Crystals and Schur $P$-Positive Expansions

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.

A Hopf Algebraic Approach to Schur Function Identities

Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.

Two Remarks on Skew Tableaux

This paper contains two results on the number $f^{\sigma/\tau}$ of standard skew Young tableaux of shape $\sigma/\tau$. The first concerns generating functions for certain classes of "periodic" shapes related to work of Gessel-Viennot and Baryshnikov-Romik. The second result gives an evaluation of the skew Schur function $s_{\lambda/\mu}(x)$ at $x=(1,1/2^{2k},1/3^{2k}, \dots)$ for $k=1,2,3$ in terms of $f^{\sigma/\tau}$ for a certain skew shape $\sigma/\tau$ depending on $\lambda/\mu$.

Lattice Path Proofs of Extended Bressoud-Wei and Koike Skew Schur Function Identities

A recent paper of the present authors provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs.

Skew quantum Murnaghan-Nakayama rule

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.

A Pieri rule for skew shapes

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.

Stable Equivalence over Symmetric Functions

By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli-type matrices for a given skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the transpose of the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This leads to an affirmative answer to a question proposed by Kuperberg.