1,667 results on '"Ring of symmetric functions"'
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2. Cylindric symmetric functions and positivity
- Author
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David Palazzo and Christian Korff
- Subjects
Ring (mathematics) ,Pure mathematics ,Structure constants ,05E05, 05E10, 14N35, 53D45, 05A10 ,Subalgebra ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Affine Lie algebra ,Symmetric function ,symbols.namesake ,Symmetric group ,Frobenius algebra ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Ring of symmetric functions ,Mathematical Physics ,Mathematics - Representation Theory ,Mathematics - Abstract
We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra $\mathfrak{\widehat{sl}}_n$ and a specialised cyclotomic Hecke algebra. Using Schur-Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the $k$-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in $\Lambda$ whose structure constants are the 3-point genus 0 Gromov-Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group $G(n,1,k)$., Comment: 63 pages, 5 figures (v3: version accepted for publication in Algebraic Combinatorics)
- Published
- 2020
3. Symmetric polynomials in tropical algebra semirings
- Author
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Davorin Lešnik and Sara Kališnik
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Fundamental theorem ,Mathematics::General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,010103 numerical & computational mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Semiring ,Kleene algebra ,Computational Mathematics ,Symmetric polynomial ,Idempotence ,FOS: Mathematics ,Tropical geometry ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Physics::Atmospheric and Oceanic Physics ,Computer Science::Formal Languages and Automata Theory ,Mathematics - Abstract
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized ( max , + ) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.
- Published
- 2019
4. Products of symmetric group characters
- Author
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Mike Zabrocki and Rosa Orellana
- Subjects
Pure mathematics ,Mathematical literature ,Root of unity ,010102 general mathematics ,Structure (category theory) ,0102 computer and information sciences ,Basis (universal algebra) ,01 natural sciences ,Theoretical Computer Science ,symbols.namesake ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Symmetric group ,Kronecker delta ,symbols ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
In [33] , the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products. In addition, we identify some applications and instances where special cases of these products have occurred elsewhere in the mathematical literature.
- Published
- 2019
5. Dual Grothendieck polynomials via last-passage percolation
- Author
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Damir Yeliussizov
- Subjects
Pure mathematics ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Probability (math.PR) ,01 natural sciences ,Schur polynomial ,Dual (category theory) ,Mathematics::K-Theory and Homology ,Percolation ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,010307 mathematical physics ,Combinatorics (math.CO) ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Probability ,Column (data store) ,Mathematics - Abstract
The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.
- Published
- 2020
- Full Text
- View/download PDF
6. ADMISSIBLE SUBSETS AND LITTELMANN PATHS IN AFFINE KAZHDAN–LUSZTIG THEORY
- Author
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Jeremie Guilhot
- Subjects
Weyl group ,Algebra and Number Theory ,010102 general mathematics ,Center (category theory) ,Basis (universal algebra) ,01 natural sciences ,Combinatorics ,symbols.namesake ,Mathematics::Quantum Algebra ,Tensor (intrinsic definition) ,0103 physical sciences ,Lie algebra ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Element (category theory) ,Mathematics::Representation Theory ,Ring of symmetric functions ,Affine Hecke algebra ,Mathematics - Abstract
The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group W0. The set of Weyl characters sλ forms a basis of the center and Lusztig showed in [11] that these characters act as translations on the Kazhdan–Lusztig basis element $$ {C}_{w_0} $$ where w0 is the longest element of W0, that is, we have $$ {C}_{w_0}{\mathrm{s}}_{\uplambda}={C}_{w_0{t}_{\uplambda}} $$ . As a consequence, the coefficients that appear when decomposing $$ {C}_{w_0{t}_{\uplambda}}{\mathrm{s}}_{\tau } $$ in the Kazhdan–Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group W0. The aim of this paper is to explain how admissible subsets and Littelmann paths, which are models to compute such multiplicities, naturally appear when working out this decomposition.
- Published
- 2018
7. On the construction of radially symmetric copulas in higher dimensions
- Author
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J.J. Arias-García, B. De Baets, and H. De Meyer
- Subjects
Pure mathematics ,Logic ,media_common.quotation_subject ,Copula (linguistics) ,Markov process ,Statistics::Other Statistics ,02 engineering and technology ,01 natural sciences ,Asymmetry ,010104 statistics & probability ,symbols.namesake ,Artificial Intelligence ,0202 electrical engineering, electronic engineering, information engineering ,Statistics::Methodology ,0101 mathematics ,Ring of symmetric functions ,Mathematics ,media_common ,Discrete mathematics ,Representation theorem ,Symmetry in biology ,Auxiliary function ,Statistics::Computation ,Symmetric function ,symbols ,020201 artificial intelligence & image processing - Abstract
We prove a representation theorem for copulas that are (simultaneously) symmetric and radially symmetric. We use this representation theorem to propose a method to construct an n-ary symmetric function that is radially symmetric, starting from an ( n − 1 ) -dimensional copula and an n-ary auxiliary function. We study the necessary and sufficient conditions on this auxiliary function that guarantee our construction method to result in a symmetric and radially symmetric n-dimensional copula. We examine several options for defining the auxiliary function in the trivariate case, inspired by the nesting of copulas, the lifting of copulas and product-like extensions. For each choice of auxiliary function, we provide several examples for different families of copulas.
