| 1. |
- Alexandersson, Per, et al.
(författare)
-
Properties of Non-symmetric Macdonald Polynomials at q=1 and q=0
- 2019
-
Ingår i: Annals of Combinatorics. - : Springer Publishing Company. - 0218-0006 .- 0219-3094. ; 23:2, s. 219-239
-
Tidskriftsartikel (refereegranskat)abstract
- We examine the non-symmetric Macdonald polynomials E at q=1, as well as the more general permuted-basement Macdonald polynomials. When q=1, we show that E(x;1,t) is symmetric and independent of t whenever is a partition. Furthermore, we show that, in general , this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on x, t, and the relative order of the entries in . We also examine the case q=0, which gives rise to the so-called permuted-basement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.
|
|
| 2. |
- Beck, Matthias, et al.
(författare)
-
Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP
- 2019
-
Ingår i: Annals of Combinatorics. - : Springer Publishing Company. - 0218-0006 .- 0219-3094. ; 23:2, s. 255-262
-
Tidskriftsartikel (refereegranskat)abstract
- In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.
|
|
| 3. |
- Liu, Fu, et al.
(författare)
-
On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity
- 2019
-
Ingår i: Annals of Combinatorics. - : Springer Berlin/Heidelberg. - 0218-0006 .- 0219-3094. ; 23:2, s. 347-365
-
Tidskriftsartikel (refereegranskat)abstract
- For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are (1) to determine if its (Ehrhart) h-polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and s-lecture hall simplices.
|
|
