scholarly journals Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions

10.37236/2248 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Tom Denton
Keyword(s):  
Schur Functions ◽  
Canonical Decomposition ◽  
Schur Function ◽  
Combinatorial Properties ◽  
Affine Permutations ◽  
New Perspective ◽  
Special Case ◽  
Arbitrary Affine

We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam.  This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements.  We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation.  As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.

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 The Plethysm $s_\lambda[s_\mu]$ at Hook and Near-Hook Shapes

10.37236/1764 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
T. M. Langley ◽  
J. B. Remmel
Keyword(s):  
Schur Functions ◽  
Schur Function ◽  
Explicit Formulas ◽  
Function Expansion ◽  
Special Case

We completely characterize the appearance of Schur functions corresponding to partitions of the form $\nu = (1^a, b)$ (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, $$s_\lambda[s_\mu] = \sum_{\nu} a_{\lambda, \mu, \nu} s_\nu.$$ Specifically, we show that no Schur functions corresponding to hook shapes occur unless $\lambda$ and $\mu$ are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm $s_{(1^c, d)}[s_{(1^a, b)}]$. We also consider the problem of adding a row or column so that $\nu$ is of the form $(1^a,b,c)$ or $(1^a, 2^b, c)$. This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.

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.

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 Karhunen–Loève decomposition of Gaussian measures on Banach spaces

2019 ◽  
Vol 39 (2) ◽  
pp. 279-297
Author(s):  
Xavier Bay ◽  
Jean-Charles Croix
Keyword(s):  
Brownian Motion ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Applied Mathematics ◽  
Canonical Decomposition ◽  
Spectral Theorem ◽  
Gaussian Measures ◽  
Covariance Operators ◽  
Active Interest ◽  
Special Case

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the socalled Karhunen–Ločve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated with Gaussian measures on Banach spaces. In the special case of the standardWiener measure, this decomposition matches with Lévy–Ciesielski construction of Brownian motion.

scholarly journals Affine Permutations of Type $A$

10.37236/1276 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Anders Björner ◽  
Francesco Brenti
Keyword(s):  
Coxeter Group ◽  
Type A ◽  
Bruhat Order ◽  
Weak Order ◽  
Combinatorial Properties ◽  
Affine Permutations

We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group $\tilde{A}_{n}$.

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.

scholarly journals On the Ornstein–Zernike equation for stationary cluster processes and the random connection model

2017 ◽  
Vol 49 (4) ◽  
pp. 1260-1287 ◽  
Author(s):  
Günter Last ◽  
Sebastian Ziesche
Keyword(s):  
Point Process ◽  
Cluster Model ◽  
Combinatorial Properties ◽  
Palm Calculus ◽  
Fourier Theory ◽  
Associated Pair ◽  
Special Case ◽  
Random Connection Model ◽  
Palm Probabilities ◽  
Cluster Processes

Abstract In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

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 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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