Skew Littlewood―Richardson rules from Hopf algebras

Thomas Lam; Aaron Lauve; Frank Sottile

doi:10.46298/dmtcs.2853

Skew Littlewood―Richardson rules from Hopf algebras

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2853 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Thomas Lam ◽

Aaron Lauve ◽

Frank Sottile

Keyword(s):

Hopf Algebras ◽

Symplectic Group ◽

Schur Functions ◽

Affine Grassmannian ◽

International Audience ◽

Pieri Rules ◽

Skew Schur Functions ◽

Nous Utilisons

International audience We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood―Richardson pour les fonctions de Schur gauches, qui implique une conjecture d'Assaf et McNamara. Nous établissons également des règles de Littlewood―Richardson gauches pour les $P-$ et $Q-$fonctions de Schur et les fonctions de Schur rubbans non commutatives, ainsi que des règles de Pieri gauches pour les $k-$fonctions de Schur, les $k-$fonctions de Schur duales, et pour l'homologie de la Grassmannienne affine du groupe symplectique.

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Pieri rules for Schur functions in superspace

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2497 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Miles Eli Jones ◽

Luc Lapointe

Keyword(s):

Generating Functions ◽

Schur Functions ◽

Macdonald Polynomials ◽

International Audience ◽

Pieri Rules

International audience The Schur functions in superspace $s_\Lambda$ and $\overline{s}_\Lambda$ are the limits $q=t= 0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We present the elementary properties of the bases $s_\Lambda$ and $\overline{s}_\Lambda$ (which happen to be essentially dual) such as Pieri rules, dualities, monomial expansions, tableaux generating functions, and Cauchy identities. Les fonctions de Schur dans le superespace $s_\Lambda$ et $\overline{s}_\Lambda$ sont les limites $q=t= 0$ et $q=t=\infty$ respectivement des polynômes de Macdonald dans le superespace. Nous présentons les propriétés élémentaires des bases $s_\Lambda$ et $\overline{s}_\Lambda$ (qui sont essentiellement duales l'une de l'autre) tels que les règles de Pieri, la dualité, le développement en fonctions monomiales, les fonctions génératrices de tableaux et les identités de Cauchy.

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A Pieri rule for skew shapes

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2886 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Sami H. Assaf ◽

Peter R. W. McNamara

Keyword(s):

Schur Functions ◽

Classical Case ◽

Combinatorial Proof ◽

Schur Function ◽

Single Row ◽

Skew Schur Function ◽

International Audience ◽

Skew Schur Functions

International audience The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case. La règle de Pieri exprime le produit d'une fonction de Schur et de la fonction de Schur d'une seule ligne en termes de fonctions de Schur. Nous étendons la règle classique de Pieri en exprimant le produit d'un fonction gauche de Schur et de la fonction de Schur d'une ligne en termes de fonctions gauches de Schur. Comme la règle classique, notre règle implique l'ajout de cases à la forme gauche initiale. Notre preuve est purement combinatoire et étend celle du cas classique.

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Skew quantum Murnaghan-Nakayama rule

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2936 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Matjaž Konvalinka

Keyword(s):

Schur Functions ◽

Schur Function ◽

Original Proof ◽

Power Sum ◽

General Identity ◽

Skew Schur Function ◽

International Audience ◽

Skew Schur Functions

International audience In this extended abstract, we extend recent results of Assaf and McNamara, the skew Pieri rule and the skew Murnaghan-Nakayama rule, to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. Dans cet article nous élargissons le cadre de résultats récents de Assaf et McNamara, la règle dissymétrique de Pieri et la règle dissymétrique de Murnaghan-Nakayama, pour obtenir une identité plus générale donnant un développement élégant du produit de la fonction de Schur dissymétrique par une somme de puissances quantiques, en termes de fonctions de Schur dissymétriques. Nous donnons deux démonstrations, la première suivant l'approche de Assaf-McNamara et la deuxième par le biais de la règle dissymétrique de Littlewood-Richardson obtenue par Lam-Lauve-Sotille.

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Hook formulas for skew shapes

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6354 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Alejandro H. Morales ◽

Igor Pak ◽

Greta Panova

Keyword(s):

General Formula ◽

Schur Functions ◽

Product Formula ◽

Young Tableaux ◽

Hook Length ◽

International Audience ◽

Border Strip ◽

Standard Young Tableaux ◽

Skew Schur Functions ◽

Reverse Plane

International audience The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.

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Schubert Polynomials and $k$-Schur Functions

The Electronic Journal of Combinatorics ◽

10.37236/4139 ◽

2014 ◽

Vol 21 (4) ◽

Cited By ~ 1

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.

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