t-Unique reductions for Mészáros’s subdivision algebra

Darij Grinberg

Research output: Contribution to journalArticlepeer-review

Abstract

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,j xj,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 Mészáros 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,j xj,k with 1 ≤ i < j < k ≤ n. The residue classes of these pathless monomials span the k-module 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,j xj,k ↦→ 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 Mészáros [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 Mészáros): If p ∈ X, and if q ∈ X is a k-linear combination of pathless monomials satisfying p ≡ q mod J, 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.

Original languageEnglish (US)
Article number078
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume14
DOIs
StatePublished - 2018

Keywords

  • Arnold relations
  • Baxter relations
  • Gröbner bases
  • Noncommutative algebra
  • Orlik
  • Subdivision algebra
  • Terao algebras
  • Yang

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