The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with more references). The subject is surprisingly rich and well-connected, so I would expect answers to come from anywhere. Part of the reason why I am posting this here is to learn more about some of these connections, more so than just to get a specific question answered.
1. The subdivision algebra
Let $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $.
Let $\left[ m\right] $ be the set $\left\{ 1,2,\ldots,m\right\} $ for each $m\in\mathbb{N}$.
Let $\mathbf{k}$ be a commutative ring. Let $\beta\in\mathbf{k}$ and $\alpha\in\mathbf{k}$. (The setup is interesting even in the case when $\beta=0$ and $\alpha=0$.)
Fix a positive integer $n$. Let $\mathcal{X}$ be the polynomial ring \begin{align} \mathbf{k}\left[ x_{i,j}\ \mid\ \left( i,j\right) \in\left[ n\right] ^{2}\text{ satisfying }i<j\right] . \end{align} This is a polynomial ring in $n\left( n-1\right) /2$ indeterminates $x_{i,j}$ over $\mathbf{k}$.
Let $\mathcal{J}$ be the ideal of $\mathcal{X}$ generated by all elements of the form \begin{align} x_{i,j}x_{j,k}-x_{i,k}\left( x_{i,j}+x_{j,k}+\beta\right) -\alpha \end{align} for $\left( i,j,k\right) \in\left[ n\right] ^{3}$ satisfying $i<j<k$.
For each $f\in\mathcal{X}$, we will let $\overline{f}$ denote the projection of $f$ onto the quotient $\mathbf{k}$-algebra $\mathcal{X}/\mathcal{J}$.
[If you are only looking for the question, scroll down to Section 2.]
The ideal $\mathcal{J}$ and the quotient algebra $\mathcal{X}/\mathcal{J}$ (known as the "$n$-th type-A subdivision algebra") have a long history. For $\alpha=0$, they originate in a 2009 paper by Karola Mészáros, but variants (noncommutative, skew-commutative, partly commutative, square-zero) appear all over the literature (I give a list of related algebras at the end of Section 4.3 of my paper). These variants include
the localization of a polynomial ring $\mathbf{k}\left[ s_{1},s_{2} ,\ldots,s_{n}\right] $ at the multiplicative subset generated by the differences $s_{i}-s_{j}$ for all $i<j$ (see below);
the cohomology of the complement of the braid arrangement (probably the oldest appearance, in Arnold 1971);
the algebra of Heaviside functions of the halfspaces of this arrangement (Gelfand/Varchenko 1987);
Kirillov's "quasi-classical Yang-Baxter algebra" (Kirillov 1997);
a recent deformation of the Orlik-Terao algebra of the braid arrangement (McBreen/Proudfoot 2015)
and probably more. There is also a mysterious similarity to the defining axiom of a Rota-Baxter algebra and a "hidden" $S_{n}$-action (which can be revealed by setting $x_{i,j} =-\beta-x_{j,i}$ for all $\left( i,j\right) \in\left[ n\right] ^{2}$ satisfying $i>j$, so that now $x_{i,j}$ is defined for any two distinct elements $i$ and $j$ of $\left[ n\right] $; this allows the symmetric group $S_{n}$ to act by $\sigma x_{i,j}=x_{\sigma\left( i\right) ,\sigma\left( j\right) }$).
One of the results in my paper (Proposition 3.4) says the following:
Theorem 1. The $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$ is free, and has a basis consisting of the projections (onto $\mathcal{X}/\mathcal{J}$) of all forkless monomials. Here, a monomial is always understood to be a formal monomial in the $x_{i,j}$ (with no coefficients attached); and such a monomial $\mathfrak{m}$ is called forkless if there exists no $\left( i,j,k\right) \in\left[ n\right] ^{3}$ with $i<j<k$ such that both $x_{i,j}$ and $x_{i,k}$ appear in $\mathfrak{m}$.
For example, the monomial $x_{1,3}x_{2,5}^{6}x_{3,4}$ is forkless, while the monomial $x_{1,3}x_{2,4}x_{2,5}$ is not (since $x_{2,4}x_{2,5}$ is a "fork").
Theorem 1, like many PBW-like theorems, has an easy and a hard part. The easy part is proving that the projections of the forkless monomials span $\mathcal{X}/\mathcal{J}$. The hard part is proving that they are $\mathbf{k} $-linearly independent. Arguably, it is not really hard, since a Gröbner basis does the whole work, but it still is an argument that makes me look for an alternative.
2. The localization
One of the reasons for looking at $\mathcal{X}/\mathcal{J}$ is its similarity to a very natural ring: the localization of a polynomial ring at the pairwise differences of its indeterminates (or, equivalently, at its Vandermonde determinant).
Let $P=\mathbf{k}\left[ s_{1},s_{2},\ldots,s_{n}\right] $ be the polynomial ring in $n$ indeterminates $s_{1},s_{2},\ldots,s_{n}$ over $\mathbf{k}$. Let $L$ be the localization of $P$ at the multiplicative subset $\left\{ s_{i}-s_{j}\ \mid\ \left( i,j\right) \in\left[ n\right] ^{2}\text{ satisfying }i<j\right\} $.
