# Mixed symmetries of S_n: immanants in the sampling of U(d) submatrices

## TL;DR

This paper investigates the statistical properties of immanants of submatrices from Haar-distributed unitary matrices, focusing on their mean and higher moments, with implications for understanding symmetries in random matrix sampling.

## Contribution

It provides new results on the moments of immanants of Haar-random submatrices, expanding the understanding of their distributional properties without relying on detailed proofs.

## Key findings

- Mean and higher moments of immanants are characterized.
- Results apply to Haar-distributed unitary matrix submatrices.
- Insights into symmetries of S_n in random matrix sampling.

## Abstract

We provide results on the mean and higher moments of immanants of submatrices of ensembles of Haar-distributed unitary matrices, mostly without proofs. This paper is based on a talk presented at ISQS29 in Prague in July 2025 by Trevor Welsh.