Cylindric skew Schur functions (Q2503365)

The cylinder \({\mathfrak C}_{vu}\), \(u,v\) integers, is defined as the quotient of the integer lattice \({\mathbb Z}^2\) modulo the shifting action which sends \((i,j)\) to \((i-u,j+v)\). This is a poset which inherits the partial order generated by the relations \((i,j)<(i+1,j)\) and \((i,j)<(i,j+1)\). A cylindric skew shape \(C\) is a finite convex subposet of \({\mathfrak C}_{vu}\). One defines semistandard cylindric tableaux of shape \(C\) and cylindric skew Schur functions \(s_C(x)\). It turns out that the \(s_C(x)\) are symmetric functions for any cylindric skew shape \(C\). The author shows that cylindric skew Schur functions arise naturally in the study of \((P,\omega)\)-partitions. Another reason to investigate them is the recent work of A. Postnikov which shows a strong connection with the problem to find a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. The author gives an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions and studies \(s_C(x)\) from the point of view of Schur-positivity. He shows that \(s_C(x)\) is Schur-positive only in the trivial case when \(C\) is isomorphic to a skew shape. Finally the author introduces the new concept of cylindric Schur-positivity and proposes the conjecture that cylindric skew Schur functions are cylindric Schur-positive.