scholarly journals DUAL EQUIVALENCE GRAPHS I: A NEW PARADIGM FOR SCHUR POSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
SAMI H. ASSAF
Keyword(s):  
Systematic Study ◽  
Generating Functions ◽  
Universal Method ◽  
New Paradigm ◽  
Quasisymmetric Functions ◽  
Dual Equivalence ◽  
Combinatorial Construction ◽  
Dual Equivalence Graphs ◽  
Schur Positive

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.

Dual Equivalence Graphs Revisited

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Macdonald Polynomials ◽  
Quasisymmetric Function ◽  
Schur Function ◽  
Connected Component ◽  
Slight Improvement ◽  
Quasisymmetric Functions ◽  
Dual Equivalence ◽  
International Audience ◽  
The Creation ◽  
Dual Equivalence Graphs

International audience In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions so that the sum of the weights of a connected component is a single Schur function. In this paper, we improve on Assaf's axiomatization of such graphs, giving locally testable criteria that are more easily verified by computers. We then demonstrate the utility of this result by giving explicit Schur expansions for a family of Lascoux-Leclerc-Thibon polynomials. This family properly contains the previously known case of polynomials indexed by two skew shapes, as was described in a 1995 paper by Christophe Carré and Bernard Leclerc. As an immediate corollary, we gain an explicit Schur expansion for a family of modified Macdonald polynomials in terms of Yamanouchi words. This family includes all polynomials indexed by shapes with less than four cells in the first row and strictly less than three cells in the second row, a slight improvement over the known two column case described in 2005 by James Haglund, Mark Haiman, and Nick Loehr.

scholarly journals A Note on Three Types of Quasisymmetric Functions

10.37236/1958 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
T. Kyle Petersen
Keyword(s):  
Generating Functions ◽  
Type A ◽  
Type B ◽  
Quasisymmetric Functions ◽  
Descent Algebra ◽  
Combinatorial Description ◽  
Partition Theorems

In the context of generating functions for $P$-partitions, we revisit three flavors of quasisymmetric functions: Gessel's quasisymmetric functions, Chow's type B quasisymmetric functions, and Poirier's signed quasisymmetric functions. In each case we use the inner coproduct to give a combinatorial description (counting pairs of permutations) to the multiplication in: Solomon's type A descent algebra, Solomon's type B descent algebra, and the Mantaci-Reutenauer algebra, respectively. The presentation is brief and elementary, our main results coming as consequences of $P$-partition theorems already in the literature.

scholarly journals P-Partitions and p-Positivity

2019 ◽  
Author(s):  
Per Alexandersson ◽  
Robin Sulzgruber
Keyword(s):  
Linear Combination ◽  
Generating Functions ◽  
Quasisymmetric Functions ◽  
Vertical Strip ◽  
Power Sums ◽  
Llt Polynomials ◽  
Tutte Polynomials

AbstractUsing the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently, any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application, we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials, and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.

scholarly journals Coxeter-Knuth Graphs and a Signed Little Map for Type B Reduced Words

10.37236/4384 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Sara Billey ◽  
Zachary Hamaker ◽  
Austin Roberts ◽  
Benjamin Young
Keyword(s):  
Structure Theorem ◽  
Type A ◽  
Axiomatic Characterization ◽  
Schubert Calculus ◽  
Type B ◽  
Dual Equivalence ◽  
Carry Over ◽  
Dual Equivalence Graphs ◽  
Reduced Words

We define an analog of David Little’s algorithm for reduced words in type B, and investigate its main properties. In particular, we show that our algorithm preserves the recording tableau of Kraśkiewicz insertion, and that it provides a bijective realization of the Type B transition equations in Schubert calculus. Many other aspects of type A theory carry over to this new setting. Our primary tool is a shifted version of the dual equivalence graphs defined by Assaf and further developed by Roberts. We provide an axiomatic characterization of shifted dual equivalence graphs, and use them to prove a structure theorem for the graph of Type B Coxeter-Knuth relations. 

scholarly journals A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
S. Assaf
Keyword(s):  
Graph Structure ◽  
Young Tableaux ◽  
Polynomial Representation ◽  
Crystal Graphs ◽  
Dual Equivalence ◽  
Weyl Duality ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Do So ◽  
Dual Equivalence Graphs

International audience For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight $0$. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the $0$-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.

Solar Filaments as Tracers of Subsurface Processes

2000 ◽  
Vol 179 ◽  
pp. 177-183
Author(s):  
D. M. Rust
Keyword(s):  
Magnetic Fields ◽  
Magnetic Flux ◽  
Coronal Mass Ejections ◽  
Interplanetary Space ◽  
Intermediate Stage ◽  
Coronal Plasma ◽  
Magnetic Helicity ◽  
The Sun ◽  
New Paradigm ◽  
Solar Filaments

AbstractSolar filaments are discussed in terms of two contrasting paradigms. The standard paradigm is that filaments are formed by condensation of coronal plasma into magnetic fields that are twisted or dimpled as a consequence of motions of the fields’ sources in the photosphere. According to a new paradigm, filaments form in rising, twisted flux ropes and are a necessary intermediate stage in the transfer to interplanetary space of dynamo-generated magnetic flux. It is argued that the accumulation of magnetic helicity in filaments and their coronal surroundings leads to filament eruptions and coronal mass ejections. These ejections relieve the Sun of the flux generated by the dynamo and make way for the flux of the next cycle.

