Composition of transpositions and equality of ribbon Schur Q-functions

TL;DR

This paper introduces a new operation on skew diagrams and explores equalities and relations among ribbon Schur Q-functions, providing a basis, irreducibility, and connections to Eulerian posets.

Contribution

It develops a composition operation on skew diagrams, derives new equalities for skew Schur Q-functions, and characterizes relations among ribbon Schur Q-functions.

Findings

Introduces composition of transpositions on skew diagrams.

Establishes a basis for skew Schur Q-functions using ribbon Schur Q-functions.

Shows non-commutative ribbon Schur Q-functions relate to Eulerian posets.

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 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.