Immanants

Immanants generalize both permanents and determinants, and they appear naturally in multiphoton interference when exchange symmetry is not purely bosonic or purely fermionic. In GroupFunctions.jl, these quantities are recovered from sums of matrix elements computed by group_function.

Tichy and Mølmer call particles governed by generalized exchange symmetries beyond the fully symmetric bosonic and fully antisymmetric fermionic cases immanons.^[1]

Definition and special cases

For an integer partition $\lambda=(\lambda_1,\ldots,\lambda_k)$ with $\sum_i \lambda_i = n$, the immanant of an $n \times n$ matrix $M$ is

\[\mathrm{Imm}^{\lambda}(M)=\sum_{\pi \in S_n}\chi^\lambda(\pi)\prod_{i=1}^n M_{i,\pi(i)},\]

where $\chi^\lambda$ is the character of the irrep $\lambda$ of the symmetric group $S_n$ – a function on permutations, distinct from the U(d) character $\chi^\lambda(U)$ of the characters page.

Two important limits are

\[\mathrm{Per}(M)=\mathrm{Imm}^{(n)}(M), \qquad \mathrm{Det}(M)=\mathrm{Imm}^{(1,\ldots,1)}(M).\]

For $3 \times 3$ matrices and mixed symmetry $(2,1)$:

\[\mathrm{Imm}^{(2,1)}(M)=2M_{11}M_{22}M_{33}-M_{12}M_{23}M_{31}-M_{13}M_{21}M_{32}.\]

Connection with group functions

A theorem of Kostant (see theorem 3 in de Guise et al.^[2]) gives the immanant directly as a sum of group functions. With $U \in U(d)$ the unitary matrix and $\Gamma^{(\lambda)}(U)$ its matrix in the irrep $\lambda$,

\[\mathrm{Imm}^{(\lambda)}(U) = \sum_t \langle t| \Gamma^{(\lambda)}(U) |t\rangle,\]

where the sum runs over the zero-weight basis states $|t\rangle$ (states with zero zweight, see basis states) of the irrep $\lambda$, and $\langle t| \Gamma^{(\lambda)}(U) |t\rangle$ is a diagonal group function. The permanent ($\lambda$ a single row) and determinant (single column) are the extreme cases. For the mixed $(2,1)$ immanant of a $3\times3$ matrix the $(11)$ irrep has two zero-weight states, so the sum has two terms – the relation verified on the tutorial page.