scholarly journals Shiraishi functor and non-Kerov deformation of Macdonald polynomials

2020 ◽  
Vol 80 (10) ◽  
Author(s):  
Hidetoshi Awata ◽  
Hiroaki Kanno ◽  
Andrei Mironov ◽  
Alexei Morozov
Keyword(s):  
Network Models ◽  
Elliptic Systems ◽  
Macdonald Polynomials ◽  
Symmetric Polynomials ◽  
Leading Order ◽  
Young Diagrams ◽  
Schur Polynomials ◽  
Pochhammer Symbol ◽  
The Family ◽  
Two Kinds Of Variables

AbstractWe suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this generalization is valid for the entire Shiraishi series, not only for its Noumi–Shiraishi part. The theta function needed in the recently suggested description of the double-elliptic systems [Awata et al. JHEP 2020:150, arXiv:2005.10563, (2020)], 6d N = 2* SYM instanton calculus and the doubly-compactified network models, is a very particular member of this huge family. The series depends on two kinds of variables, $$\vec {x}$$ x → and $$\vec {y}$$ y → , and on a set of parameters, which becomes infinitely large now. Still, one of the parameters, p is distinguished by its role in the series grading. When $$\vec {y}$$ y → are restricted to a discrete subset labeled by Young diagrams, the series multiplied by a monomial factor reduces to a polynomial at any given order in p. All this makes the map from functions to the hypergeometric-like series very promising, and we call it Shiraishi functor despite it remains to be seen, what are exactly the morphisms that it preserves. Generalized Noumi–Shiraishi (GNS) symmetric polynomials inspired by the Shiraishi functor in the leading order in p can be obtained by a triangular transform from the Schur polynomials and possess an interesting grading. They provide a family of deformations of Macdonald polynomials, as rich as the family of Kerov functions, still very different from them, and, in fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov case, these polynomials do not depend on the ordering of Young diagrams in the triangular expansion.

scholarly journals Invariant valuations on quaternionic vector spaces

2012 ◽  
Vol 11 (3) ◽  
pp. 467-499 ◽  
Author(s):  
Andreas Bernig
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Young Diagrams ◽  
Schur Polynomials ◽  
Combinatorial Dimension ◽  
Translation Invariant

AbstractThe spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.

scholarly journals NONPLANAR INTEGRABILITY: BEYOND THE SU(2) SECTOR

2011 ◽  
Vol 26 (26) ◽  
pp. 4553-4583 ◽  
Author(s):  
ROBERT DE MELLO KOCH ◽  
BADR AWAD ELSEID MOHAMMED ◽  
STEPHANIE SMITH
Keyword(s):  
Harmonic Oscillators ◽  
Projection Operators ◽  
Young Diagrams ◽  
Schur Polynomials ◽  
Recursion Relations ◽  
Anomalous Dimensions ◽  
Dilatation Operator ◽  
Analytic Expressions ◽  
The One ◽  
Kravchuk Polynomials

We compute the one-loop anomalous dimensions of restricted Schur polynomials with a classical dimension Δ~O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

scholarly journals A comparison theorem for cosmological lightcones

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
Mauro Carfora ◽  
Francesca Familiari
Keyword(s):  
Scalar Curvature ◽  
Variational Analysis ◽  
Length Scale ◽  
Leading Order ◽  
Spacetime Geometry ◽  
The Family ◽  
Physical Spacetime ◽  
The Given ◽  
Natural Scale ◽  
Cosmological Spacetime

