scholarly journals A Schur-Like Basis of NSym Defined by a Pieri Rule

10.37236/3857 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
John Maxwell Campbell ◽  
Karen Feldman ◽  
Jennifer Light ◽  
Pavel Shuldiner ◽  
Yan Xu
Keyword(s):  
Symmetric Functions ◽  
Arbitrary Element ◽  
Dual Algebra ◽  
Natural Analogues ◽  
Elementary Basis ◽  
Homogeneous Basis

Recent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions. We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule. The commutative image of an element of this basis indexed by a partition equals the element of the Schur basis indexed by the same partition and the commutative image is $0$ otherwise.  We establish a rule for right-multiplying an arbitrary element of this basis by an arbitrary element of the ribbon basis, and a Murnaghan-Nakayama-like rule for this new basis.  Elements of this new basis indexed by compositions of the form $(1^n, m, 1^r)$ are evaluated in terms of the complete homogeneous basis and the elementary basis.

scholarly journals Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Laura Colmenarejo
Keyword(s):  
Symmetric Functions ◽  
Symmetric Groups ◽  
Linear Groups ◽  
New Approach ◽  
Stability Properties ◽  
Integer Points ◽  
General Linear Groups ◽  
International Audience ◽  
Geometric Techniques ◽  
Homogeneous Basis

International audience Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope. Les coefficients du pléthysme sont des constantes de structure importantes de la théorie des fonctions symétriques, ainsi que de la théorie de la représentation des groupes symétriques et des groupes généraux linéaires. En 1950, Foulkes a observé pour ces coefficients de phénomènes de stabilité: certaines suites de coefficients du pléthysme sont stationnaires. De telles propriétés ont été démontrées par Brion, au moyen de techniques géométriques, et par Thibon et Carré, au moyen d’opérateurs vertex. Dans ce travail, nous présentons une nouvelle approche, purement combinatoire, pour démontrer des propriétés de stabilité de ce type. Nous décomposons les coefficients du pléthysme comme somme alternées de coefficients de pléthysme d’un autre type (liés à la base des fonctions symétriques sommes complètes), qui comptent les points entiers dans des polytopes. Nous démontrons la stabilité des suites de ces coefficients en exhibant des bijections entres les ensembles de points entiers des polytopes correspondants.

scholarly journals Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Adam Doliwa
Keyword(s):  
Power Series ◽  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Algebra Structure ◽  
Natural Basis ◽  
Dual Algebra ◽  
Power Sum ◽  
Hopf Algebra Structure

We introduce a coloured generalization  $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions  described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables.  We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.

scholarly journals Generalized Pattern-Matching Conditions for

2013 ◽  
Vol 2013 ◽  
pp. 1-20 ◽  
Author(s):  
Sergey Kitaev ◽  
Andrew Niedermaier ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Pattern Matching ◽  
Infinite Number ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Matching Condition ◽  
Natural Analogues ◽  
Generalized Pattern Matching ◽  
Product Order

We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of . Our research leads to connections to many known objects/structures yet to be explained combinatorially.

A UNIQUE FINDING OF ZIRCON INTO BADDELEYITE TRANSFORMATION (THE ICHETJU ORE OCCURRENCE, THE MIDDLE TIMAN)

2018 ◽  
pp. 27-35
Author(s):  
S. G. Skublov ◽  
A. O. Krasotkina ◽  
A. B. Makeyev ◽  
O. L. Galankina ◽  
A. E. Melnik
Keyword(s):  
High Temperature ◽  
Experimental Studies ◽  
Middle Timan ◽  
Natural Analogues ◽  
Opposite Situation ◽  
Formation Conditions

Findings of the growth relationships between baddeleyite and zircon are rare, due to significant differences in the formation conditions of the minerals. A reaction replacement (partial to complete) of baddeleyite by zircon is possible during metamorphism accompanied by the interaction with high-Si fluids. The opposite situation, when zircon is replaced by baddeleyite, is extremely rare in the nature. Transformation of zircon from polymineral (compound) ore occurrence Ichetju (the Middle Timan) with the formation of microaggregates of baddeleyite, ratile and florencite has been found out. The size of the largest segregations of baddeleyite does not exceed 10 microns in diameter. Microaggregates are unevenly related to the rim of zircon with a thickness of 10 to 50 rfn, voids and cracks across the grain. Altered zircon rim (a mixture of newly formed minerals) is characterized by sharply increased composition of REE (especially LREE), Y, Nb, Ca, Ti. The composition of Th and U also increases. An overview of the experimental studies on the reaction between zircon and baddeleyite and single natural analogues allows to make a conclusion that the most likely mechanism of the transformation of zircon from ore occurrence Ichetju to baddeleyite (intergrowth with ratile and florencite) is due to the effect of interaction of primary zircon with high-temperature (higher than 500—600°C) alkaline fluids transporting HFSE (REE, Y, Nb, Ti). This is indirectly confirmed by the findings of zircon with anomalous high composition of Y and REE up to 100000 and 70000 ppm respectively.

