Fourier-Deligne transform and representations of the symmetric group

TL;DR

This paper computes the Fourier-Deligne transform of certain local systems related to symmetric group representations and geometric configurations, revealing connections to secant varieties and Kronecker coefficients.

Contribution

It provides an explicit calculation of the Fourier-Deligne transform for local systems associated with symmetric group representations, linking algebraic and geometric structures.

Findings

Explicit Fourier-Deligne transform formulas for local systems on cones over configuration spaces.

Connection between Young diagram manipulations and secant varieties in projective space.

Application to rederiving results on Kronecker coefficients.

Abstract

We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal L}_{\Lambda}$ on the cone over $\Conf_n({\P}^1)$ associated to a representation $\Lambda$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $\Lambda$ is at least $\frac{|\Lambda|-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal R}_{\lambda}$ on the cone over the $k$-th secant variety of the rational normal curve in ${\P}^n$, where ${\mathcal R}_{\lambda}$ corresponds to the representation $\lambda$ of $S_k$, the Young diagram of which is obtained from the Young diagram of $\Lambda$ by deleting its first row. We also prove an analogous statement for $S_n$-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.