- Published
- 2018
8. Graded semisimple algebras are symmetric
- Author
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Laura Nastasescu, Constantin Nastasescu, and Sorin Dascalescu
- Subjects
Symmetric algebra ,Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Triple system ,Mathematics::Rings and Algebras ,010102 general mathematics ,Graded ring ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Graded Lie algebra ,Filtered algebra ,Mathematics::Category Theory ,Differential graded algebra ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.
- Published
- 2017
9. A proof of the shuffle conjecture
- Author
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Erik Carlsson and Anton Mellit
- Subjects
Graded vector space ,Conjecture ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Diagonal ,Zero (complex analysis) ,0102 computer and information sciences ,Space (mathematics) ,01 natural sciences ,Combinatorics ,Character (mathematics) ,010201 computation theory & mathematics ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki [Canad. J. Math., 64 (2012), 822–844], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by Haglund, Haiman, Loehr, Remmel, and Ulyanov [Duke Math. J., 126 (2005), 195–232]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space V ∗ V_* whose degree zero part is the ring of symmetric functions Sym [ X ] \operatorname {Sym}[X] over Q ( q , t ) \mathbb {Q}(q,t) . We then extend these operators to an action of an algebra A ~ \tilde {\mathbb A} acting on this space, and interpret the right generalization of the ∇ \nabla using an involution of the algebra which is antilinear with respect to the conjugation ( q , t ) ↦ ( q − 1 , t − 1 ) (q,t)\mapsto (q^{-1},t^{-1}) .
- Published
- 2017
10. Z2-contractions of classical Lie algebras and symmetric polynomials
- Author
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Oksana S. Yakimova
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Power sum symmetric polynomial ,010102 general mathematics ,Stanley symmetric function ,Complete homogeneous symmetric polynomial ,Killing form ,01 natural sciences ,Schur polynomial ,0103 physical sciences ,Freudenthal magic square ,Elementary symmetric polynomial ,010307 mathematical physics ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
We consider Z 2 -contractions of classical Lie algebras and the behaviour of the symmetric invariants under these contractions. It is demonstrated on three different examples how the theory of symmetric polynomials works in this invariant-theoretic problem.
- Published
- 2017
11. SYMMETRIC PROPERTIES FOR DEGENERATE EULER POLYNOMIALS OF THE SECOND KIND
- Author
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C. S. Ryoo
- Subjects
Classical orthogonal polynomials ,Pure mathematics ,Algebra and Number Theory ,Power sum symmetric polynomial ,Discrete orthogonal polynomials ,Mathematical analysis ,Orthogonal polynomials ,Elementary symmetric polynomial ,Complete homogeneous symmetric polynomial ,Ring of symmetric functions ,Schur polynomial ,Mathematics - Published
- 2017
12. Comparison of complete and simplified models in terms of eigenvalues of cyclically symmetric ring structures
- Author
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Penghui Zhang, Wenjia Sun, and Shiyu Wang
- Subjects
Pure mathematics ,Work (thermodynamics) ,Ring (mathematics) ,Mechanical Engineering ,Aerospace Engineering ,02 engineering and technology ,01 natural sciences ,Parametric instability ,Algebra ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Mechanics of Materials ,Parametric vibration ,0103 physical sciences ,Automotive Engineering ,General Materials Science ,Ring of symmetric functions ,010301 acoustics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This work examines the free and parametric vibrations of constrained cyclically symmetric ring structures and compares the restrictions from different modeling assumptions. A complete model and two simplified models are developed by introducing different restrictions on deflections. The eigenvalues are formulated by direct perturbation and mode superposition. The results verify that they are comparable on the condition that the stiffness ratio of the rotating support to ring bending is relatively small. The eigenvalue splitting, parametric instability and their relationships are determined in closed form. The simplified models can correctly predict the splitting behavior before or after the loci veering. Two types of models obtain the same rules governing the instability, but the simplified models cannot accurately approach the time-evolution of the instability. Hence, it needs to be cautious to use the simplified models to estimate the instability behaviors for a cyclically symmetric ring structure with larger stiffness ratios especially within the loci veering range. Main results are verified by numerical calculations.
