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.

Quantum algebras and parity-dependent spectra

2007 ◽  
Vol 463 (2086) ◽  
pp. 2415-2427 ◽  
Author(s):  
C.V Sukumar ◽  
Andrew Hodges
Keyword(s):  
Quantum Optics ◽  
Harmonic Oscillator ◽  
Combinatorial Theory ◽  
Squeezed States ◽  
Quantum Algebras ◽  
Matrix Elements ◽  
Euler Numbers ◽  
Simple Harmonic Oscillator ◽  
Non Gaussian ◽  
Special Case

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

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 Orthogonal Polynomials of Compact Simple Lie Groups

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Maryna Nesterenko ◽  
Jiří Patera ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Uniform Method ◽  
Symmetric Polynomials ◽  
Semisimple Lie Group ◽  
Algebraic Construction ◽  
Special Case

Recursive algebraic construction of two infinite families of polynomials innvariables is proposed as a uniform method applicable to every semisimple Lie group of rankn. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of typeA1. The obtained not Laurent-type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of typesA1,A2,A3,C2,C3,G2, andB3together with lowest polynomials.

NORMALIZED RICCI FLOW, SURGERY, AND SEIBERG–WITTEN INVARIANTS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450005
Author(s):  
MASASHI ISHIDA
Keyword(s):  
Existence Theorem ◽  
Euler Characteristic ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Nonsingular Solution ◽  
Ricci Flows ◽  
Normalized Ricci Flow ◽  
Four Manifolds ◽  
Special Case

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's [Formula: see text] invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

scholarly journals Compare the ratio of symmetric polynomials of odds to one and stop

2017 ◽  
Vol 54 (1) ◽  
pp. 12-22
Author(s):  
Tomomi Matsui ◽  
Katsunori Ano
Keyword(s):  
Lower Bound ◽  
Optimal Stopping ◽  
Optimization Technique ◽  
Secretary Problem ◽  
Symmetric Polynomials ◽  
Threshold Strategy ◽  
Optimal Stopping Rule ◽  
Special Cases ◽  
Unified View ◽  
Special Case

AbstractIn this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last ℓ successes, given a sequence of independent Bernoulli trials of length N, where k and ℓ are predetermined integers satisfying 1≤k≤ℓ<N. This problem includes some odds problems as special cases, e.g. Bruss’ odds problem, Bruss and Paindaveine’s problem of selecting the last ℓ successes, and Tamaki’s multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton’s inequalities and optimization technique, which gives a unified view to the previous works.

scholarly journals Schur Polynomials, Banded Toeplitz Matrices and Widom's Formula

10.37236/2651 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Per Alexandersson
Keyword(s):  
Characteristic Equation ◽  
Toeplitz Matrices ◽  
Linear Recurrence ◽  
Schur Polynomials ◽  
Determinant Formula ◽  
Special Case

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot\mathbf{1}^c)/(\mathbf{\lambda} + k\cdot\mathbf{1}^r)}(x_1,\dots,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices.In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

scholarly journals Schubert complexes and degeneracy loci

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Steven V Sam
Keyword(s):  
Euler Characteristic ◽  
Vector Bundles ◽  
Homology Class ◽  
Schur Polynomials ◽  
Degeneracy Loci ◽  
Schubert Polynomials ◽  
Conceptual Proof ◽  
Degeneracy Locus ◽  
Alternating Sums ◽  
International Audience

International audience The classical Thom―Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears. \par La formule classique de Thom―Porteous exprime la classe d'homologie du locus de la dégénérescence d'une fonction générique entre deux fibrés vectoriels comme une somme alternée des polynômes de Schur. Un preuve de cette formule a été donnée par Pragacz en exprimant ce alternant somme comme la caractéristique d'Euler d'un complexe de Schur, ce qui donne une explication pour les signes. Fulton puis généralisée cette formule à la situation des drapeaux de fibrés vectoriels à l'aide alternant des sommes de polynômes de Schubert. S'appuyant sur le Schubert foncteurs de Kraśkiewicz et Pragacz, nous introduisons les complexes de Schubert et montrent que la somme alternée de Fulton peuvent être réalisées en tant que Euler caractéristique de ce complexe, fournissant ainsi une preuve conceptuelle pour lesquelles une somme alternée appara\^ıt.

Bar-Natan’s geometric complex and a categorification of the Dye–Kauffman–Miyazawa polynomial

2016 ◽  
Vol 25 (01) ◽  
pp. 1550076
Author(s):  
Keiji Tagami
Keyword(s):  
Equivalence Class ◽  
Euler Characteristic ◽  
Khovanov Homology ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Link Homology ◽  
Stable Equivalence ◽  
Special Case

In this paper, we construct a categorification of the two-variable Dye–Kauffman–Miyazawa polynomial by utilizing Bar-Natan’s construction of the Khovanov homology and homotopy quantum field theories (HQFTs) given by Turaev. In particular, for any stable equivalence class, we construct a [Formula: see text]-graded link homology over [Formula: see text] whose graded Euler characteristic is the two-variable Dye–Kauffman–Miyazawa polynomial. Moreover, we show that it is isomorphic to a special case of Dye–Kauffman–Manturov’s categorification. In this sense, we explain the special case of Dye–Kauffman–Manturov’s homology in terms of Bar-Natan’s construction.

scholarly journals Symmetric Polynomials in the Symplectic Alphabet and the Change of Variables $z_j = x_j + x_j^{-1}$

10.37236/9354 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Per Alexandersson ◽  
Luis Angel González-Serrano ◽  
Egor Maximenko ◽  
Mario Alberto Moctezuma-Salazar
Keyword(s):  
Toeplitz Matrices ◽  
Symmetric Polynomial ◽  
Laurent Polynomial ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Schur Polynomials ◽  
Power Sums ◽  
Change Of Variables ◽  
Schur Polynomial ◽  
Elementary Symmetric Polynomials

Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that\[P(x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1})=Q(x_1+x_1^{-1},\ldots,x_n+x_n^{-1}).\] We denote this polynomial $Q$ by $\Phi_n(P)$ and show that $\Phi_n$ is an epimorphism of algebras. We compute $\Phi_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomials of the form $\Phi_n(\operatorname{s}_{\lambda/\mu}^{(2n)})$, where $\operatorname{s}_{\lambda/\mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,\ldots,x_n,x^{-1}_1,\ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.

Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

2018 ◽  
Vol 41 ◽  
Author(s):  
Daniel Crimston ◽  
Matthew J. Hornsey
Keyword(s):  
General Theory ◽  
Biological Markers ◽  
Natural Environment ◽  
Relevant Dimension ◽  
Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

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