Newton polytopes of dual \(k\)-Schur polynomials (Q6632114)

Context: Given a polynomial \(f\) with real coefficients, the Newton polytope of \(f\) is the convex hull of its exponent vectors. Newton polytopes has been extensively studied in various areas of mathematics because they provide visual tools to analyze structure of polynomials and theirs associated algebraic varieties. A polynomial \(f\) has saturated Newton polytope (SNP) if the set of lattice points in its Newton polytope coincides with the set of its exponent vectors. Monical-Tokcan-Yong has showed and conjectured various polynomials has SNP. Motivated by this work, the SNP property of many other polynomials are studied. \(M\)-convexity is another interesting property stronger than SNP property: A homogeneous polynomial \(f\) is \(M\)-convex if and only if \(f\) has SNP and its Newton polytope is a generalized permutahedron.\N\NResult: The purpose of this paper is to study Newton polytopes and \(M\)-convexity of dul \(k\)-Schur polynomials, affine Stanley symmetric polynomials, and cylindric skew Schur polynomials. They show that dual \(k\)-Schur polynomial is \(M\)-convex and its Newton polytope is a \(\lambda\)-permutahedron (Theorem 3.8). The affine Stanley symmetric polynomials, cylindric skew Schur polynomials are \(M\)-convex (Theorem 4.6, Corollary 4.9).\N\NKey: The key in the proof of Theorem 3.8 is that the support of the dual \(k\)-Schur polynomial indexed by a \(k\)-bounded partition and the support of the Schur polynomial indexed by the same partition coincide. The key in the proof of Theorem 4.6 is that affine Stanley symmetric functions expand positively in terms of dual \(k\)-Schur functions (Theorems 4.3, 4.4) and the criterion for determining whether linear combinations of dual \(k\)-Schur polynomials are SNP or not is similar to those of Schur functions (Theorems 4.1, 4.5). The key in the proof of Corollary 4.9 is that cylindric skew Schur polynomials are considered as affine Stanley symmetric polynomials (Theorem 4.8).\N\NStructure: In Sect. 2, they review some notations and facts on dual \(k\)-Schur functions. In Sect. 3, they prove Theorem 3.8. In Sect. 4, they prove Theorem 4.6 and Corollary 4.9. In Sect.5, they present some open problems and conjectures for further research.