- Published
- 2017
13. ON THE SYMMETRIC PROPERTIES FOR THE GENERALIZED DEGENERATE TANGENT POLYNOMIALS
- Author
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C. S. Ryoo
- Subjects
Classical orthogonal polynomials ,Physics ,Pure mathematics ,Algebra and Number Theory ,Power sum symmetric polynomial ,Orthogonal polynomials ,Mathematical analysis ,Elementary symmetric polynomial ,Complete homogeneous symmetric polynomial ,Tangent vector ,Ring of symmetric functions ,Schur polynomial - Published
- 2017
14. Introduction to Sublinear Analysis — 2: Symmetric Case
- Author
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I. V. Orlov and I. V. Baran
- Subjects
Statistics and Probability ,Pure mathematics ,Power sum symmetric polynomial ,Triple system ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Complete homogeneous symmetric polynomial ,01 natural sciences ,010305 fluids & plasmas ,Symmetric closure ,Symmetric function ,Representation theory of the symmetric group ,0103 physical sciences ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
The advanced theory of the first and higher symmetric Frechet differentials and K-sub-differentials is constructed including the mean value theorem and the Taylor formula. We give simple sufficient conditions for symmetric K-subdifferentiability and consider some applications to Fourier series and variational functionals.
- Published
- 2017
15. Properties of m-complex symmetric operators
- Author
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Ji Eun Lee, Eungil Ko, and Muneo Cho
- Subjects
Algebra ,Pure mathematics ,General Mathematics ,Nilpotent operator ,Order (ring theory) ,Complete homogeneous symmetric polynomial ,Algebraic number ,Operator theory ,Type (model theory) ,Ring of symmetric functions ,Mathematics ,Symmetric operator - Abstract
In this paper, we study several properties of $m$-complex symmetric operators. In particular, we prove that if $T\in{\cal L(H)}$ is an $m$-complex symmetric operator and $N$ is a nilpotent operator of order $n>2$ with $TN=NT$, then $T+N$ is a $(2n+m-2)$-complex symmetric operator. Moreover, we investigate the decomposability of $T+A$ and $TA$ where $T$ is an $m$-complex symmetric operator and $A$ is an algebraic operator. Finally, we provide various spectral relations of such operators. As some applications of these results, we discuss Weyl type theorems for such operators.
- Published
- 2017
16. CHARACTERISTIC FUNCTION OF SYMMETRIC DAMEK-RICCI SPACE
- Author
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Sinhwi Kim and JeongHyeong Park
- Subjects
Power sum symmetric polynomial ,Triple system ,Symmetric bilinear form ,General Mathematics ,Symmetric space ,Elementary symmetric polynomial ,Symmetric tensor ,Ring of symmetric functions ,Mathematics ,Symmetric closure ,Mathematical physics - Published
- 2017
17. A POLYNOMIAL WITH COEFFICIENTS IN CERTAIN ELEMENTARY SYMMETRIC POLYNOMIALS
- Author
-
Yasuhiko Kamiyama
- Subjects
Pure mathematics ,Algebra and Number Theory ,Symmetric polynomial ,Difference polynomials ,Power sum symmetric polynomial ,Elementary symmetric polynomial ,Complete homogeneous symmetric polynomial ,Newton's identities ,Ring of symmetric functions ,Schur polynomial ,Mathematics - Published
- 2017
18. The algebra of symmetric analytic functions on L∞
- Author
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Andriy Zagorodnyuk, Pablo Galindo, and T. V. Vasylyshyn
- Subjects
Power sum symmetric polynomial ,Triple system ,General Mathematics ,010102 general mathematics ,Subalgebra ,Stanley symmetric function ,Complete homogeneous symmetric polynomial ,01 natural sciences ,010101 applied mathematics ,Algebra ,Symmetric polynomial ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.
- Published
- 2017
19. On the Ring of Local Unitary Invariants for Mixed X-States of Two Qubits
- Author
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Vladimir P. Gerdt, Arsen Khvedelidze, and Yuri Palii
- Subjects
Statistics and Probability ,Discrete mathematics ,Reduced ring ,Principal ideal ring ,Pure mathematics ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Boolean ring ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Primitive ring ,Simple ring ,0103 physical sciences ,0101 mathematics ,010306 general physics ,Ring of symmetric functions ,Quotient ring ,Mathematics - Abstract
Entangling properties of a mixed two-qubit system can be described by local homogeneous unitary invariant polynomials in the elements of the density matrix. The structure of the corresponding ring of invariant polynomials for a special subclass of states, the so-called mixed X-states, is established. It is shown that for the X-states there is an injective ring homomorphism of the quotient ring of SU(2)×SU(2)-invariant polynomials modulo its syzygy ideal to the SO(2) × SO(2)-invariant ring freely generated by five homogeneous polynomials of degrees 1, 1, 1, 2, 2.
