The Murnaghan―Nakayama rule for k-Schur functions

• Published in: Discrete Mathematics & Theoretical Computer Science
• Main Authors: Bandlow, Jason, Schilling, Anne, Zabrocki, Mike
• Other Authors: Department of Mathematics Philadelphia, University of Pennsylvania, Department of Mathematics Univ California Davis (MATH - UC Davis), University of California Davis (UC Davis), University of California (UC)-University of California (UC), Department of Mathematics and Statistics Toronto, York University Toronto, Bousquet-Mélou, Mireille and Wachs, Michelle and Hultman, Axel
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2011
• Subjects: Murnaghan―Nayakama rule; symmetric functions; noncommutative symmetric functions; k-Schur functions; [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]; [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]; Fomin; Greene; Morse; Iceland; morse
• Online Access: https://hal.inria.fr/hal-01215093; https://hal.inria.fr/hal-01215093/document; https://hal.inria.fr/hal-01215093/file/dmAO0110.pdf; https://doi.org/10.46298/dmtcs.2894

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.