# Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

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 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.

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 (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 $$i such that both $$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.

[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 $$i) to $$\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 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.