- Published
- 2017
20. Rank-r decomposition of symmetric tensors
- Author
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Jie Wen, Qin Ni, and Wenhuan Zhu
- Subjects
Tensor contraction ,Power sum symmetric polynomial ,Rank (linear algebra) ,010102 general mathematics ,010103 numerical & computational mathematics ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Combinatorics ,Mathematics (miscellaneous) ,Tensor product ,Symmetric tensor ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Computer Science::Databases ,Mathematics - Abstract
An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.
- Published
- 2017
21. Mathematical Model of Interaction of a Symmetric Top with an Axially Symmetric External Field
- Author
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V. S. Lyashko, S.I. Zub, S. I. Lyashko, N. I. Lyashko, and Stanislav S. Zub
- Subjects
021103 operations research ,General Computer Science ,Power sum symmetric polynomial ,Triple system ,010102 general mathematics ,0211 other engineering and technologies ,Geometry ,02 engineering and technology ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Computer Science::Computers and Society ,Symmetric closure ,Nonlinear Sciences::Chaotic Dynamics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Representation theory of the symmetric group ,Physics::Atomic and Molecular Clusters ,Elementary symmetric polynomial ,0101 mathematics ,Axial symmetry ,Ring of symmetric functions ,Mathematics ,Mathematical physics - Abstract
A symmetric top is considered, which is a particular case of a mechanical top that is usually described by the canonical Poisson structure on T*SE (3). This structure is invariant under the right action of the rotation group SO(3), but the Hamiltonian of the symmetric top is invariant only under the right action of the subgroup S 1, which corresponds to the rotation of the symmetric top around its axis of symmetry. This Poisson structure is obtained as the reduction T* SE (3) / S 1. A Hamiltonian and motion equations are proposed that describe a wide class of interaction models of the symmetric top with an axially symmetric external field.
- Published
- 2017
22. Two More Symmetric Properties of Odd Numbers
- Author
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Xingbo Wang
- Subjects
Combinatorics ,Power sum symmetric polynomial ,Elementary symmetric polynomial ,Ring of symmetric functions ,Mathematics - Published
- 2017
23. Subspace filters and polar bases for spaces of symmetric multilinear functions
- Author
-
Thomas H. Pate
- Subjects
Symmetric algebra ,Pure mathematics ,Algebra and Number Theory ,Basis (linear algebra) ,Triple system ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Symmetric function ,Linear form ,0101 mathematics ,Ring of symmetric functions ,Subspace topology ,Mathematics ,Vector space - Abstract
Let V be a finite dimensional complex vector space, and for each positive integer n let denote the vector space whose elements are the complex valued symmetric n-linear functions on (n-copies). If , that is, f is a linear functional on V, then we define the symmetric n-linear function by for all . It is well known that the elements , known as polar elements, span . A basis for consisting entirely of polar elements is a polar basis. We present a surprisingly simple recursive algorithm for the generation of polar bases. Explicit stand alone formulas for polar bases are also presented. One such explicit basis is so natural that it should probably be called the canonical polar basis for . We introduce the notion of subspace filter and show that each subspace filter of order n provides polar bases for all of the spaces simultaneously, and we extend this result to the entire symmetric algebra .
- Published
- 2017
24. Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials
- Author
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Jiří Patera, Severin Pošta, Agata Bezubik, and Jiří Hrivnák
- Subjects
Pure mathematics ,Power sum symmetric polynomial ,General Mathematics ,Discrete orthogonal polynomials ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Exponential polynomial ,010305 fluids & plasmas ,Classical orthogonal polynomials ,0103 physical sciences ,Orthogonal polynomials ,Elementary symmetric polynomial ,010306 general physics ,Ring of symmetric functions ,Mathematics - Abstract
The common exponential functions whose exponents are the scalar products 〈λ,x〉, where x is a real variable and λ is an integer, admit two generalizations to any higher dimension, the symmetric and the antisymmetric ones [KLIMYK, A.—PATERA, J.: (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms, J. Phys. A: Math. Theor. 40 (2007), 10473–10489]. Restriction in the paper to the three variables only allows us to work out many specific properties of the symmetric and antisymmetric functions useful in applications. Such are (i) the orthogonalities, both the continuous one and the discrete one on the 3D lattice of any density; (ii) corresponding discrete and continuous Fourier transforms; (iii) generating functions for the related polynomials in three variables, and others. Rapidly increasing precision of the interpolation with increasing density of the 3D lattice is shown in an example.
