A Schur-Like Basis of NSym Defined by a Pieri Rule

  • John Maxwell Campbell
  • Karen Feldman
  • Jennifer Light
  • Pavel Shuldiner
  • Yan Xu
Keywords: Non-commutative symmetric functions, Pieri rules, Schur basis


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.
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