AbstractLet (M, g) denote a cosmological spacetime describing the evolution of a universe which is isotropic and homogeneous on large scales, but highly inhomogeneous on smaller scales. We consider two past lightcones, the first, $${{\mathcal {C}}_{L}^{-}}(p, g)$$ C L - ( p , g ) , is associated with the physical observer $$p\in \,M$$ p ∈ M who describes the actual physical spacetime geometry of (M, g) at the length scale L, whereas the second, $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ C L - ( p , g ^ ) , is associated with an idealized version of the observer p who, notwithstanding the presence of local inhomogeneities at the given scale L, wish to model (M, g) with a member $$(M, \hat{g})$$ ( M , g ^ ) of the family of Friedmann–Lemaitre–Robertson–Walker spacetimes. In such a framework, we discuss a number of mathematical results that allows a rigorous comparison between the two lightcones $${\mathcal {C}_{L}^{-}}(p, g)$$ C L - ( p , g ) and $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ C L - ( p , g ^ ) . In particular, we introduce a scale-dependent (L) lightcone-comparison functional, defined by a harmonic type energy, associated with a natural map between the physical $${\mathcal {C}_{L}^{-}}(p, g)$$ C L - ( p , g ) and the FLRW reference lightcone $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ C L - ( p , g ^ ) . This functional has a number of remarkable properties, in particular it vanishes iff, at the given length-scale, the corresponding lightcone surface sections (the celestial spheres) are isometric. We discuss in detail its variational analysis and prove the existence of a minimum that characterizes a natural scale-dependent distance functional between the two lightcones. We also indicate how it is possible to extend our results to the case when caustics develop on the physical past lightcone $${\mathcal {C}_{L}^{-}}(p, g)$$ C L - ( p , g ) . Finally, by exploiting causal diamond theory, we show how the distance functional is related (to leading order in the scale L) to spacetime scalar curvature in the causal past of the two lightcones, and briefly illustrate a number of its possible applications.

scholarly journals Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Cristian Lenart
Keyword(s):  
Weyl Group ◽  
Type A ◽  
Macdonald Polynomials ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Affine Weyl Group ◽  
Littlewood Polynomials ◽  
Combinatorial Formulas ◽  
International Audience ◽  
Type C

International audience A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.

Behind Family Trees

2014 ◽  
Vol 4 (2) ◽  
pp. 189-221 ◽  
Author(s):  
Jyri Lehtinen ◽  
Terhi Honkola ◽  
Kalle Korhonen ◽  
Kaj Syrjänen ◽  
Niklas Wahlberg ◽  
...  
Keyword(s):  
Network Models ◽  
Language Family ◽  
Tree Model ◽  
Language History ◽  
Network Methods ◽  
The Family ◽  
Tree Models ◽  
Basic Vocabulary ◽  
Tree Building ◽  
Computational Analyses

Although it has long been recognized that the family tree model is too simplistic to account for historical connections between languages, most computational studies of language history have concentrated on tree-building methods. Here, we employ computational network methods to assess the utility of network models in comparison with tree models in studying the subgrouping of Uralic languages. We also compare basic vocabulary data with words that are more easily borrowed and replaced cross-linguistically (less basic vocabulary) in order to find out how secondary connections affect computational analyses of this language family. In general, the networks support a treelike pattern of diversification, but also provide information about conflicting connections underlying some of the ambiguous divergences in the trees. These are seen as reflections of unclear divergence patterns (either in ancestral protolanguages or between languages closely related at present), which pose problems for a tree model. The networks also show that the relationships of closely related present-day languages are more complex than what the tree models suggest. When comparing less basic with basic vocabulary, we can detect the effect of borrowing between different branches (horizontal transfer) mostly between and within the Finnic and Saami subgroups. We argue that the trees obtained with basic vocabulary provide the primary pattern of the divergence of a language family, whereas networks, especially those constructed with less basic vocabulary, add reality to the picture by showing the effect of more complicated developments affecting the connections between the languages.

scholarly journals Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Hidetoshi Awata ◽  
Hiroaki Kanno ◽  
Andrei Mironov ◽  
Alexei Morozov
Keyword(s):  
Partition Function ◽  
Elliptic Function ◽  
Gauge Theories ◽  
Network Models ◽  
Symmetric Polynomials ◽  
Conformal Blocks ◽  
Gamma Functions ◽  
Elliptic Genera ◽  
Surface Operator ◽  
Insufficient Number