- Published
- 2017
25. New families of balanced symmetric functions and a generalization of Cusick, Li and Stǎnicǎ’s conjecture
- Author
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Luis A. Medina, Oscar E. González, Ivelisse Rubio, Francis N. Castro, and Rafael A. Arce-Nazario
- Subjects
Discrete mathematics ,Conjecture ,Generalization ,Applied Mathematics ,Stanley symmetric function ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Computer Science Applications ,Symmetric function ,Finite field ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Elementary symmetric polynomial ,Ring of symmetric functions ,Boolean function ,Mathematics - Abstract
In this paper we provide new families of balanced symmetric functions over any finite field. We also generalize a conjecture of Cusick, Li, and Stǎnicǎ about the non-balancedness of elementary symmetric Boolean functions to any finite field and prove part of our conjecture.
- Published
- 2017
26. Regular quaternionic polynomials and their properties
- Author
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Yu. M. Grigor’ev
- Subjects
Numerical Analysis ,Mathematics::Complex Variables ,Applied Mathematics ,Discrete orthogonal polynomials ,010102 general mathematics ,Hypercomplex analysis ,01 natural sciences ,Classical orthogonal polynomials ,Algebra ,Computational Mathematics ,Difference polynomials ,0103 physical sciences ,Wilson polynomials ,Orthogonal polynomials ,Elementary symmetric polynomial ,010307 mathematical physics ,0101 mathematics ,Ring of symmetric functions ,Analysis ,Mathematics - Abstract
Unlike in complex analysis, in all hypercomplex function theories a hypercomplex variable is not monogenic (regular) and there exists a problem to define analogues of positive and negative powers of the complex variable. R. Fueter firstly introduces a system of symmetric regular quaternion polynomials as analogues of positive powers of a complex variable and proves the Taylor theorem in his theory. In Clifford analysis an analogical idea is used. The Fueter symmetric polynomials are both left- and right-regular, the symmetric polynomials in Clifford analysis are also both left- and right-monogenic. In this paper we construct only left-regular quaternion polynomials and show that the theory of regular quaternion functions of a vector valued quaternion variable can be developed using these polynomials.
- Published
- 2017
27. Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz matrix and the Chebyshev polynomials
- Author
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Ahmet Zahid Küçük and Murat Düz
- Subjects
Chebyshev polynomials ,Power sum symmetric polynomial ,Tridiagonal matrix ,lcsh:T57-57.97 ,Discrete orthogonal polynomials ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Toeplitz matrix ,Mathematics::Numerical Analysis ,Combinatorics ,Classical orthogonal polynomials ,lcsh:Applied mathematics. Quantitative methods ,Computer Science::Mathematical Software ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Published
- 2017
28. Two Classes of 1-Resilient Prime-Variable Rotation Symmetric Boolean Functions
- Author
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Xuan Guang, Fang-Wei Fu, and Lei Sun
- Subjects
Discrete mathematics ,Parity function ,Applied Mathematics ,Two-element Boolean algebra ,Stanley symmetric function ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,Boolean algebras canonically defined ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Prime (order theory) ,Combinatorics ,010201 computation theory & mathematics ,Signal Processing ,0202 electrical engineering, electronic engineering, information engineering ,Electrical and Electronic Engineering ,Symmetric difference ,Boolean function ,Ring of symmetric functions ,Mathematics - Published
- 2017
29. Аналог больших полиномов $q$-Якоби
- Author
-
Grigorii Iosifovich Olshanskii
- Subjects
Discrete mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Symmetric function ,Algebra ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Jacobi polynomials ,0101 mathematics ,Algebra over a field ,Ring of symmetric functions ,Mathematics - Published
- 2017
30. Local symmetric square L-factors of representations of general linear groups
- Author
-
Shunsuke Yamana
- Subjects
Pure mathematics ,Representation theory of the symmetric group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Elementary symmetric polynomial ,010307 mathematical physics ,0101 mathematics ,Ring of symmetric functions ,01 natural sciences ,Covering groups of the alternating and symmetric groups ,Square (algebra) ,Mathematics - Published
- 2017
31. Symmetric identities for degenerate (h,q)-tangent polynomials associated with the p-adic integral on Zp
- Author
-
Cheon Seoung Ryoo
- Subjects
Combinatorics ,Power sum symmetric polynomial ,General Mathematics ,Degenerate energy levels ,Elementary symmetric polynomial ,Tangent ,Complete homogeneous symmetric polynomial ,Ring of symmetric functions ,Schur polynomial ,Mathematics - Published
- 2017
32. On the symmetric identities for the second kind generalized q-Euler polynomials
- Author
-
Cheon Seoung Ryoo
- Subjects
