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 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 Interval positroid varieties and a deformation of the ring of symmetric functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Allen Knutson ◽  
Mathias Lederer
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Dimensional Subspace ◽  
Schubert Varieties ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Two Parameter ◽  
Parameter Deformation

International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

scholarly journals Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions

10.37236/5737 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Pavel Galashin ◽  
Darij Grinberg ◽  
Gaku Liu
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
The Other ◽  
Countable Family ◽  
Plane Partitions ◽  
K Theory ◽  
Reverse Plane ◽  
Grothendieck Polynomials

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.

scholarly journals Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Pavel Galashin ◽  
Darij Grinberg ◽  
Gaku Liu
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
The Other ◽  
Countable Family ◽  
Plane Partitions ◽  
International Audience ◽  
K Theory ◽  
Reverse Plane ◽  
Grothendieck Polynomials

International audience The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.

scholarly journals Noncommutative Symmetric Hall-Littlewood Polynomials

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Monomial Basis ◽  
Noncommutative Symmetric Functions ◽  
New Family ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Noncommutative Setting

International audience Noncommutative symmetric functions have many properties analogous to those of classical (commutative) symmetric functions. For instance, ribbon Schur functions (analogs of the classical Schur basis) expand positively in noncommutative monomial basis. More of the classical properties extend to noncommutative setting as I will demonstrate introducing a new family of noncommutative symmetric functions, depending on one parameter. It seems to be an appropriate noncommutative analog of the Hall-Littlewood polynomials. Les fonctions symétriques non commutatives ont de nombreuses propriétés analogues à celles des fonctions symétriques classiques (commutatives). Par exemple, les fonctions de Schur en rubans (analogues de la base de Schur classique) admettent des développements à coefficients positifs dans la base des monômes non commutatifs. La plupart des propriétés classiques s'étendent au cas non commutatif, comme je le montrerai en introduisant une nouvelle famille de fonctions symétriques non commutatives, dépendant d'un paramètre. Cette famille semble être un analogue non commutatif approprié de la famille des polynômes de Hall-Littlewood.

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.

SYMMETRIC FUNCTIONS OF BINARY PRODUCTS OF TRIBONACCI LUCAS NUMBERS AND ORTHOGONAL POLYNOMIALS

2021 ◽  
Vol 21 (2) ◽  
pp. 461-478
Author(s):  
HIND MERZOUK ◽  
ALI BOUSSAYOUD ◽  
MOURAD CHELGHAM
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Symmetric Functions ◽  
Lucas Numbers

In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

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

scholarly journals Comultiplication Rules for the Double Schur Functions and Cauchy Identities

10.37236/102 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
A. I. Molev
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Primitive Elements ◽  
Expansion Formula ◽  
New Family ◽  
Dual Object ◽  
Multiplication Rule ◽  
Distinguished Basis ◽  
Natural Way

The double Schur functions form a distinguished basis of the ring $\Lambda(x\!\parallel\!a)$ which is a multiparameter generalization of the ring of symmetric functions $\Lambda(x)$. The canonical comultiplication on $\Lambda(x)$ is extended to $\Lambda(x\!\parallel\!a)$ in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood–Richardson coefficients provide a multiplication rule for the dual Schur functions.

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