Representation Stability Computations

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

Partition $\lambda$, a non-increasing list of positive integers with sum $\leq 23$:
Maximum degree $N$, an integer in the range $[1,50]$:

Show Young diagram

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. A larger source of prior computations is due to Bergström, who computed $H^i(\mathcal{M}_{0,n+1})$ as a sum of irreducible $S_{n+1}$ representations for $n \leq 22$ based on formulas due to Getzler. See his GitHub for data and related paper for discussion. This is relevant because there is an $S_{n}$-equivariant homotopy equivalence $$\text{PConf}_n(\mathbb{C}) \cong S^1 \times \mathcal{M}_{0, n+1}.$$ Along with a sharp bound on stability proved by Hersh-Reiner, Bergström's data can be used to determine the stable decomposition of $H^i(\text{PConf}(\mathbb{C}); \mathbb{C})$ for $0 \leq i \leq 6$.

Input

Degree of cohomology $i$:

Output

Stable cohomology of $H^i(\text{PConf}_n(\mathbb{C}); \mathbb{C})$:

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