On Antipodes of Immaculate Functions

Abstract

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.