Composition of Transpositions and Equality of Ribbon Schur $Q$-Functions

Farzin Barekat, Stephanie van Willigenburg

Abstract


We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur $Q$-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur $Q$-functions. Moreover, we determine all relations between ribbon Schur $Q$-functions; show they supply a ${\Bbb Z}$-basis for skew Schur $Q$-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur $Q$-functions is the flag $h$-vector of Eulerian posets.


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