Representation Stability Computations

Computes the multiplicty of a family of irreducibles in the stable cohomology of complex configuration space. Borrowing notation from Farb, the function YoungToPoly takes a Young diagram $\lambda$ and nonnegative integer $maxDegree$ as input and returns $d_i(\lambda)$ for $0 \leq i \leq maxDegree$ as the coefficients of a formal power series in $z^{-1}$.

The Data folder includes the results.csv file of the coefficients $d_i(\lambda)$ for $0 \leq i \leq 50$ and all Young diagrams $\lambda$ with 23 or fewer boxes. It also includes the stable.csv file of the stable decomposition of $H^i(\text{PConf}_n(\mathbb{C});\mathbb{C})$ for $0 \leq i \leq 11$. For an interactive lookup table of this data visit my website. For implementation details and background, see my preprint.

Code Details

The code was prepared targeting .NET 8.0 in Visual Studio 2022. MainMethod/Main.cs is the execution point and contains a helper method for saving the computational output to results.csv. Note that the code has been optimized through memoization to compute YoungToPoly for all $|\lambda| \leq n$ for some $n$. In particular, once $\text{YoungToPoly}(\lambda)$ has been computed for some $\lambda$, it is fast to compute $\text{YoungToPoly}(\lambda')$ for any $|\lambda'| < |\lambda|$.

Below we briefly describe each project file.

Integer Methods

Defines the Partition class which stores a partition of $n$ as a non-decreasing list of positive integers whose sum is equal to $n$. We implement a simple hash function for memoization. Defines the BigRational class which handles rational number arithmetic and stores the numerator and denominator as BigIntegers to avoid overflow errors.

Polynomials

Defines the LaurentPolynomial, Polynomial, and CharacterPolynomial classes. Also defines SymmetricPolynomial and RationalPolynomial classes, although these two are not needed for current functionality. Note that character polynomials are stored as rational linear combinations in the binomial basis.

RepStability

Defines YoungToChar.PartToCharPoly which takes in a partition $\lambda$ and returns the character polynomial $\chi^\lambda$ of the family of irreducibles $V(\lambda)$ based on example 1.7.14 in Macdonald. Note that while computing $\chi^\lambda$ we determine the character table of $S_n$ for all $n \leq |\lambda|$ via the recursive Murnaghan-Nakayama rule.

This project also contains PolyStatistics.CharPolyToPowSeries which takes as input a characteristic polynomial $\chi$ and returns the convergent polynomial statistic $\lim_{n\to\infty}q^{-n}\sum_{f \in \text{Conf}_n(\mathbb{F}_q)} \chi(f)$ as a formal power series in $z = q^{-1}$ via equation 2.11 from Chen. We memoize wherever possible for performance.