scholarly journals Crystal Analysis of type C Stanley Symmetric Functions

10.37236/6952 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Graham Hawkes ◽  
Kirill Paramonov ◽  
Anne Schilling
Keyword(s):  
Coxeter Group ◽  
Nonnegative Integer ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Combinatorial Description ◽  
Highest Weight ◽  
Type C ◽  
Stanley Symmetric Functions

Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.

scholarly journals Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
K Theory ◽  
Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

scholarly journals Schur $P$-positivity and Involution Stanley Symmetric Functions

2017 ◽  
Vol 2019 (17) ◽  
pp. 5389-5440 ◽  
Author(s):  
Zachary Hamaker ◽  
Eric Marberg ◽  
Brendan Pawlowski
Keyword(s):  
Power Series ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Pattern Avoidance ◽  
Dominance Order ◽  
Schubert Polynomials ◽  
Stanley Symmetric Functions ◽  
Skew Schur Functions

Abstract The involution Stanley symmetric functions$\hat{F}_y$ are the stable limits of the analogs of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words and are indexed by the involutions in the symmetric group. By construction, each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila–Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials.

scholarly journals Crystal Structures for Double Stanley Symmetric Functions

10.37236/8872 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Graham Hawkes
Keyword(s):  
Crystal Structures ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Type A ◽  
Combinatorial Objects ◽  
Type C ◽  
Stanley Symmetric Functions ◽  
Relationship Of

We relate the combinatorial definitions of the type $A_n$ and type $C_n$ Stanley symmetric functions, via a combinatorially defined "double Stanley symmetric function," which gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives the type $C$ case at $(\mathbf{x},\mathbf{x})$.  We induce a  type $A$ bicrystal structure on the underlying combinatorial objects of this function which has previously been done in the type $A$ and type $C$ cases.  Next we prove a few statements about the algebraic relationship of these three Stanley symmetric functions. We conclude with some conjectures about what happens when we generalize our constructions to type $C$.

scholarly journals The Murnaghan―Nakayama rule for k-Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Anne Schilling ◽  
Mike Zabrocki
Keyword(s):  
Explicit Formula ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Schur Function ◽  
Noncommutative Symmetric Functions ◽  
Power Sum ◽  
International Audience

International audience We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene. Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

scholarly journals Bottom Schur Functions

10.37236/1820 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Peter Clifford ◽  
Richard P. Stanley
Keyword(s):  
Generating Function ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Power Sums ◽  
Power Sum ◽  
Border Strip ◽  
Ramanujan Identity

We give a basis for the space spanned by the sum $\hat{s}_\lambda$ of the lowest degree terms in the expansion of the Schur symmetric functions $s_\lambda$ in terms of the power sum symmetric functions $p_\mu$, where deg$(p_i)=1$. These lowest degree terms correspond to minimal border strip tableaux of $\lambda$. The dimension of the space spanned by $\hat{s}_\lambda$, where $\lambda$ is a partition of $n$, is equal to the number of partitions of $n$ into parts differing by at least 2. Applying the Rogers-Ramanujan identity, the generating function also counts the number of partitions of $n$ into parts $5k+1$ and $5k-1$. We also show that a symmetric function closely related to $\hat{s}_\lambda$ has the same coefficients when expanded in terms of power sums or augmented monomial symmetric functions.

scholarly journals Product of Stanley symmetric functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Nan Li
Keyword(s):  
Nous Avons ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Nous Obtenons ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Natural Expansion ◽  
Natural Way

International audience We study the problem of expanding the product of two Stanley symmetric functions $F_w·F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, and study the behavior of the expansion of $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. Nous étudions le problème de développement du produit de deux fonctions symétriques de Stanley $F_w·F_u$ en fonctions symétriques de Stanley de façon naturelle. Notre méthode consiste à considérer une fonction symétrique de Stanley comme un polynôme du Schubert stabilisè $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, et à étudier le comportement de développement de $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ en polynômes de Schubert lorsque $n$ augmente. Nous prouvons que cette développement se stabilise et donc nous obtenons une développement naturelle pour le produit de deux fonctions symétriques de Stanley. Dans le cas où l'une des permutations est Grassmannienne, nous avons une meilleure compréhension de cette stabilité.

scholarly journals Symmetric Fundamental Expansions to Schur Positivity

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Positive Integer ◽  
Symmetric Function ◽  
Schur Functions ◽  
Quasisymmetric Functions ◽  
Combinatorial Description ◽  
International Audience ◽  
Sum Of Functions

International audience We consider families of quasisymmetric functions with the property that if a symmetric function f is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of f. We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families.

scholarly journals Schur and $e$-Positivity of Trees and Cut Vertices

10.37236/8930 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Samantha Dahlberg ◽  
Adrian She ◽  
Stephanie Van Willigenburg
Keyword(s):  
Linear Combination ◽  
Connected Graph ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Perfect Matching ◽  
Schur Functions ◽  
Connected Components ◽  
Elementary Symmetric Functions ◽  
Chromatic Symmetric Function ◽  
Positive Linear Combination

We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geqslant \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geqslant\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.

scholarly journals Combinatorial description of the cohomology of the affine flag variety

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Seung Jin Lee
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Flag Variety ◽  
Schubert Calculus ◽  
Power Sum ◽  
Schubert Polynomials ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Affine Flag Variety ◽  
Affine Flag

International audience We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.

Stanley Symmetric Functions and Peterson Algebras

2014 ◽  
pp. 133-168 ◽  
Author(s):  
Thomas Lam ◽  
Luc Lapointe ◽  
Jennifer Morse ◽  
Anne Schilling ◽  
Mark Shimozono ◽  
...  
Keyword(s):  
Symmetric Functions ◽  
Stanley Symmetric Functions
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