Equality of skew Schur functions in noncommuting variables
Emma Yu Jin, Stephanie van Willigenburg
TL;DR
The paper addresses the longstanding question of when two skew Schur functions in noncommuting variables are equal by classifying all equalities of the form $s_{(\delta,\mathcal{D})}=s_{(\tau,\mathcal{T})}$ for connected skew diagrams with $\mathcal{D}\neq\mathcal{T}$. It achieves a complete characterization: equality occurs exactly when $\mathcal{D}$ is a nonsymmetric ribbon, $\mathcal{T}=\mathcal{D}^*$, and the bijection $\overline{\tau^{-1}\delta}$ preserves each block of the set partition $[\alpha]$ determined by the row-length composition $\alpha=\alpha(\mathcal{D})$; equivalently, $\tau^{-1}\delta$ maps blocks to the reversed blocks of $[\alpha^*]$. The proof combines the noncommutative Jacobi–Trudi determinant framework with a reduction to connected ribbons and leverages the relationship between $\operatorname{NCSym}$ and $\operatorname{Sym}$ through $\rho$, together with necessary overlap conditions for equality. This yields a succinct combinatorial criterion enabling exact equality checks and advances understanding of skew Schur functions in noncommuting variables with potential implications for basis theory in noncommutative symmetric function theory.
Abstract
The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, $s_{(δ,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $δ$ is a permutation in the symmetric group $S_n$. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(δ,D)} = s_{(τ,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$ and $\overline{τ^{-1}δ}$ is an explicit bijection between two set partitions determined by $D$.