On the Support of Grothendieck Polynomials

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Annals of Combinatorics Pub Date : 2024-08-08 DOI:10.1007/s00026-024-00712-3

Karola Mészáros, Linus Setiabrata, Avery St. Dizier

引用次数: 0

Abstract

Grothendieck polynomials \(\mathfrak {G}_w\) of permutations \(w\in S_n\) were introduced by Lascoux and Schützenberger (C R Acad Sci Paris Sér I Math 295(11):629–633, 1982) as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of \(\mathbb {C}^n\). We conjecture that the exponents of nonzero terms of the Grothendieck polynomial \(\mathfrak {G}_w\) form a poset under componentwise comparison that is isomorphic to an induced subposet of \(\mathbb {Z}^n\). When \(w\in S_n\) avoids a certain set of patterns, we conjecturally connect the coefficients of \(\mathfrak {G}_w\) with the Möbius function values of the aforementioned poset with \(\hat{0}\) appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations

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论格罗内迪克多项式的支持度

Lascoux和Schützenberger（C R Acad Sci Paris Sér I Math 295(11):629-633, 1982）介绍了排列\(w\in S_n\)的格罗根迪克多项式（Grothendieck polynomials \(\mathfrak {G}_w\) of permutations \(w\in S_n\)），作为K理论中舒伯特循环的K理论类的一组杰出代表。我们猜想格罗登第克多项式 \(\mathfrak {G}_w\)的非零项的指数构成了一个分量比较下的正集，这个正集与\(\mathbb {Z}^n\) 的诱导子集同构。当\(w\in S_n\)避免了一组特定的模式时，我们猜想\(\mathfrak {G}_w\)的系数与上述附加了\(\hat{0}\)的poset的莫比乌斯函数值有关。我们证明了格拉斯曼和烟花排列猜想的特例

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来源期刊

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches