D-functions and immanants of unitary matrices and submatrices

TL;DR

This paper generalizes Kostant's result on immanants of unitary matrices to their submatrices, linking them to sums of diagonal D-functions, with implications for multiphoton interferometry and group representations.

Contribution

It extends Kostant's theorem to submatrices of unitary matrices and explores potential extensions to non-principal submatrices and different su(m) representations.

Findings

Immanants of principal submatrices are sums of D-functions.

Evidence suggests extension to non-principal submatrices.

Discussion on extending results to different su(m) representations.

Abstract

Motivated by recent results in multiphoton interferometry, we expand a result of Kostant on immanants of an arbitrary $m\times m$ unitary matrix $T\in$ su$(m)$ to the submatrices of $T$. Specifically, we show that immanants of principal submatrices of a unitary matrix $T$ are a sum $\sum_{t} D^{(\lambda)}_{tt}(\Omega)$ of the diagonal $D$-functions of group element $\Omega$, with $t$ determined {by} the choice of submatrix, and the irrep $(\lambda)$ determined by the immanant under consideration. We also provide evidence that this result extends to some submatrices that are not principal diagonal, and we discuss how this result can be extended to cases where $T$ carries an su$(m)$ representation that is different from the defining representation.