1,141 research outputs found
Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers
This is the first of three papers that develop structures which are counted
by a "parabolic" generalization of Catalan numbers. Fix a subset R of
{1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are
determined by R. These are the "inverses" of (parabolic) multipermutations
whose multiplicities are determined by R. The standard forms of the ordered
partitions are refered to as "R-permutations". The notion of 312-avoidance is
extended from permutations to R-permutations. Let lambda be a partition of N
such that the set of column lengths in its shape is R or R union {n}. Fix an
R-permutation pi. The type A Demazure character (key polynomial) in x_1, ..,
x_n that is indexed by lambda and pi can be described as the sum of the weight
monomials for some of the semistandard Young tableau of shape lambda that are
used to describe the Schur function indexed by lambda. Descriptions of these
"Demazure" tableaux developed by the authors in earlier papers are used to
prove that the set of these tableaux is convex in Z^N if and only if pi is
R-312-avoiding if and only if the tableau set is the entire principal ideal
generated by the key of pi. These papers were inspired by results of Reiner and
Shimozono and by Postnikov and Stanley concerning coincidences between Demazure
characters and flagged Schur functions. This convexity result is used in the
next paper to deepen those results from the level of polynomials to the level
of tableau sets. The R-parabolic Catalan number is defined to be the number of
R-312-avoiding permutations. These special R-permutations are reformulated as
"R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless
R-tuples"; the latter n-tuples arise in multiple contexts in these papers.Comment: 20 pp with 2 figs. Identical to v.3, except for the insertion of the
publication data for the DMTCS journal (dates and volume/issue/number). This
is one third of our "Parabolic Catalan numbers ..", arXiv:1612.06323v
Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)
Fix an integer partition lambda that has no more than n parts. Let beta be a
weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur
function indexed by lambda and beta is a polynomial generating function in x_1,
.., x_n for certain semistandard tableaux of shape lambda. Let pi be an
n-permutation. The type A Demazure character (key polynomial, Demazure
polynomial) indexed by lambda and pi is another such polynomial generating
function. Reiner and Shimozono and then Postnikov and Stanley studied
coincidences between these two families of polynomials. Here their results are
sharpened by the specification of unique representatives for the equivalence
classes of indexes for both families of polynomials, extended by the
consideration of more general beta, and deepened by proving that the polynomial
coincidences also hold at the level of the underlying tableau sets. Let R be
the set of lengths of columns in the shape of lambda that are less than n.
Ordered set partitions of {1,..,n} with block sizes determined by R, called
R-permutations, are used to describe the minimal length representatives for the
parabolic quotient of the nth symmetric group specified by the set
{1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to
these set partitions. The R-parabolic Catalan number is defined to be the
number of these. Every flagged Schur function arises as a Demazure polynomial.
Those Demazure polynomials are precisely indexed by the R-312-avoiding
R-permutations. Hence the number of flagged Schur functions that are distinct
as polynomials is shown to be the R-parabolic Catalan number. The projecting
and lifting processes that relate the notions of 312-avoidance and of
R-312-avoidance are described with maps developed for other purposes.Comment: 27 pages, 2 figures. Identical to v.2, except for the insertion of
the publication data for the DMTCS journal (dates and volume/issue/number).
