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t-Unique Reductions for Meszaros's Subdivision Algebra
Journal article   Open access   Peer reviewed

t-Unique Reductions for Meszaros's Subdivision Algebra

Darij Grinberg and University of Minnesota, USA
Symmetry, integrability and geometry, methods and applications, v 14, 078
01 Jan 2018
DOI: https://doi.org/10.3842/SIGMA.2018.078
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https://doi.org/10.3842/SIGMA.2018.078View
Published, Version of Record (VoR) Open

Abstract

Physical Sciences Physics Physics, Mathematical Science & Technology
Fix a commutative ring k, two elements beta is an element of k and alpha is an element of k and a positive integer n. Let X be the polynomial ring over k in the n(n - 1)/2 indeterminates x(i,j) for all 1 <= i < j <= n. Consider the ideal J of X generated by all polynomials of the form x(i,j)x(j),(k) - x(i,k)(x(i,j)+x(j,k)+beta)-alpha for 1 <= i < j < k <= n. The quotient algebra X/J (at least for a certain choice of k, beta and alpha has been introduced by Karola Meszaros 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 x(i,j)x(j,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 is an element of X modulo the ideal J by applying replacements of the form x(x,j)x(j,k) -> x(i,k)(x(i,j)+ x(j,k) + beta) + alpha 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 Meszaros [Algebraic Combin. 1 (2018), 395-414] to defining a k-algebra homomorphism D from X into the polynomial ring k[t(1),t(2),...,t(n-1)] that sends each x(i,j) to t(i). We show the following fact (generalizing a conjecture of Meszaros): If p is an element of X, and if q is an element of X is a k-linear combination of pathless monomials satisfying p equivalent to q mod J(,) then D(q) does not depend on q (as long as beta, alpha and p are fixed). Thus, the above way of reducing a p is an element of 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.

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