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

SIGMA 14 (2018), 078, 34 pages      arXiv:1704.00839      https://doi.org/10.3842/SIGMA.2018.078

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

Darij Grinberg
School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA

Received November 22, 2017, in final form July 15, 2018; Published online July 26, 2018

Abstract
Fix a commutative ring $\mathbf{k}$, two elements $\beta \in\mathbf{k}$ and $\alpha\in\mathbf{k}$ and a positive integer $n$. Let $\mathcal{X}$ be the polynomial ring over $\mathbf{k}$ in the $n(n-1) /2$ indeterminates $x_{i,j}$ for all $1\leq i$<$j\leq n$. Consider the ideal $\mathcal{J}$ of $\mathcal{X}$ 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 $1\leq i$<$j$<$k\leq n$. The quotient algebra $\mathcal{X}/\mathcal{J}$ (at least for a certain choice of $\mathbf{k}$, $\beta$ and $\alpha$) 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 $\mathcal{X}$ is said to be pathless if it has no divisors of the form $x_{i,j}x_{j,k}$ with $1\leq i$<$j$<$k\leq n$. The residue classes of these pathless monomials span the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, but (in general) are $\mathbf{k}$-linearly dependent. More combinatorially: reducing a given $p\in\mathcal{X}$ modulo the ideal $\mathcal{J}$ by applying replacements of the form $x_{i,j}x_{j,k}\mapsto x_{i,k} ( x_{i,j}+x_{j,k}+\beta ) +\alpha$ always eventually leads to a $\mathbf{k}$-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 $\mathbf{k}$-algebra homomorphism $D$ from $\mathcal{X}$ into the polynomial ring $\mathbf{k} [ t_{1},t_{2},\ldots,t_{n-1} ] $ that sends each $x_{i,j}$ to $t_{i}$. We show the following fact (generalizing a conjecture of Mészáros): If $p\in\mathcal{X}$, and if $q\in\mathcal{X}$ is a $\mathbf{k}$-linear combination of pathless monomials satisfying $p\equiv q\operatorname{mod}\mathcal{J}$, then $D(q) $ does not depend on $q$ (as long as $\beta$, $\alpha$ and $p$ are fixed). Thus, the above way of reducing a $p\in\mathcal{X}$ modulo $\mathcal{J}$ may lead to different results, but all of them become identical once $D$ is applied. We also find an actual basis of the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, using what we call forkless monomials.

Key words: subdivision algebra; Yang-Baxter relations; Gröbner bases; Arnold relations; Orlik-Terao algebras; noncommutative algebra.

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