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The Orlik-Terao algebra and the cohomology of configuration space. (English) Zbl 1373.52030

In the present paper, the authors provide a recursive algorithm for computing the Orlik-Terao algebra of the Coxeter arrangement \(A_{n-1}\) as a graded representation of \(S_{n}\) using a link with the cohomology of the configuration space of \(n\) points in \(SU(2)\) modulo translations. Let us shortly present an outline of the core of this paper.
For a given positive integer \(n\), denote by \(OT_{n}\) the subalgebra of rational functions on \(\mathbb{C}^{n}\) generated by the elements \(e_{ij}:= \frac{1}{x_{i}-x_{j}}\) for \(i \neq j\).This algebra is known as the Orlik-Terao algebra of the Coxeter arrangement of type \(A_{n-1}\). We regard \(OT_{n}\) as a graded ring with \(\deg(e_{ij}) = 2\). Let \(R_{n}:= \mathbb{C}[z_{1},\dots, z_{n}] / \langle z_{1} + \dots + z_{n} \rangle\) with its natural \(S_{n}\) action, and graded by putting \(\deg(z_{i}) = 2\). Consider the \(S_{n}\)-equivariant graded algebra homomorphism \(\phi_{n} : R_{n} \rightarrow OT_{n}\) taking \(z_{i}\) to \(\sum_{i \neq j} e_{ij}\). This gives \(OT_{n}\) the structure of a graded module over \(R_{n}\), and it is in fact a free module. If we define \(M_{n}:= OT_{n} \otimes_{R_{n}}\mathbb{C}\) to be the ring obtained by setting \(\phi_{n}(z_{i})\) equal to zero for all \(i\), then there exists an isomorphism of graded \(R_{n}\)-modules \[ OT_{n} \cong R_{n} \otimes_{\mathbb{C}} M_{n}. \] The above isomorphism is not canonical, and is not compatible with the ring structure on the two sides. However, it is compatible with the action of \(S_{n}\) on both sides.
Now consider the configuration space \(\text{Conf}(n, \mathbb{R}^{3})\) of \(n\) labeled points in \(\mathbb{R}^{3}\), which admits an action of \(S_{n}\) given by permutations of labels. Let \[ C_{n} := H^{*}( \text{Conf} (n, \mathbb{R}^{3}); \mathbb{C}), \] which is a graded representation of \(S_{n}\). Let \(G = SU(2)\), and consider the configuration space \(\text{Conf}(n,G) / G\) of \(n\) labeled points in \(G\) up to simultaneous translation by left multiplication. This space admits an action of \(S_{n}\) by permuting the labels. Finally, let us define \[ D_{n} := H^{*}(\text{Conf}(n,G)/G ; \mathbb{C}), \]
and \(W_{n} := R_{n} / \langle z_{i}z_{j} \rangle\). Firstly, we can observe the following.
Theorem. There exists an isomorphism \[ C_{n} \cong D_{n} \otimes_{\mathbb{C}} W_{n} \] of graded representations of \(S_{n}\).
The above result allows to recover \(D_{n}\) from \(C_{n}\), and this is important due to the fact that there are explicit formulaes for \(C_{n}\). The main conjecture can be formulated as follows.
Conjecture. There exists an isomorphism of graded \(S_{n}\) representations \(M_{n} \cong D_{n}\).
The authors claim that they verified this conjecture on a computer up to \(n=10\). The recursive formulae (see Theorem 3.2 therein) which allows to verify the above statement is obtained using the theory of hypertoric varieties – from my point of view, this as a really interesting approach. Notice also that the authors generalized the conjecture to the case of graphical arrangements given by simple connected graphs (see Conjecture 2.16 therein).

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
55R80 Discriminantal varieties and configuration spaces in algebraic topology
20C30 Representations of finite symmetric groups
55N33 Intersection homology and cohomology in algebraic topology
14N20 Configurations and arrangements of linear subspaces
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