scholarly journals On Algebraic Basis of the Algebra of Symmetric Polynomials on lp(Cn)

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Victoriia Kravtsiv ◽  
Taras Vasylyshyn ◽  
Andriy Zagorodnyuk
Keyword(s):  
Symmetric Polynomials ◽  
Algebraic Basis

We consider polynomials on spaces lpCn,1≤p<+∞, of p-summing sequences of n-dimensional complex vectors, which are symmetric with respect to permutations of elements of the sequences, and describe algebraic bases of algebras of continuous symmetric polynomials on lpCn.

scholarly journals Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$

2020 ◽  
Vol 12 (1) ◽  
pp. 5-16
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Linear Combination ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Symmetric Polynomial ◽  
Complex Numbers ◽  
Symmetric Polynomials ◽  
Algebraic Basis

This work is devoted to study algebras of continuous symmetric, that is, invariant with respect to permutations of coordinates of its argument, polynomials and $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers resp., where $1\leq p < +\infty.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n).$ Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$ Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n),$ and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$

SOME PROPERTIES OF ELEMENTARY SYMMETRIC POLYNOMIALS ON THE CARTESIAN SQUARE OF THE COMPLEX BANACH SPACE L∞ [ 0,1 ]

2018 ◽  
pp. 9-16
Author(s):  
T. V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Symmetric Polynomials ◽  
Bounded Functions ◽  
Elementary Symmetric Polynomials ◽  
Algebraic Basis

We construct the element of the Cartesian square of the complex Banach space L ∞ [ 0,1 ] of all Lebesgue measurable essentially bounded functions on [ 0,1 ] by the predefined values of elementary symmetric polynomials on this element. Results of this work can be applied to the investigation of an algebraic basis of the algebra of continuous symmetric polynomials on the Cartesian square of the complex Banach space L ∞ [ 0,1 ] .

scholarly journals Algebraic basis of the algebra of block-symmetric polynomials on $\ell_1 \oplus \ell_{\infty}$

2019 ◽  
Vol 11 (1) ◽  
pp. 89-95 ◽  
Author(s):  
V.V. Kravtsiv
Keyword(s):  
Sequence Spaces ◽  
Symmetric Polynomials ◽  
Algebraic Basis ◽  
Continuous Block

We consider so called block-symmetric polynomials on sequence spaces $\ell_1\oplus \ell_{\infty}, \ell_1\oplus c, \ell_1\oplus c_0,$ that is, polynomials which are symmetric with respect to permutations of elements of the sequences. It is proved that every continuous block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ can be uniquely represented as an algebraic combination of some special block-symmetric polynomials, which form an algebraic basis. It is interesting to note that the algebra of block-symmetric polynomials is infinite-generated while $\ell_{\infty}$ admits no symmetric polynomials. Algebraic bases of the algebras of block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ and $\ell_1\oplus c_0$ are described.

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 $L_^{(\mathbb{R})}\infty[0,1]$.

scholarly journals Milne’s volume function and vector symmetric polynomials

2009 ◽  
Vol 44 (5) ◽  
pp. 583-590 ◽  
Author(s):  
Emmanuel Briand ◽  
Mercedes Rosas
Keyword(s):  
Symmetric Polynomials ◽  
Volume Function

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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