ASSESSMENT OF GRAY FOREST SOIL ENZYMATIC ACTIVITY IN AGRICULTURAL SYSTEMS

2019 ◽  
Keyword(s):  
Enzymatic Activity ◽  
Environmental Sustainability ◽  
Hydrolytic Enzymes ◽  
Invertase Activity ◽  
Phosphorus Cycle ◽  
Soil Enzymatic Activity ◽  
Urease Enzyme ◽  
Natural Analogues ◽  
Activity Of Enzymes ◽  
The Impact

On the grey forest medium-loamy soil of Vladimir Opolye region we have studied the impact of various methods of basic cultivation and fertilizer systems on the activity of redox and hydrolytic enzymes: ure-ase (nitrogen cycle), invertase (carbon cycle), phosphatase (phosphorus cycle), and catalase, involved in the cycle of carbon in the soil. The second humus horizon with capacity of 19-24cm was found at the depth of 20 - 21 cm on the experimental field. We have studied three modes of basic soil cultivation: an-nual shallow flat plowing (6-8 cm), annual deep flat plowing (20-22 cm), and annual moldboard plowing (20-22 cm) with normal and intensive application of fertilizers. The most enzymatically active layer is 0-20 cm. No relevant difference has been found in the level of enzymes activity between variants of basic soil treatment. Activity of enzymes increases with application of fertilizers on the intensive background. In agrogenic soils, soil enzymatic activity is lower on average by 16-22% compared to the soil of the res-ervoir. The biggest negative transformation of activity has been observed at the urease enzyme (up to 50%). With annual moldboard plowing on the intensive backgroung, enzyme activity has been close to the natural level – 98.4%. Catalise and invertase activity in this case were found to be higher (105 and 116% respectively) than that of natural analogues. Activity of enzymes increases with intensive application of fertilizers as compared with normal background. This is particularly evident with 6-8cm deep beardless plowing and 20-22cm deep moldboard plowing. In general, the obtained biochemical indicators charac-terize the highest environmental sustainability of this variation within our research.

CLASSES OF WEIGHTED SYMMETRIC FUNCTIONS

1988 ◽  
Vol 14 (2) ◽  
pp. 429
Author(s):  
Tran
Keyword(s):  
Symmetric Functions

WEIGHTED SYMMETRIC FUNCTIONS

1989 ◽  
Vol 15 (1) ◽  
pp. 313
Author(s):  
Tran
Keyword(s):  
Symmetric Functions

scholarly journals Applying Ateb–Gabor Filters to Biometric Imaging Problems

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 717
Author(s):  
Mariia Nazarkevych ◽  
Natalia Kryvinska ◽  
Yaroslav Voznyi
Keyword(s):  
Wavelet Transformation ◽  
Symmetric Functions ◽  
Gabor Filter ◽  
Signal To Noise Ratio ◽  
The Other ◽  
Mean Square ◽  
Image Transformation ◽  
Gabor Filtering ◽  
Signal To Noise ◽  
Filtration Method

This article presents a new method of image filtering based on a new kind of image processing transformation, particularly the wavelet-Ateb–Gabor transformation, that is a wider basis for Gabor functions. Ateb functions are symmetric functions. The developed type of filtering makes it possible to perform image transformation and to obtain better biometric image recognition results than traditional filters allow. These results are possible due to the construction of various forms and sizes of the curves of the developed functions. Further, the wavelet transformation of Gabor filtering is investigated, and the time spent by the system on the operation is substantiated. The filtration is based on the images taken from NIST Special Database 302, that is publicly available. The reliability of the proposed method of wavelet-Ateb–Gabor filtering is proved by calculating and comparing the values of peak signal-to-noise ratio (PSNR) and mean square error (MSE) between two biometric images, one of which is filtered by the developed filtration method, and the other by the Gabor filter. The time characteristics of this filtering process are studied as well.

scholarly journals A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

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