Abstract
Recent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions. We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule. The commutative image of an element of this basis indexed by a partition equals the element of the Schur basis indexed by the same partition and the commutative image is $0$ otherwise. We establish a rule for right-multiplying an arbitrary element of this basis by an arbitrary element of the ribbon basis, and a Murnaghan-Nakayama-like rule for this new basis. Elements of this new basis indexed by compositions of the form $(1^n, m, 1^r)$ are evaluated in terms of the complete homogeneous basis and the elementary basis.
Highlights
The Hopf algebra Sym of symmetric functions is a subalgebra of QSym, the Hopf algebra of quasi-symmetric functions
Recent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions
We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule
Summary
The Hopf algebra Sym of symmetric functions is a subalgebra of QSym, the Hopf algebra of quasi-symmetric functions. The position of the fundamental and ribbon bases as “the” Schur analogues of QSym and NSym was called into question in the exploration of the quasi-symmetric function expansion of Macdonald polynomials [HHL]. Computer explorations of NSym and QSym using the software Sage [sage] lead us to consider other related possible bases of NSym and their dual bases of QSym. We are ideally searching for a basis that simplifies proofs of Schur-positivity properties of quasisymmetric function expansions. We prove a combinatorial formula for the product of an element of the shin basis and an element of the ribbon basis, and in Section 5 we prove a combinatorial formula for the product of an element of the shin basis and an element Ψn, which is an analogue of the power sum generator of degree n for NSym. In Section 6, we give a formula for the shin function evaluated at compositions of the form (1n, m, 1r). We show the close relationship between the shin function and the Schur basis of Sym
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