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

Abstract

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.

Supersymmetric Polynomials on the Space of Absolutely Convergent Series

“…Spectra of algebras H bs ( p ) were investigated also in [13,14]. Polynomials which are symmetric with respect to some other representations of the group of permutations of natural numbers were considered in [15][16][17].…”

Rings of Multisets and Integer Multinumbers

“…Such a ring of multisets M 0 was constructed in [6] using symmetric and supersymmetric polynomials on a Banach space (see also [7]). More details about algebras of symmetric polynomials on Banach spaces can be found in [8][9][10][11][12][13][14]. The combinatorial approach to symmetric polynomials can be found in [15].…”

Spectra of algebras of block-symmetric analytic functions of bounded type

“…Note that such kinds of algebras are much more complicated and in the general case have no algebraic basis (see e.g. [21,22,[24][25][26]37]). Note that if dim X < ∞, then block-symmetric polynomials are investigated in the classical theory of invariants and combinatorics [18,32,36].…”

“…The algebra P vs (ℓ p (C s )) was considered in [6,25]. Note that in combinatorics, blocksymmetric polynomials on finite-dimensional spaces are called MacMahon symmetric polynomials (see e.g.…”