Classical orthogonal polynomials ,Combinatorics ,Power sum symmetric polynomial ,Orthogonal polynomials ,Wilson polynomials ,Elementary symmetric polynomial ,Complete homogeneous symmetric polynomial ,Ring of symmetric functions ,Schur polynomial ,Mathematics - Published
- 2017
33. Symmetric properties for generalized Euler polynomials of the second kind
- Author
-
Cheon Seoung Ryoo
- Subjects
Classical orthogonal polynomials ,Power sum symmetric polynomial ,Discrete orthogonal polynomials ,Orthogonal polynomials ,Mathematical analysis ,Elementary symmetric polynomial ,Complete homogeneous symmetric polynomial ,Ring of symmetric functions ,Schur polynomial ,Mathematics - Published
- 2017
34. Complex symmetric operators, skew symmetric operators and reflexivity
- Author
-
Sen Zhu
- Subjects
Pure mathematics ,Algebra and Number Theory ,Power sum symmetric polynomial ,Triple system ,010102 general mathematics ,010103 numerical & computational mathematics ,Spectral theorem ,Complete homogeneous symmetric polynomial ,Operator theory ,01 natural sciences ,Reflexivity ,Skew-symmetric matrix ,0101 mathematics ,Ring of symmetric functions ,Analysis ,Mathematics - Published
- 2017
35. Representation of spectra of algebras of block-symmetric analytic functions of bounded type
- Author
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Andriy Zagorodnyuk and V. V. Kravtsiv
- Subjects
Discrete mathematics ,Symmetric algebra ,Power sum symmetric polynomial ,algebraic basis ,Triple system ,lcsh:Mathematics ,General Mathematics ,symmetric intertwining ,010102 general mathematics ,block-symmetric polynomials ,symmetric convolution ,Complete homogeneous symmetric polynomial ,lcsh:QA1-939 ,01 natural sciences ,Bounded type ,spectrum ,010305 fluids & plasmas ,Symmetric function ,0103 physical sciences ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,block-symmetric analytic functions ,Mathematics - Abstract
The paper contains a description of symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\ell_{1}$-sum of the space $\mathbb{C}^{2}.$ We show that the specrum of such algebra does not coincide of point evaluation functionals and described characters of the algebra as functions of exponential type with plane zeros.
- Published
- 2016
36. Constructions ofp-variable 1-resilient rotation symmetric functions overGF(p)
- Author
-
Shaojing Fu, Shanqi Pang, Chao Li, and Jiao Du
- Subjects
Discrete mathematics ,Power sum symmetric polynomial ,Computer Networks and Communications ,Computer science ,Triple system ,Stanley symmetric function ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Symmetric function ,symbols.namesake ,Jacobi rotation ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Elementary symmetric polynomial ,Ring of symmetric functions ,Information Systems - Abstract
Rotation symmetric Boolean functions have been extensively studied in the recent years because of their applications in cryptography. In this study, a novel method to construct p-variable 1-resilient rotation symmetric functions over GF(p) is proposed based on a Latin square with maximum cycle structure, which is not required to solve any equation system. And a lower bound on the number of p-variable 1-resilient rotation symmetric functions is given. At last, an equivalent characterization of p-variable 1-resilient rotation symmetric functions over GF(p) is demonstrated, as a direct corollary, the number of p-variable 1-resilient rotation symmetric functions is represented by all the solutions of the equation system. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2016
37. Hamming weights of symmetric Boolean functions
- Author
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Thomas W. Cusick
- Subjects
Discrete mathematics ,Symmetric Boolean function ,Parity function ,Applied Mathematics ,Stanley symmetric function ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Combinatorics ,Symmetric function ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Discrete Mathematics and Combinatorics ,Elementary symmetric polynomial ,Boolean function ,Hamming weight ,Ring of symmetric functions ,Mathematics - Abstract
It has been known since 1996 (work of Cai et al.) that the sequence of Hamming weights { w t ( f n ( x 1 , ź , x n ) ) } , where f n is a symmetric Boolean function of degree d in n variables, satisfies a linear recurrence with integer coefficients. In 2011, Castro and Medina used this result to show that a 2008 conjecture of Cusick, Li and Stźnicź about when an elementary symmetric Boolean function can be balanced is true for all sufficiently large n . Quite a few papers have been written about this conjecture, but it is still not completely settled. Recently, Guo, Gao and Zhao proved that the conjecture is equivalent to the statement that all balanced elementary symmetric Boolean functions are trivially balanced. This motivates the further study of the weights of symmetric Boolean functions f n . In this paper we prove various new results on the trivially balanced functions. We also determine a period (sometimes the minimal one) for the sequence of weights w t ( f n ) modulo any odd prime p , where f n is any symmetric function, and we prove some related results about the balanced symmetric functions.
