Stable Multiplicities of Irreducible Families
In recent work, I implemented an algorithm for computing the representation stable cohomology of the ordered configuration space of the plane, denoted \(\text{PConf}(\mathbb{C})\). Using this algorithm we computed a table of new stable multiplicities. The raw data is available as a csv file and the code is available on GitHub. Below we include an interactive lookup table for ease of access.
Input
Output
Formal power series in \(z\) with coefficients \(\{(-1)^id_i(\lambda)z^i\}_{i=0}^N\) where \(d_i(\lambda)\) is the limiting multiplicity of \(V(\lambda)\) in \(H^i(\text{PConf}(\mathbb{C});\mathbb{C})\):
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Stable Cohomology of Configuration Space
Informed by the table of stable multiplicities \(d_i(\lambda)\), we proved the following theorem which gives a bound on when \(d_i(\lambda)\) is non-zero. \[\textbf{Theorem A. } \text{Let $\lambda$ be a partition of $n$. Then for $0 \leq i < n/2$, $d_i(\lambda) = 0$.}\] Theorem A combined with our computational results of \(d_i(\lambda)\) describes the stable cohomology \(H^i(\text{PConf}(\mathbb{C}; \mathbb{C}))\) for \(0 \leq i \leq 11\) as a sum of irreducible families \(V(\lambda)\). In addition to the coefficients \(d_i(\lambda)\), we provide a csv file with the data of these stable decompositions, and below we include a lookup table. The case of \(i = 4\) was previously computed by Farb.
Input
Output
Stable cohomology of \(H^i(\text{PConf}_n(\mathbb{C}; \mathbb{C}))\):
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