TY - JOUR

T1 - t-Unique reductions for Mészáros’s subdivision algebra

AU - Grinberg, Darij

N1 - Publisher Copyright:
© 2018, Institute of Mathematics. All rights reserved.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Arnold relations

KW - Baxter relations

KW - Gröbner bases

KW - Noncommutative algebra

KW - Orlik

KW - Subdivision algebra

KW - Terao algebras

KW - Yang

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UR - http://www.scopus.com/inward/citedby.url?scp=85051846391&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2018.078

DO - 10.3842/SIGMA.2018.078

M3 - Article

AN - SCOPUS:85051846391

VL - 14

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

M1 - 078

ER -