Abstract As a development of [1], we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in 6d SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular U(1) case, we find an explicit plethystic formula for the 6d partition function, which is a non-trivial elliptic generalization of the (q, t) Nekrasov-Okounkov formula from 5d.

scholarly journals Analysis of Religious Socialization Based on Interviews Conducted with Young Adults

Religions ◽  
2019 ◽  
Vol 10 (6) ◽  
pp. 365 ◽  
Author(s):  
Gabriella Pusztai ◽  
Zsuzsanna Demeter-Karászi
Keyword(s):  
Young Adults ◽  
Social Network ◽  
Qualitative Interviews ◽  
Network Models ◽  
Religious Practice ◽  
Constructivist Approach ◽  
Religious Socialization ◽  
The Social ◽  
The Family ◽  
Religious Transmission

The term religious socialization has become a pressing issue in the context of religious socialization research. It also raises the question whether religious transmission can be interpreted through the reproduction or constructivist approach. Previously, the reconstruction models determined the approach of studying religious socialization. According to these models, socialization meant the adoption of the patterns of religious practice in the family. In this sense, socialization is periodical. The constructivist and the social network models of sociology significantly changed our conception of religious socialization. The earlier model was replaced by a model which rests on activity, correlation and open-endedness. In this paper, 18 qualitative interviews were analyzeanalyzed. Because religiosity is a multidimensional phenomenon, we wanted to analyze development in each dimension, which is why we relied on Glock and Stark’s model. Based on our results, seven types emerged and these findings have strengthened the constructivist approach.

scholarly journals Relation between large dimension operators and oscillator algebra of Young diagrams

2015 ◽  
Vol 12 (04) ◽  
pp. 1550047 ◽  
Author(s):  
Hai Lin
Keyword(s):  
Large Dimension ◽  
Harmonic Oscillators ◽  
Oscillator Algebra ◽  
Young Diagrams ◽  
Schur Polynomials ◽  
Large N

The operators with large scaling dimensions can be labeled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large N but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu–Wigner contraction of the u(2) algebra inside of the u(p) algebra of p giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large N limit.

scholarly journals Polynomials from Combinatorial -theory

2019 ◽  
pp. 1-34 ◽  
Author(s):  
Cara Monical ◽  
Oliver Pechenik ◽  
Dominic Searles
Keyword(s):  
Euler Characteristic ◽  
Combinatorial Theory ◽  
Symmetric Polynomials ◽  
Schur Polynomials ◽  
Fundamental Particle ◽  
Ring Of Polynomials ◽  
Special Case

Abstract We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$ -theoretic deformation of the quasi-key basis and also a lift of the $K$ -analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$ -analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $K$ -theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $K$ -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.

scholarly journals Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$

2020 ◽  
Vol 53 (2) ◽  
pp. 192-205
Author(s):  
T. Vasylyshyn ◽  
A. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Analytic Functions ◽  
Symmetric Functions ◽  
Real Banach Space ◽  
Symmetric Polynomials ◽  
The Real ◽  
Cartesian Power ◽  
The Family ◽  
Algebraic Basis

We construct an algebraic basis of the algebra of symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) continuous polynomials on the $n$th Cartesian power of the real Banachspace $L_^{(\mathbb{R})}\infty[0,1]$ of Lebesgue measurable essentially bounded real valued functions on $[0,1].$ Also we describe the spectrum of the Fr\'{e}chet algebra $A_s(L_^{(\mathbb{R})}\infty[0,1])$ of symmetric real-valued functions on the space $L_^{(\mathbb{R})}\infty[0,1]$, which is the completion of the algebra of symmetric continuous real-valued polynomials on  $L_^{(\mathbb{R})}\infty[0,1]$ with respect to the family of norms of uniform convergence of complexifications of polynomials. We show that $A_s(L_^{(\mathbb{R})}\infty[0,1])$ contains not only analytic functions. Results of the paper can be used for investigations of algebras of symmetric functions on the $n$th Cartesian power of the Banach space $L_^{(\mathbb{R})}\infty[0,1]$.

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