scholarly journals A Flag Whitney Number Formula for Matroid Kazhdan-Lusztig Polynomials

10.37236/6120 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Max Wakefield
Keyword(s):  
Intersection Cohomology ◽  
Combinatorial Formula ◽  
Whitney Numbers ◽  
Number Formula ◽  
Whitney Number ◽  
Insight Into ◽  
Lusztig Polynomials

For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main result in this note is a combinatorial formula for the coefficients of these matroid Kazhdan-Lusztig polynomials in terms of flag Whitney numbers. This formula gives insight into some vanishing behavior of the matroid Kazhdan-Lusztig polynomials.

scholarly journals Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Francesco Brenti ◽  
Fabrizio Caselli
Keyword(s):  
Combinatorial Formula ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Bruhat Graph ◽  
Lusztig Polynomials

International audience We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions. On montre une formule combinatoire pour les polynômes de Kazhdan-Lusztig qui est valable en toute généralité. Cette formule est plus simple et plus explicite que toutes les autres formules connues; de plus, elle ne peut pas être simplifiée linéairement. La preuve utilise une nouvelle base pour la sous-algèbre des sommets de l’algèbre des fonctions quasi-symmetriques.

scholarly journals A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Kyungyong Lee ◽  
George D. Nasr ◽  
Jamie Radcliffe
Keyword(s):  
Young Tableaux ◽  
Combinatorial Formula ◽  
Combinatorial Interpretation ◽  
Special Cases ◽  
Almost All ◽  
Lusztig Polynomials

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.

scholarly journals Deodhar Elements in Kazhdan-Lusztig Theory

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Brant Jones
Keyword(s):  
Coxeter Groups ◽  
Pattern Avoidance ◽  
Weyl Groups ◽  
Schubert Varieties ◽  
Combinatorial Formula ◽  
Nous Obtenons ◽  
Integer Coefficients ◽  
Nous Montrons ◽  
International Audience ◽  
Lusztig Polynomials

International audience The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered. Les polynômes de Kazhdan-Lusztig $P_{x,w}(q)$ des groupes de Weyl finis apparaissent en théorie des représentations, ainsi qu’en géométrie des variétés de Schubert. Il a été démontré peu après leur introduction qu’ils avaient des coefficients entiers positifs, mais on ne connaît toujours pas d’interprétation combinatoire simple de cette propriété dans le cas général. Deodhar a proposé un cadre donnant un algorithme, en général récursif, calculant des formules attractives pour les polynômes de Kazhdan-Lusztig. Billey-Warrington ont démontré que cet algorithme est non récursif lorsque$w$ évite les hexagones et les $321$ et qu’il donne des formules combinatoires simples. Nous introduisons une notion d’évitement de schémas dansles groupes de Coxeter quelconques nous permettant de généraliser les résultats de Billey-Warrington à tout groupe de Weyl fini. Nous montrons que le coefficient de tête $\mu (x,w)$ de ces polynômes de Kazhdan-Lusztig est toujours $0$ ou $1$. Cela généralise aussi des résultats de Fan-Greenqui identifient les groupes de Coxeter complètement serrés. Enfin, en type $A$, nous obtenons une classe plus large de permutations évitant la récursion.

scholarly journals A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $\rho$-Removed Uniform Matroids

10.37236/9435 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Kyungyong Lee ◽  
George D. Nasr ◽  
Jamie Radcliffe
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Young Tableaux ◽  
Combinatorial Formula ◽  
Integer Coefficients ◽  
Lusztig Polynomials

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan–Lusztig polynomials of $\rho$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.

scholarly journals A Combinatorial Formula for the Coefficient of q in Kazhdan–Lusztig Polynomials

2019 ◽  
Author(s):  
Leonardo Patimo
Keyword(s):  
Weyl Groups ◽  
Combinatorial Formula ◽  
Combinatorial Interpretation ◽  
Lusztig Polynomials

Abstract We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan–Lusztig polynomials and we prove it for finite simply-laced Weyl groups.

