$t$-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 $n(n-1)\mathrm{/}2$ indeterminates $x_{i,j}$ for all $1\leq i<j\leq n$\mathrm{.}Considertheideal$$\mathcal{J}$$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 $1\le i<j<k\le n$. The quotient algebra $\mathcal{X}\mathrm{/}\mathcal{J}$ (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 $1\le i<j<k\le n$. The residue classes of these pathless monomials span the $\mathbf{k}$-module $\mathcal{X}\mathrm{/}\mathcal{J}$, 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 $\mathbf{k}[t_1,t_2,\dots ,t_{n-1}]$ 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}$$,then$D(q)$doesnotdependon$q$(aslongas$\beta$,$\alpha$and$p$arefixed)\mathrm{.}Thus,reducinga$p\in$\mathcal{X}$$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 $\mathcal{X}\mathrm{/}\mathcal{J}$, using what we call forkless monomials.