[If you are only looking for the question, scroll down to Section 3.]
The following fact, which I surprisingly could not find anywhere in the literature, describes $L$ as a $\mathbf{k}$-module:
Theorem 2. Consider the family of all elements of the form $\prod _{i=1}^{n}g_{i}\in L$, where each $g_{i}$ has either the form $\dfrac {1}{\left( s_{i}-s_{j}\right) ^{m}}$ for some $j\in\left\{ i+1,i+2,\ldots ,n\right\} $ and $m>0$ or the form $s_{i}^{k}$ for some $k\in\mathbb{N}$. This family is a basis of the $\mathbf{k}$-module $L$.
This is proven by an iterated partial-fractial decomposition (which appears to have a historical precedent -- the "method of Elliott" in MacMahon's partition analysis), inducting on $n$.
The basis in Theorem 2 is similar to the "forkless monomials" basis in Theorem 1, if we forget for a moment about the possibility of the $g_{i}$ being $s_{i}^{k}$. Indeed, if the $g_{i}$ are always of the form $\dfrac{1}{\left( s_{i}-s_{j}\right) ^{m}}$, then the resulting product $\prod_{i=1}^{n}g_{i}$ is a "forkless monomial" in the variables $\dfrac{1}{s_{i}-s_{j}}$. Does this mean that $\mathcal{X}/\mathcal{J}$ embeds into $L$ as a $\mathbf{k}$-algebra?
At least in a particular case, it does:
Proposition 3. Assume that $\beta=0$ and $\alpha=0$. Then, there is a $\mathbf{k}$-algebra homomorphism $H:\mathcal{X}/\mathcal{J}\rightarrow L$ sending each $\overline{x_{i,j}}$ (with $i<j$) to $\dfrac{1}{s_{i}-s_{j}}$. This homomorphism $H$ is injective (and, in fact, sends the "forkless monomial" basis from Theorem 1 to a subfamily of the basis from Theorem 2).
The image of $H$ is the algebra $\mathbf{K}\left[ \alpha_{\mathcal{A}} ^{-1}\right] $ in Definition 1.1 of Orlik/Terao 1994. That said, for some reason, everyone in the hyperplane arrangement community seems to care mostly about a finite-dimensional quotient of this algebra.
But back in the general case ($\beta$ and $\alpha$ arbitrary), the homomorphism $H$ from Proposition 3 does not seem to generalize. Instead, there is a more complicated homomorphism, which we will now construct.
3. The injectivity question
We define a $\mathbf{k} $-algebra homomorphism $A:\mathcal{X}/\mathcal{J}\rightarrow L$ by \begin{align} A\left( \overline{x_{i,j}}\right) =\dfrac{s_{i}s_{j}+\beta s_{j}+\alpha }{s_{i}-s_{j}}\qquad\text{for all }\left( i,j\right) \in\left[ n\right] ^{2}\text{ satisfying }i<j. \end{align} This $A$ is well-defined (easy to check). (My notations $s_{i}$ and $A$ here correspond to the notations $\widetilde{q}_{i}$ and $\widetilde{A}$ from my paper.)
Question 1. Is $A$ injective in the general case?
Note that I cannot answer this even in the case when $\alpha=0$ and $\beta=0$, since $A$ (unlike $H$) does not send forkless monomials to elements of the basis from Theorem 2.
To me, the nicest way to answer Question 1 (positively, of course) would be by proving that $A$ sends forkless monomials to $\mathbf{k}$-linearly independent elements of $L$. This would then yield that the forkless monomials are themselves $\mathbf{k}$-linearly independent, thus reproving the hard part of Theorem 1. Another option seems to be by extending the $\mathbf{k}$-algebra $\mathcal{X}/\mathcal{J}$ somehow and showing that $\mathcal{X}/\mathcal{J}$ embeds into the new algebra, then finding an isomorphism from that algebra to $L$.
Quite possibly, it is easier to work in a ring of Laurent series than to work in $L$. Fortunately, $L$ embeds into the ring of Laurent series in the variables $s_{1}/s_{2},\ s_{2}/s_{3},\ \ldots,\ s_{n-1}/s_{n},\ s_{n}$ (see Section 2.2 of my paper for a formal definition of this ring; for the analytically minded among us, this corresponds to the "regime" $\left| s_{1}\right| \ll\left| s_{2}\right| \ll\cdots\ll\left| s_{n}\right| \ll1$); the fraction $\dfrac{s_{i}s_{j}+\beta s_{j}+\alpha }{s_{i}-s_{j}}$ needs to be interpreted as $\dfrac{s_{i}+\beta+\alpha/s_{j} }{s_{i}/s_{j}-1}$ in this case. Working in this latter ring allows one to compare coefficients, which ideally should help proving linear independence. However, I have not been successful with this approach.
