Abstract
Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur- positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.
Highlights
Given any subset A of the symmetric group Sn, define the quasi-symmetric functionQ(A) := Fn,Des(π), π∈A where Des(π) := {i : π(i) > π(i + 1)} is the descent set of π and Fn,Des(π) is Gessel’s fundamental quasi-symmetric function defined by Fn,D(x) :=xi1 xi2 · · · xin .i1≤i2 ≤...≤in ij
Even though the proof details are not included in this extended abstract due to lack of space, they appear in the full version of this paper [8], where we provide a representation-theoretic proof that involves Solomon’s descent representations, Kronecker products of symmetric functions and Stanley’s shuffling theorem
Consider the map from the set of standard Young tableaux (SYT for short) of all zigzag shapes of size n to permutations in Sn defined by listing the entries of the SYT, starting from the southwest corner and moving along the shape. The restriction of this map to the set SYT(Zn,J ) of tableaux of a fixed zigzag shape Zn,J is a bijection to permutations in Sn with descent set J
Summary
Given any subset A of the symmetric group Sn, define the quasi-symmetric function. Q(A) := Fn,Des(π), π∈A where Des(π) := {i : π(i) > π(i + 1)} is the descent set of π and Fn,Des(π) is Gessel’s fundamental quasi-symmetric function defined by. Schur-positivity of various set and multiset products of grid classes follows from the above two results.
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