On Antipodes of Immaculate Functions

The immaculate basis of the Hopf algebra \(\textsf {NSym}\) of noncommutative symmetric functions is a Schur-like basis of \(\textsf {NSym}\) that has been applied in many areas in the field of algebraic combinatorics. The problem of determining a cancellation-free formula for the antipode of \(\textsf {NSym}\) evaluated at an arbitrary immaculate function \( {\mathfrak {S}}_{\alpha } \) remains open, letting \(\alpha \) denote an integer composition. However, for the cases whereby we let \(\alpha \) be a hook or consist of at most two rows, Benedetti and Sagan (J Combin Theory Ser A 148:275–315, 2017) have determined cancellation-free formulas for expanding \(S({\mathfrak {S}}_{\alpha })\) in the \({\mathfrak {S}}\)-basis. According to a Jacobi–Trudi-like formula for expanding immaculate functions in the ribbon basis that we had previously proved bijectively (Discrete Math 340(7):1716–1726, 2017), by applying the antipode S of \(\textsf {NSym}\) to both sides of this formula, we obtain a cancellation-free formula for expressing \(S({\mathfrak {S}}_{(m^{n})})\) in the R-basis, for an arbitrary rectangle \((m^{n})\). We explore the idea of using this R-expansion, together with sign-reversing involutions, to determine combinatorial interpretations of the \({\mathfrak {S}}\)-coefficients of antipodes of rectangular immaculate functions. We then determine cancellation-free formulas for antipodes of immaculate functions much more generally, using a Jacobi–Trudi-like formula recently introduced by Allen and Mason that generalizes Campbell’s formulas for expanding \({\mathfrak {S}}\)-elements into the R-basis, and we further explore how new families of composition tableaux may be used to obtain combinatorial interpretations for expanding \(S({\mathfrak {S}}_{\alpha })\) into the \({\mathfrak {S}}\)-basis.