D-functions and immanants of unitary matrices and submatrices

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 ${\sum }_t{D}_{tt}^{(\lambda )}(\Omega )$ 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.

13 pages. Updated to published version. Comments welcome

References in corpus (6)

• Sampling of partially distinguishable bosons and the relation to the multidimensional permanent
• Generalized multi-photon quantum interference
• Partial indistinguishability theory for multi-photon experiments in multiport devices
• Dual pairing of symmetry groups and dynamical groups in physics
• Coincidence Landscapes for Three-Channel Linear Optical Networks
• Algorithms for SU(n) boson realizations and D-functions

Cited by in corpus (5)

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• Discriminating distinguishability
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• Coincidence landscapes for polarized bosons
• Sum rules in multiphoton coincidence rates