Matching works

3 works

math.CO — Combinatorics

Signed seeds and G-gradings on cluster algebras

Contributed by Lauren Williams, Alan Yan

The theory of cluster algebras is closely connected to the theory of total positivity; indeed, the desire to better understand total positivity was one of the main motivations for Fomin and Zelevinsky’s introduction of cluster algebras [FZ02]. In particular, any cluster variety whose coordinate ring has a cluster structure has a natural notion of positive part: the subset of the variety where all cluster variables are positive. In this paper, we explain that there are other signed cells contained in cluster varieties that are equally natural from a cluster-theoretic point of view. These come from signed seeds, which can be thought of as a Z/2Z-grading on cluster variables, and which were introduced in [EZLP+ 23] in the context of the amplituhedron. More generally, given any abelian group G, we introduce the notion of a G-graded seed for a cluster algebra, which is a way of assigning elements of G to each cluster variable which is compatible with the cluster structure. When G is the multiplicative group {−1, 1}, this recovers the above notion of signed seed; when G = C∗ , this recovers the notion of cluster automorphism group [GSV10] or cluster dilation group [NS26]; and when G = Zd , this recovers the notion of graded cluster algebra studied by Grabowski-Launois [GL14], Grabowski [Gra15] and Gekhtman-Shapiro-Vainshtein [GSV10, Section 5.2] (which had previously appeared in special cases in work of Fomin-Zelevinsky [FZ07]). The examples we study include the space of square matrices, symmetric matrices, skew-symmetric matrices, positroid varieties, and amplituhedron tiles. We also connect this notion to tropical mutation when G = R or Z.

Primarily human-written textHuman understanding: all partsGrassmanniansamplituhedroncluster algebras

Sep 30, 2026PDF ↗View work →

hexagon:2609.00135v1

math.CO — Combinatorics

A counterexample to the Burman--Kulishov conjecture on Lie elements

Contributed by Darij Grinberg

Burman and Kulishov defined Lie elements in the group algebra $\mathbf k[S_n]$ by comparing, on every exterior power of the reflection representation $V$ , the usual action of $\mathbf k[S_n]$ with the infinitesimal action induced by its action on $V$ . They conjectured that the Lie algebra $\mathcal L_n$ of all Lie elements is generated by the Kirchhoff differences $1-(ij)$ . We disprove this conjecture for $n=4$ by exhibiting an explicit counterexample arising from the $(2,2)$ -block of $\mathbf k[S_4]$ . More generally, we describe $\mathcal L_n$ in terms of the Artin--Wedderburn decomposition of $\mathbf k[S_n]$ : its hook blocks are determined by the action on $V$ , whereas its non-hook blocks are arbitrary. Consequently, the primitive central idempotents of the non-hook blocks yield linearly independent obstructions to the conjecture. We also identify the Lie algebra generated by the Kirchhoff differences in terms of the derived algebra of the Lie algebra generated by transpositions. Along the way, we give an integral, and hence characteristic-free, proof that the exterior powers of $V$ are the hook-shaped Specht modules.

A mix of human-written and AI-generated textHuman understanding: all partsLie algebrasymmetric group algebrasymmetric group representations

Sep 29, 2026PDF ↗View work →

hexagon:2609.00116v1

math.CO — Combinatorics

A Counterexample to a Conjecture on Fused Specht Polynomials

Contributed by Darij Grinberg

Lafay, Peltola and Roussillon conjectured that their realization of simple modules of the fused Hecke algebra by fused Specht polynomials extends from Young diagrams with two columns to Young diagrams of arbitrary shape. We give a counterexample for \[ n=8,\qquad \lambda=(3,2,2,1),\qquad \varsigma=(2,2,2,2). \] More precisely, we exhibit an explicit linear dependence among three fused Specht polynomials indexed by row-strict Young tableaux. The same example also yields an infinite family of counterexamples.

Primarily AI-generated textHuman understanding: some partsSpecht modulesSpecht polynomialsYoung tableaux

Sep 28, 2026PDF ↗View work →

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory