Characters associated to the canonical character of QSym

We write all of these characters as elements of the algebra NSym of noncommutative symmetric finctions. This is the free associative algebra generated by elements H_n for nonnegative integers n, with H₀=1, the multiplicative identity. Each of the characters is the infinite sum of its homogeneous pieces and we exhibit these homogeneous pieces for each character up to degree 10.

The canonical character H is just the sum of the homogeneous generators H_n of NSym. We compute its image S(H_n) under the antipode S. It has an even and odd part, represented respectively as E and O, and we check that it is the product of the two. There are also two other characters associated to the canonical character, namely the Euler character chi and the Odd character nu. We record the first few homogeneous components of each of these five characters here.