Multiplicative structures of the immaculate basis of non-commutative symmetric functions

Abstract

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising relation develops among non-commutative Littlewood-Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope.

30 pages: we cleaned and fixed many details in the proofs. The interested reader may toggle \specialcomments in the TeX file to reveal 6 added pages of details and ideas (in red)

References in corpus (5)

• Quasisymmetric Schur functions
• A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions
• Indecomposable modules for the dual immaculate basis of quasi-symmetric functions
• The Pieri rule for dual immaculate quasi-symmetric functions
• Structure Constants for Immaculate Functions

Cited by in corpus (8)

• Structure Constants for Immaculate Functions
• The immaculate basis of the non-commutative symmetric functions
• Extended Schur functions and bases related by involutions
• A quantum Murnaghan--Nakayama rule for the flag manifold
• A generalization of the dual immaculate quasisymmetric functions in partially commutative variables
• The pre-Pieri rules
• Noncommutative Shifted Symmetric Functions
• Quasisymmetric Schur $Q$-functions and peak Young quasisymmetric Schur functions