Calibration problems in hydrogen luminosities

1966 ◽  
Vol 24 ◽  
pp. 322-330
Author(s):  
A. Beer
Keyword(s):  
Galactic Structure ◽  
B Stars ◽  
Insight Into

The investigations which I should like to summarize in this paper concern recent photo-electric luminosity determinations of O and B stars. Their final aim has been the derivation of new stellar distances, and some insight into certain patterns of galactic structure.

Stochasting modeling of planetary ring systems

1984 ◽  
Vol 75 ◽  
pp. 461-469 ◽  
Author(s):  
Robert W. Hart
Keyword(s):  
Maximum Entropy ◽  
Ring System ◽  
The Sun ◽  
Ultimate State ◽  
Interparticle Interactions ◽  
Ring Systems ◽  
First Order ◽  
Planetary Ring ◽  
Insight Into

ABSTRACTThis paper models maximum entropy configurations of idealized gravitational ring systems. Such configurations are of interest because systems generally evolve toward an ultimate state of maximum randomness. For simplicity, attention is confined to ultimate states for which interparticle interactions are no longer of first order importance. The planets, in their orbits about the sun, are one example of such a ring system. The extent to which the present approximation yields insight into ring systems such as Saturn's is explored briefly.

Skeletal elements of crinoid echinoderms: High-resolution structural and microanalytical results

1983 ◽  
Vol 41 ◽  
pp. 194-195
Author(s):  
D. F. Blake ◽  
L. F. Allard ◽  
D. R. Peacor
Keyword(s):  
High Resolution ◽  
Marine Invertebrates ◽  
Sea Urchins ◽  
Structural Features ◽  
Sea Stars ◽  
High Resolution Tem ◽  
Relationship Of ◽  
The Relationship ◽  
Insight Into

Echinodermata is a phylum of marine invertebrates which has been extant since Cambrian time (c.a. 500 m.y. before the present). Modern examples of echinoderms include sea urchins, sea stars, and sea lilies (crinoids). The endoskeletons of echinoderms are composed of plates or ossicles (Fig. 1) which are with few exceptions, porous, single crystals of high-magnesian calcite. Despite their single crystal nature, fracture surfaces do not exhibit the near-perfect {10.4} cleavage characteristic of inorganic calcite. This paradoxical mix of biogenic and inorganic features has prompted much recent work on echinoderm skeletal crystallography. Furthermore, fossil echinoderm hard parts comprise a volumetrically significant portion of some marine limestones sequences. The ultrastructural and microchemical characterization of modern skeletal material should lend insight into: 1). The nature of the biogenic processes involved, for example, the relationship of Mg heterogeneity to morphological and structural features in modern echinoderm material, and 2). The nature of the diagenetic changes undergone by their ancient, fossilized counterparts. In this study, high resolution TEM (HRTEM), high voltage TEM (HVTEM), and STEM microanalysis are used to characterize tha ultrastructural and microchemical composition of skeletal elements of the modern crinoid Neocrinus blakei.

Structure and function of neural networks in the retina

1988 ◽  
Vol 46 ◽  
pp. 26-27
Author(s):  
Peter Sterling
Keyword(s):  
Ganglion Cells ◽  
Three Dimensional ◽  
Structure And Function ◽  
Synaptic Connections ◽  
Visual Axis ◽  
One Dimensional ◽  
Cat Retina ◽  
Ultrathin Sections ◽  
And Function ◽  
Insight Into

The synaptic connections in cat retina that link photoreceptors to ganglion cells have been analyzed quantitatively. Our approach has been to prepare serial, ultrathin sections and photograph en montage at low magnification (˜2000X) in the electron microscope. Six series, 100-300 sections long, have been prepared over the last decade. They derive from different cats but always from the same region of retina, about one degree from the center of the visual axis. The material has been analyzed by reconstructing adjacent neurons in each array and then identifying systematically the synaptic connections between arrays. Most reconstructions were done manually by tracing the outlines of processes in successive sections onto acetate sheets aligned on a cartoonist's jig. The tracings were then digitized, stacked by computer, and printed with the hidden lines removed. The results have provided rather than the usual one-dimensional account of pathways, a three-dimensional account of circuits. From this has emerged insight into the functional architecture.

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