$t$-Unique Reductions for Mészáros's Subdivision Algebra

Symmetry Integrability and Geometry-Methods and Applications

article

Darij Grinberg¹ 

[1] School of Mathematics, University of Minnesota

关键词: subdivision algebra;    Yang–Baxter relations;    Gr¨obner bases;    Arnold relations;    Orlik–Terao algebras;    noncommutative algebra;   

DOI  :  10.3842/SIGMA.2018.078

来源: National Academy of Science of Ukraine

【 摘 要 】

Fix a commutative ring k, two elements β ∈ k and α ∈ k and a positive integer n. Let X be the polynomial ring over k in the n(n − 1)/2 indeterminates xi,j for all 1 ≤ i < j ≤ n. Consider the ideal J of X generated by all polynomials of the form xi,jxj,k − xi,k(xi,j+xj,k+β)−α for 1 ≤ i < j < k ≤ n. The quotient algebra X /J (at least for a certain choice of k, β and α) has been introduced by Karola M´esz´aros in [Trans. Amer. Math. Soc. 363 (2011), 4359–4382] as a commutative analogue of Anatol Kirillov’s quasi-classical Yang– Baxter algebra. A monomial in X is said to be pathless if it has no divisors of the form xi,jxj,k with 1 ≤ i < j < k ≤ n. The residue classes of these pathless monomials span the kmodule X /J , but (in general) are k-linearly dependent. More combinatorially: reducing a given p ∈ X modulo the ideal J by applying replacements of the form xi,jxj,k 7→ xi,k(xi,j+ xj,k + β) + α always eventually leads to a k-linear combination of pathless monomials, but the result may depend on the choices made in the process. More recently, the study of Grothendieck polynomials has led Laura Escobar and Karola M´esz´aros [Algebraic Combin. 1 (2018), 395–414] to defining a k-algebra homomorphism D from X into the polynomial ring k[t1, t2, . . . , tn−1] that sends each xi,j to ti . We show the following fact (generalizing a conjecture of M´esz´aros): If p ∈ X , and if q ∈ X is a k-linear combination of pathless monomials satisfying p ≡ q modJ , then D(q) does not depend on q (as long as β, α and p are fixed). Thus, the above way of reducing a p ∈ X modulo J may lead to different results, but all of them become identical once D is applied. We also find an actual basis of the k-module X /J , using what we call forkless monomials.

【 授权许可】

Unknown