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.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.