scholarly journals Continuous block-symmetric polynomials of degree at most two on the space $(L_\infty)^2$

2016 ◽  
Vol 8 (1) ◽  
pp. 38-43 ◽  
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Symmetric Polynomial ◽  
Symmetric Polynomials ◽  
Continuous Block

We introduce block-symmetric polynomials on $(L_\infty)^2$ and prove that every continuous block-symmetric polynomial of degree at most two on $(L_\infty)^2$ can be uniquely represented by some "elementary" block-symmetric polynomials.

scholarly journals A categorification of the chromatic symmetric polynomial

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Radmila Sazdanović ◽  
Martha Yip
Keyword(s):  
Symmetric Function ◽  
Symmetric Polynomial ◽  
Chromatic Polynomial ◽  
Khovanov Homology ◽  
Symmetric Polynomials ◽  
Nous Obtenons ◽  
Combinatorial Properties ◽  
International Audience ◽  
Chromatic Symmetric Function ◽  
Frobenius Series

International audience The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$<sub>*</sub>($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology. Le polynôme chromatique symétrique d’un graphe $G$ est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie $H$<sub>*</sub>($G$) de modules gradués $S_n$, dont la série bigraduée de Frobeniusse $Frob_G(q,t)$ réduit au polynôme chromatique symétrique à $q=t=1$. Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie.

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 $\ell_p(\mathbb{C}^n).$

scholarly journals Symmetric $*$-polynomials on $\mathbb C^n$

2018 ◽  
Vol 10 (2) ◽  
pp. 395-401
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Vector Space ◽  
Symmetric Polynomial ◽  
Vector Spaces ◽  
Symmetric Polynomials ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Cartesian Power ◽  
Elementary Symmetric Polynomials

$*$-Polynomials are natural generalizations of usual polynomials between complex vector spaces. A $*$-polynomial is a function between complex vector spaces $X$ and $Y,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for nonnegative integers $p$ and $q,$ a $(p,q)$-polynomial is a function between $X$ and $Y,$ which is the restriction to the diagonal of some mapping, acting from the Cartesian power $X^{p+q}$ to $Y,$ which is linear with respect to every of its first $p$ arguments, antilinear with respect to every of its last $q$ arguments and invariant with respect to permutations of its first $p$ arguments and last $q$ arguments separately. In this work we construct formulas for recovering of $(p,q)$-polynomial components of $*$-polynomials, acting between complex vector spaces $X$ and $Y,$ by the values of $*$-polynomials. We use these formulas for investigations of $*$-polynomials, acting from the $n$-dimensional complex vector space $\mathbb{C}^n$ to $\mathbb{C},$ which are symmetric, that is, invariant with respect to permutations of coordinates of its argument. We show that every symmetric $*$-polynomial, acting from $\mathbb{C}^n$ to $\mathbb{C},$ can be represented as an algebraic combination of some "elementary" symmetric $*$-polynomials. Results of the paper can be used for investigations of algebras, generated by symmetric $*$-polynomials, acting from $\mathbb{C}^n$ to $\mathbb{C}.$

scholarly journals Differential operators and Higher Specht polynomials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012096
Author(s):  
Ibrahim Nonkané ◽  
Léonard Todjihounde
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Symmetric Groups ◽  
Symmetric Polynomial ◽  
Symmetric Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential ◽  
Moser System

Abstract In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

scholarly journals Enumerating Partial Latin Rectangles

10.37236/9093 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Raúl M. Falcón ◽  
Rebecca J. Stones
Keyword(s):  
Algebraic Geometry ◽  
Computational Methods ◽  
Latin Squares ◽  
Symmetric Polynomial ◽  
Chromatic Polynomial ◽  
Numerical Results ◽  
Polynomial Method ◽  
Symmetric Polynomials ◽  
Exact Size ◽  
Geometry Method

This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

Elementary Symmetric Polynomials in Numbers of Modulus 1

2002 ◽  
Vol 54 (2) ◽  
pp. 239-262 ◽  
Author(s):  
Donald I. Cartwright ◽  
Tim Steger
Keyword(s):  
Symmetric Polynomial ◽  
Complex Polynomial ◽  
Complex Numbers ◽  
Symmetric Polynomials ◽  
Elementary Symmetric Polynomial ◽  
Elementary Symmetric Polynomials

AbstractWe describe the set of numbers σk(z1,…,zn+1), where z1,…,zn+1 are complex numbers of modulus 1 for which z1z2 … zn+1 = 1, and σk denotes the k-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type Ãn.

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 Nonsymmetric Macdonald Superpolynomials

2021 ◽  
Author(s):  
Charles F. Dunkl ◽  
Keyword(s):  
Classical Group ◽  
Young Tableau ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Symmetric Polynomial ◽  
Inner Product ◽  
Symmetric Polynomials ◽  
Special Points ◽  
Standard Young Tableau

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

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.

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
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