theory operators which take the form of a product of a trace
with a Schur polynomial, and their string theory duals. These states represent
strings excited on bubbling AdS geometries which are dual to the Schur
polynomials. These geometries generically take the form of multiple annuli in
the phase space plane. We study the coherent state wavefunction of the lattice,
which labels the trace part of the operator, for a general Young tableau and
their dual description on the droplet plane with a general concentric ring
pattern. In addition we identify a density matrix over the coherent states on
all the geometries within a fixed constraint. This density matrix may be used
to calculate the entropy of a given ensemble of operators. We finally recover
the BMN string spectrum along the geodesic near any circle from the ansatz of
the coherent state wavefunction.Comment: 41 pages, 12 figures, published version in JHE
SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries
We show how the density-matrix numerical renormalization group (DM-NRG)
method can be used in combination with non-Abelian symmetries such as SU(N),
where the decomposition of the direct product of two irreducible
representations requires the use of a so-called outer multiplicity label. We
apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze
the finite size spectrum, determine local fermionic, spin, superconducting, and
trion spectral functions, and also compute the temperature dependence of the
conductance. Our calculations reveal a rich Fermi liquid structure.Comment: 18 pages, 9 figure
The weighted hook-length formula II: Complementary formulas
Recently, a new weighted generalization of the branching rule for the hook
lengths, equivalent to the hook formula, was proved. In this paper, we
generalize the complementary branching rule, which can be used to prove
Burnside's formula. We present three different proofs: bijective, via weighted
hook walks, and via the ordinary weighted branching rule.Comment: 20 pages, 9 figure
Antipode formulas for some combinatorial Hopf algebras
Motivated by work of Buch on set-valued tableaux in relation to the K-theory
of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras
that can be thought of as K-theoretic analogues of the Hopf algebras of
symmetric functions, quasisymmetric functions, noncommutative symmetric
functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They
described the bialgebra structure in all cases that were not yet known but left
open the question of finding explicit formulas for the antipode maps. We give
combinatorial formulas for the antipode map for the K-theoretic analogues of
the symmetric functions, quasisymmetric functions, and noncommutative symmetric
functions.Comment: 26 page
On character generators for simple Lie algebras
We study character generating functions (character generators) of simple Lie
algebras. The expression due to Patera and Sharp, derived from the Weyl
character formula, is first reviewed. A new general formula is then found. It
makes clear the distinct roles of ``outside'' and ``inside'' elements of the
integrity basis, and helps determine their quadratic incompatibilities. We
review, analyze and extend the results obtained by Gaskell using the Demazure
character formulas. We find that the fundamental generalized-poset graphs
underlying the character generators can be deduced from such calculations.
These graphs, introduced by Baclawski and Towber, can be simplified for the
purposes of constructing the character generator. The generating functions can
be written easily using the simplified versions, and associated Demazure
expressions. The rank-two algebras are treated in detail, but we believe our
results are indicative of those for general simple Lie algebras.Comment: 50 pages, 11 figure
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