This is two-thirds of our preprint "Parabolic Catalan numbers count flagged
Schur functions; Convexity of tableau sets for Demazure characters",
arXiv:1612.06323v
Large-to small-scale dynamo in domains of large aspect ratio: kinematic regime
The Sunâs magnetic field exhibits coherence in space and time on much larger scales than
the turbulent convection that ultimately powers the dynamo. In this work, we look for numerical
evidence of a large-scale magnetic field as the magnetic Reynolds number, Rm, is
increased. The investigation is based on the simulations of the induction equation in elongated
periodic boxes. The imposed flows considered are the standard ABC flow (named after
Arnold, Beltrami & Childress) with wavenumber ku = 1 (small-scale) and a modulated ABC
flow with wavenumbers ku = m, 1, 1 ± m, where m is the wavenumber corresponding to
the long-wavelength perturbation on the scale of the box. The critical magnetic Reynolds
number Rcrit
m decreases as the permitted scale separation in the system increases, such that
Rcrit
m â [Lx /Lz]
â1/2. The results show that the α-effect derived from the mean-field theory
ansatz is valid for a small range of Rm after which small scale dynamo instability occurs and the
mean-field approximation is no longer valid. The transition from large- to small-scale dynamo
is smooth and takes place in two stages: a fast transition into a predominantly small-scale
magnetic energy state and a slower transition into even smaller scales. In the range of Rm
considered, the most energetic Fourier component corresponding to the structure in the long
x-direction has twice the length-scale of the forcing scale. The long-wavelength perturbation
imposed on the ABC flow in the modulated case is not preserved in the eigenmodes of the
magnetic field
The Age, Metallicity and Alpha-Element Abundance of Galactic Globular Clusters from Single Stellar Population Models
Establishing the reliability with which stellar population parameters can be
measured is vital to extragalactic astronomy. Galactic GCs provide an excellent
medium in which to test the consistency of Single Stellar Population (SSP)
models as they should be our best analogue to a homogeneous (single) stellar
population. Here we present age, metallicity and -element abundance
measurements for 48 Galactic globular clusters (GCs) as determined from
integrated spectra using Lick indices and SSP models from Thomas, Maraston &
Korn, Lee & Worthey and Vazdekis et al. By comparing our new measurements to
independent determinations we are able to assess the ability of these SSPs to
derive consistent results -- a key requirement before application to
heterogeneous stellar populations like galaxies.
We find that metallicity determinations are extremely robust, showing good
agreement for all models examined here, including a range of enhancement
methods. Ages and -element abundances are accurate for a subset of our
models, with the caveat that the range of these parameters in Galactic GCs is
limited. We are able to show that the application of published Lick index
response functions to models with fixed abundance ratios allows us to measure
reasonable -element abundances from a variety of models. We also
examine the age-metallicity and [/Fe]-metallicity relations predicted
by SSP models, and characterise the possible effects of varied model horizontal
branch morphology on our overall results.Comment: 22 pages, 19 figures, accepted for publication in MNRA
New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi
Nine (= 2 x 2 x 2 + 1) product identities for certain one-variable generating functions of certain families of plane partitions are presented in a unified fashion. The first two of these identities are originally due to MacMahon, Bender, Knuth, Gordon and Andrews and concern symmetric plane partitions. All nine identities are derived from tableaux descriptions of weights of especially nice representations of Lie groups, eight of them for the âright end nodeâ representations of SOË(2n+1) and Sp(2n). The two newest identities come from a tableaux description which originally arose in work of De Concini and Procesi on classical invariant theory. All of the identities are of the most interest when viewed, in the context of plane partitions with symmetries contained in three-dimensional boxes
Product evaluations of lefschetz determinants for grassmannians and of determinants of multinomial coefficients
A general result which produces product evaluations of determinants of certain raising operators for sl(2) representations is obtained. The most combinatorially interesting cases occur for self-dual raising operators of Peck posets. Applications include the following: A nice product expression is found for the determinant of the Lefschetz duality linear transformation on the cohomology of a Grassmannian. Known product expressions for the cardinalities of two sets of plane partitions are re-derived. The appearance of rising factorials for the hooks in one of these product expressions is âexplainedâ by the appearance of rising factorials in sl(2) determinants. A higher dimensional generalization in a certain sense of MacMahon's famous product enumeration result for Ferrers diagrams contained in a box is stated in the context of nonintersecting lattice paths
Minuscule Elements of Weyl Groups, the Numbers Game, andd-Complete Posets
Certain posets associated to a restricted version of the numbers game of Mozes are shown to be distributive lattices. The posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties, which form the definition ofd-complete poset. In representation theoretic language, the top âminuscule portionsâ of weight diagrams for integrable representations of simply laced KacâMoody algebras are shown to be distributive lattices. These lattices form a certain family of intervals of weak Bruhat orders. These Bruhat lattices are useful in studying reduced decompositions of λ-minuscule elements of Weyl groups and their associated Schubert varieties. Thed-complete posets have recently been proven to possess both the hook length and the jeu de taquin properties
Equivalence of the combinatorial and the classical definitions of schur functions
A short elementary proof of the title is presented
Reflection and algorithm proofs of some more lie group dual pair identities
Recently developed reflection techniques of Gessel and Zeilberger and Schensted algorithm techniques of Benkart and Stroomer are used to give new proofs of some dual pair (or Cauchy-type) symmetric function identifies first found by A. O. Morris long ago and recently found anew by Hasegawa in the context of dual pairs of representations of Lie algebras
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