Equality of skew Schur functions in noncommuting variables

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 (, ₃), where D is a connected skew diagram with n boxes and is a permutation in the symmetric group Sₙ. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: s (, ₃) = s (, ₓ) such that D T if and only if D is a nonsymmetric ribbon, T is the antipodal rotation of D and ^-1 is an explicit bijection between two set partitions determined by D.