scholarly journals An Orthosymplectic Pieri Rule

10.37236/7387 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Anna Stokke
Keyword(s):  
Linear Combination ◽  
Schur Functions ◽  
Symmetric Polynomial ◽  
Schur Function ◽  
Product Rule ◽  
Integer Coefficients ◽  
Pieri Formula ◽  
Symplectic Characters ◽  
Homogeneous Symmetric Polynomial

The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundaram's Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule. 

scholarly journals Powers of the Vandermonde determinant, Schur functions, and the dimension game

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Cristina Ballantine
Keyword(s):  
Young Diagram ◽  
Schur Functions ◽  
Symmetric Polynomial ◽  
Nous Allons ◽  
Vandermonde Determinant ◽  
Schur Function ◽  
Combinatorial Properties ◽  
International Audience

International audience Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, I will give a recursive approach for computing the coefficient of the Schur function $s_μ$ in the decomposition of an even power of the Vandermonde determinant in $n+1$ variables in terms of the coefficient of the Schur function $s_λ$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $μ$ is obtained from the Young diagram of $λ$ by adding a tetris type shape to the top or to the left. Comme toute puissance paire du déterminant de Vandermonde est un polynôme symétrique, nous voulons comprendre sa décomposition dans la base des fonctions de Schur. Nous allons étudier plusieurs propriétés combinatoires des coefficients de la décomposition. En particulier, nous allons donner une approche récursive pour le calcul du coefficient de la fonction de Schur $s_μ$ dans la décomposition d'une puissance paire du déterminant de Vandermonde en $n+1$ variables, en fonction du coefficient de la fonction de Schur $s_λ$ dans la décomposition de la même puissance paire du déterminant de Vandermonde en $n$ variables, lorsque le diagramme de Young de $μ$ est obtenu à partir du diagramme de Young de $λ$ par l'addition d'une forme de type tetris vers le haut ou vers la gauche.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

On Cyclotomic Numbers of Order Sixteen

1954 ◽  
Vol 6 ◽  
pp. 449-454 ◽  
Author(s):  
Emma Lehmer
Keyword(s):  
Linear Combination ◽  
Primitive Root ◽  
Number Of Solutions ◽  
Integer Coefficients ◽  
Cyclotomic Numbers ◽  
Image Position ◽  
Mod P

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of (mod p),then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where,anda ≡ b ≡ 1 (mod 4),while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

scholarly journals Enumerations for Compositions and Complete Homogeneous Symmetric Polynomial

2015 ◽  
Vol 7 (2) ◽  
pp. 1
Author(s):  
Soumendra Bera
Keyword(s):  
Positive Integer ◽  
Symmetric Polynomial ◽  
Enumeration Problems ◽  
Homogeneous Symmetric Polynomial

<p class="abstract">We count the number of occurrences of <em>t </em>as the summands<em> </em>(i) in the compositions of a positive integer <em>n</em> into <em>r</em> parts; and (ii) in all compositions of <em>n</em>; and subsequently obtain other results involving compositions. The initial counting further helps to solve the enumeration problems for complete homogeneous symmetric polynomial.</p>

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 Quasisymmetric Schur functions

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
James Haglund ◽  
Sarah Mason ◽  
Kurt Luoto ◽  
Steph van Willigenburg
Keyword(s):  
Schur Functions ◽  
Quasisymmetric Function ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Quasisymmetric Schur Functions

International audience We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions. Nous étudions une nouvelle base des fonctions quasisymétriques, les fonctions de quasiSchur. Ces fonctions sont obtenues en spécialisant les fonctions de Macdonald dissymétrique. Nous décrivons les compositions que donne une simple fonction quasisymétriques. Nous décrivons aussi une règle par certaines fonctions de Schur.

QUANTUM SCHUBERT POLYNOMIALS AND QUANTUM SCHUR FUNCTIONS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 385-404
Author(s):  
ANATOL N. KIRILLOV
Keyword(s):  
Schur Functions ◽  
Macdonald Polynomials ◽  
Schur Function ◽  
Quantum Double ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
Quantum Analog ◽  
Double Schubert Polynomials

We introduce the quantum multi–Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations, the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nägelsbach–Kostka and Jacobi–Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove also an analog of the Giambelli and the Billey–Jockusch–Stanley formula for quantum Schubert polynomials. Finally we formulate two conjectures about the structure of quantum double and quantum Schubert polynomials for 321–avoiding permutations.

scholarly journals On integers which are representable as sums of large squares

2015 ◽  
Vol 11 (08) ◽  
pp. 2505-2511 ◽  
Author(s):  
Alessio Moscariello
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Integer Coefficients ◽  
Representation Of Integers

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n2, (n + 1)2, …} is asymptotically O(n2), verifying thus a conjecture of Dutch and Rickett. Furthermore, we ask a question on the representation of integers as sum of four large squares.

scholarly journals Bijective proofs for Schur function identities which imply Dodgson's condensation formula and Plücker relations

10.37236/1560 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Markus Fulmek ◽  
Michael Kleber
Keyword(s):  
Schur Functions ◽  
Schur Function ◽  
Plücker Relations

We present a "method" for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi–Trudi identity. We illustrate this "method" by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson's condensation formula, Plücker relations and a recent identity of the second author.

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