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math.RA — Rings and Algebras
math.RA — Rings and Algebras
The augmentation of the Markov cluster algebra is explicit but not inducible as a rooted cluster morphism. This answers a question of Chang and Zhu negatively.
math.RA — Rings and Algebras
We give a finite criterion for total sign skew symmetry of integer matrices of order three.
math.QA — Quantum Algebra
We prove Conjecture 11 of Lee, Li, Rupel, and Zelevinsky on the support of triangular basis elements in rank-two quantum cluster algebras.
math.QA — Quantum Algebra
We prove Conjecture 12 of Lee, Li, Rupel, and Zelevinsky: for arbitrary positive integers , every strongly positive basis of the coefficient-free rank-two quantum cluster algebra contains a unique element pointed at each pair in . These pointed elements exhaust the basis.
math.CO — Combinatorics
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.
math.RA — Rings and Algebras
Let be a commutative ring. We show that every homogeneous polynomial of degree splits into a product of homogeneous linear forms over a suitable noncommutative ring extension of (that is, over a noncommutative -algebra whose structure morphism is injective). The algebra is universal for such factorizations and is free as an -module. The proof uses Bergman's Diamond Lemma: we define by generators and relations, and the relations form a terminating reduction system that has no ambiguities and thus is confluent. An analogous result is also shown for inhomogeneous polynomials (with inhomogeneous factors). This easily follows from the homogeneous case by homogenizing and then setting the homogenizing variable equal to .
math.QA — Quantum Algebra
We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank.
math.AC — Commutative Algebra
We show that each univariate polynomial over a commutative ring can be factored into linear factors over a suitable commutative ring extension of . The proof proceeds by universal construction: is defined as the tensor product , where is the polynomial ring $\ZZ[a_1, b_1, a_2, b_2, \ldots, a_m, b_m]$, and where is its subring generated by its ``homogenized elementary symmetric polynomials'' for all . The injectivity of the structure homomorphism is deduced from a combinatorial study of the diagonal subring of . In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.