- Published
- 2016
38. Property of Rational Functions Related to Band-Pass Transformation With Application to Symmetric Filters Design
- Author
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Igor M. Filanovsky
- Subjects
Discrete mathematics ,Pure mathematics ,Power sum symmetric polynomial ,020208 electrical & electronic engineering ,02 engineering and technology ,Complete homogeneous symmetric polynomial ,020202 computer hardware & architecture ,Symmetric function ,Filter design ,0202 electrical engineering, electronic engineering, information engineering ,Elementary symmetric polynomial ,Electrical and Electronic Engineering ,Newton's identities ,Network synthesis filters ,Ring of symmetric functions ,Mathematics - Abstract
This paper considers the condition when a rational function $F(s)$ may be represented as the function of the argument $s+(1/s)$ . If this condition is satisfied then $F(s)$ is the ratio of recursive (symmetric) polynomials. This paper investigates the network properties of such rational functions and their realization. Then the symmetric polynomials are applied for synthesis of symmetric band-pass filters. Substituting $p=s+(1/s)$ into symmetric band-pass filter transfer function one obtains its low-pass generating filter. The slew rate and overshoot of generating filter step-response is closely connected with the step-response duration of symmetric band-pass filter. The choice of generating filter becomes an additional factor of symmetric band-pass filter design. As the generating filters the paper proposes using Lommel polynomial filters which have easy control of overshoot and slew rate. An example of six order symmetric band-pass filter is given.
- Published
- 2016
39. Hall–Littlewood symmetric functions via the chip-firing game
- Author
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Pasquale Petrullo, Robert Cori, and D. Senato
- Subjects
Discrete mathematics ,Power sum symmetric polynomial ,Triple system ,010102 general mathematics ,Stanley symmetric function ,0102 computer and information sciences ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Symmetric closure ,Symmetric function ,Combinatorics ,010201 computation theory & mathematics ,Discrete Mathematics and Combinatorics ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
Via the chip-firing game, a class of Schur positive symmetric functions depending on four parameters is introduced for any labeled connected simple graph. Tableaux formulae are stated to expand such symmetric functions in terms of the complete symmetric functions. In the case of a simple path, the resulting symmetric functions reduce to the transformed Hall-Littlewood symmetric functions when the parameters are suitable specialized.
- Published
- 2016
40. Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
- Author
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Joel Brewster Lewis and Eric Marberg
- Subjects
Power series ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Skew ,K-Theory and Homology (math.KT) ,Hopf algebra ,01 natural sciences ,Omega ,Set (abstract data type) ,0103 physical sciences ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Ring of symmetric functions ,Mathematics - Representation Theory ,Mathematics - Abstract
We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse's shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a "$K$-theoretic" Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new explicit formulas for the involution $\omega$ on the ring of symmetric functions., Comment: 51 pages; v2: one conjecture upgraded to a theorem, another new conjecture added, some minor edits; v3: minor edits, fixed typos, added references
- Published
- 2019
41. Positive specializations of symmetric Grothendieck polynomials
- Author
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Damir Yeliussizov
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Schur polynomial ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,010307 mathematical physics ,Combinatorics (math.CO) ,0101 mathematics ,Ring of symmetric functions ,Parametrization ,Mathematics - Abstract
It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei–Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, K-theoretic deformations of Schur polynomials.
- Published
- 2019
42. Hopf algebra structure of symmetric and quasisymmetric functions in superspace
- Author
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Luc Lapointe, Maria Elena Pinto, and Susanna Fishel
- Subjects
Monomial ,Pure mathematics ,0102 computer and information sciences ,Superspace ,01 natural sciences ,Theoretical Computer Science ,General Relativity and Quantum Cosmology ,High Energy Physics::Theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Ring of symmetric functions ,Mathematics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,010102 general mathematics ,Mathematics::Rings and Algebras ,Basis (universal algebra) ,Hopf algebra ,Noncommutative geometry ,Symmetric function ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Combinatorics (math.CO) - Abstract
We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.