Representation of information in SEM by equal signal lines

1978 ◽  
Vol 36 (1) ◽  
pp. 144-145
Author(s):  
E. Rau ◽  
N. Karelin ◽  
V. Dukov ◽  
M. Kolomeytsev ◽  
S. Gavrikov ◽  
...  
Keyword(s):  
Electronic System ◽  
Space Distribution ◽  
Specimen Surface ◽  
Universal Method ◽  
Base Level ◽  
Signal Value ◽  
Information Signal ◽  
Fixed Base ◽  
Scanning Line ◽  
Amplitude Analyzer

There are different methods and devices for the increase of the videosignal information in SEM. For example, with the help of special pure electronic [1] and opto-electronic [2] systems equipotential areas on the specimen surface in SEM were obtained. This report generalizes quantitative universal method for space distribution representation of research specimen parameter by contour equal signal lines. The method is based on principle of comparison of information signal value with the fixed levels.Transformation image system for obtaining equal signal lines maps was developed in two versions:1)In pure electronic system [3] it is necessary to compare signal U (see Fig.1-a), which gives potential distribution on specimen surface along each scanning line with fixed base level signals εifor obtaining quantitative equipotential information on solid state surface. The amplitude analyzer-comparator gives flare sport videopulses at any fixed coordinate and any instant time when initial signal U is equal to one of the base level signals ε.

Quantitative parallel EELS spectrum imaging developed as a new tool in AEM

1992 ◽  
Vol 50 (2) ◽  
pp. 1202-1203
Author(s):  
Gianluigi Botton ◽  
Gilles L'espérance
Keyword(s):  
Statistical Analysis ◽  
Spatial Resolution ◽  
Personal Computer ◽  
Systematic Study ◽  
Present Author ◽  
Chemical Elements ◽  
Detection Limits ◽  
Research Groups ◽  
Quantitative Performance ◽  
Spectrum Imaging

As interest for parallel EELS spectrum imaging grows in laboratories equipped with commercial spectrometers, different approaches were used in recent years by a few research groups in the development of the technique of spectrum imaging as reported in the literature. Either by controlling, with a personal computer both the microsope and the spectrometer or using more powerful workstations interfaced to conventional multichannel analysers with commercially available programs to control the microscope and the spectrometer, spectrum images can now be obtained. Work on the limits of the technique, in terms of the quantitative performance was reported, however, by the present author where a systematic study of artifacts detection limits, statistical errors as a function of desired spatial resolution and range of chemical elements to be studied in a map was carried out The aim of the present paper is to show an application of quantitative parallel EELS spectrum imaging where statistical analysis is performed at each pixel and interpretation is carried out using criteria established from the statistical analysis and variations in composition are analyzed with the help of information retreived from t/γ maps so that artifacts are avoided.

Does the Motor System Facilitate Spatial Imagery?

2017 ◽  
Vol 49 (3) ◽  
pp. 127-137 ◽  
Author(s):  
Markus Krüger ◽  
Horst Krist
Keyword(s):  
Motor System ◽  
Third Graders ◽  
External Support ◽  
New Paradigm ◽  
Perceptual Condition ◽  
Imagery Condition ◽  
Passive Rotation ◽  
Perception Condition ◽  
Motor Condition ◽  
Dynamic Imagery

Abstract. Recent studies have ascertained a link between the motor system and imagery in children. A motor effect on imagery is demonstrated by the influence of stimuli-related movement constraints (i. e., constraints defined by the musculoskeletal system) on mental rotation, or by interference effects due to participants’ own body movements or body postures. This link is usually seen as qualitatively different or stronger in children as opposed to adults. In the present research, we put this interpretation to further scrutiny using a new paradigm: In a motor condition we asked our participants (kindergartners and third-graders) to manually rotate a circular board with a covered picture on it. This condition was compared with a perceptual condition where the board was rotated by an experimenter. Additionally, in a pure imagery condition, children were instructed to merely imagine the rotation of the board. The children’s task was to mark the presumed end position of a salient detail of the respective picture. The children’s performance was clearly the worst in the pure imagery condition. However, contrary to what embodiment theories would suggest, there was no difference in participants’ performance between the active rotation (i. e., motor) and the passive rotation (i. e., perception) condition. Control experiments revealed that this was also the case when, in the perception condition, gaze shifting was controlled for and when the board was rotated mechanically rather than by the experimenter. Our findings indicate that young children depend heavily on external support when imagining physical events. Furthermore, they indicate that motor-assisted imagery is not generally superior to perceptually driven dynamic imagery.

Sign in / Sign up

Export Citation Format

Share Document