$t$-Unique Reductions for M\'esz\'aros's Subdivision Algebra

Fix a commutative ring $\mathbf{k}$, two elements $\beta,\alpha\in\mathbf{k}$ 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 M\'esz\'aros 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. Recently, the study of Grothendieck polynomials has led Laura Escobar and Karola M\'esz\'aros 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 M\'esz\'aros): 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, reducing a $p\in\mathcal{X}$ modulo $\mathcal{J}$ may lead to different results depending on the choices made in the reduction process, 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.