- Published
- 2019
- Full Text
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43. The Hopf structure of symmetric group characters as symmetric functions
- Author
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Mike Zabrocki and Rosa Orellana
- Subjects
Pure mathematics ,05E05, 05E10, 20C30 ,010102 general mathematics ,Coproduct ,Structure (category theory) ,0102 computer and information sciences ,Basis (universal algebra) ,01 natural sciences ,Symmetric function ,Character (mathematics) ,010201 computation theory & mathematics ,Symmetric group ,Product (mathematics) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis., arXiv admin note: text overlap with arXiv:1605.06672; remark 1: irreducible character basis of the rook algebra
- Published
- 2018
44. Analysis of symmetric function ideals
- Author
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Abukuse Mbirika
- Subjects
Combinatorics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,Polynomial ring ,Ideal (ring theory) ,Complete homogeneous symmetric polynomial ,Ring of symmetric functions ,Quotient ring ,Cohomology ring ,Mathematics ,Hessenberg variety - Abstract
Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties. Let R be the polynomial ring Z[x1, . . . , xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal Ih generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi’s quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Grobner basis, which is a useful property. Using the Grobner basis for Jh, we identify a basis for the quotient R/Jh. We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h > h′, we have Ih ⊂ Ih′ and Jh ⊂ Jh′ . We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Grobner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety. The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter
- Published
- 2018
45. A GENERALIZATION OF SYMMETRIC RING PROPERTY
- Author
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Sang Jo Yun, Seung Ick Lee, Hong Kee Kim, Tai Keun Kwak, Hyo Jin Sung, Sung Ju Ryu, and Yang Lee
- Subjects
Discrete mathematics ,Principal ideal ring ,Ring (mathematics) ,Pure mathematics ,Power sum symmetric polynomial ,General Mathematics ,010102 general mathematics ,Boolean ring ,0102 computer and information sciences ,Complete homogeneous symmetric polynomial ,01 natural sciences ,010201 computation theory & mathematics ,Elementary symmetric polynomial ,Zero ring ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Published
- 2016
46. Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique
- Author
-
Guyan Ni, Bing Hua, and Mengshi Zhang
- Subjects
Discrete mathematics ,021103 operations research ,Optimization problem ,Power sum symmetric polynomial ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Quantum entanglement ,Complete homogeneous symmetric polynomial ,Management Science and Operations Research ,Squashed entanglement ,01 natural sciences ,Measure (mathematics) ,Elementary symmetric polynomial ,Applied mathematics ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue (US-eigenvalue) for symmetric complex tensors, which can be taken as a multilinear optimization problem in complex number field. In this paper, we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem. Then we use Jacobian semidefinite relaxation method to solve it. Some numerical examples are presented.
- Published
- 2016
47. NEW THEOREMS ON CARLITZ’S TWISTED q-BERNOULLI POLYNOMIALS UNDER THE SYMMETRIC GROUP OF DEGREE n
- Author
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Ugur Duran and Mehmet Acikgoz
- Subjects
Discrete mathematics ,Pure mathematics ,Degree (graph theory) ,Power sum symmetric polynomial ,Stanley symmetric function ,General Medicine ,Complete homogeneous symmetric polynomial ,Bernoulli polynomials ,symbols.namesake ,Symmetric group ,symbols ,Elementary symmetric polynomial ,Ring of symmetric functions ,Mathematics - Published
- 2016
48. New convolutions for complete and elementary symmetric functions
- Author
-
Mircea Merca
- Subjects
Discrete mathematics ,Power sum symmetric polynomial ,Applied Mathematics ,010102 general mathematics ,Stanley symmetric function ,0102 computer and information sciences ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Symmetric function ,010201 computation theory & mathematics ,Elementary symmetric polynomial ,0101 mathematics ,Newton's identities ,Ring of symmetric functions ,Bernoulli number ,Analysis ,Mathematics - Abstract
In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.
- Published
- 2016
49. SOME SYMMETRIC IDENTITIES INVOLVING CARLITZ’S TWISTED q-BERNOULLI POLYNOMIALS UNDER S_5
- Author
-
Ugur Duran and Mehmet Acikgoz
- Subjects
Discrete mathematics ,symbols.namesake ,Power sum symmetric polynomial ,symbols ,Elementary symmetric polynomial ,Complete homogeneous symmetric polynomial ,Ring of symmetric functions ,Bernoulli polynomials ,Mathematics - Published
- 2016
50. Dual bases for noncommutative symmetric and quasi-symmetric functions via monoidal factorization
- Author
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Gérard Duchamp, Van Chiên Bui, V. Hoang Ngoc Minh, L. Kane, and Christophe Tollu
- Subjects
Algebra and Number Theory ,Triple system ,010102 general mathematics ,Stanley symmetric function ,0102 computer and information sciences ,Complete homogeneous symmetric polynomial ,01 natural sciences ,Noncommutative geometry ,Symmetric closure ,Symmetric function ,Algebra ,Computational Mathematics ,010201 computation theory & mathematics ,Elementary symmetric polynomial ,0101 mathematics ,Ring of symmetric functions ,Mathematics - Abstract
We propose effective constructions of dual bases for the noncommutative symmetric and quasi-symmetric functions. To this end, we use an effective variation of Schutzenberger's factorization adapted to the diagonal pairing between a graded space and its dual.
- Published
- 2016
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