Basis states and Gelfand-Tsetlin patterns

This library computes matrix elements of U(d) irreducible representations. In quantum optics, for example, these elements describe how passive linear networks transform multiphoton states. Gelfand-Tsetlin (GT) patterns label the basis states between which these matrix elements are evaluated and sometimes give those states a physical interpretation.

Young diagrams commonly identify U(d) irreps. A GT pattern records both the irrep and a state within it. More precisely, its rows record the integer partitions obtained by restricting the irrep along the chain $U(d) \supset U(d-1) \supset \cdots \supset U(1)$. GT patterns thus provide canonical labels for basis vectors, making them useful in subsequent computations.

The following standard definitions establish the basic picture.

Definitions. Fix a positive integer $d$. The group $U(d)$ consists of the $d\times d$ unitary matrices. A representation is a homomorphism $\rho: U(d) \rightarrow GL(V)$ satisfying $\rho(u \cdot u')=\rho(u)\rho(u')$, where $V$ is a finite-dimensional complex vector space and $GL(V)$ is the group of invertible linear maps from $V$ to itself.

A representation is irreducible (an irrep) if the zero subspace and $V$ itself are its only subspaces invariant under every $\rho(U)$. A partition is a nonincreasing integer list $\lambda=[\lambda_1 \ge \cdots \ge \lambda_d]$, or equivalently, a Young diagram with $\lambda_j$ boxes in row $j$. Every finite-dimensional irrep of $U(d)$ corresponds to exactly one partition $\lambda$. We denote the representation by $\rho_\lambda$ and its representation space by $V_\lambda$; the partition becomes the top row of a GT pattern.

GT-pattern origin

To enumerate an orthonormal basis of $V_\lambda$ with particularly useful properties, we restrict $U(d)$ first to $U(d-1)$ and then recursively to $U(d-2)$, continuing until we reach $U(1)$. Because the irreducible representations of $U(1)$ are one-dimensional, their representation spaces define the desired basis vectors.

To restrict $\rho_\lambda$ to $U(d-1)$, evaluate it only on block-diagonal matrices of the form

\[u = \begin{pmatrix} u' & 0 \\ 0& 1\end{pmatrix}=u'\oplus (1), \]

where $u'$ is a unitary matrix of dimension $d-1$. The restricted representation sends $u'$ to $\rho(u' \oplus (1))$ and generally decomposes into irreducible representations of $U(d-1)$. This decomposition is multiplicity-free:

\[V_\lambda\big|_{U(d-1)} \cong \bigoplus_{\mu\,\prec\,\lambda} V_\mu.\]

The standard Gelfand–Tsetlin branching rule identifies the labels in this decomposition as exactly those $\mu=(\mu_1,\ldots,\mu_{d-1})$ that satisfy the betweenness condition^[1]:

\[\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots \geq \lambda_{d-1} \geq \mu_{d-1} \geq \lambda_d.\]

Each admissible $\mu$ therefore appears exactly once. For example, for the $U(3)$ highest weight $\lambda=(2,1,0)$, the admissible $U(2)$ labels $\mu=(\mu_1,\mu_2)$ satisfy

\[2 \geq \mu_1 \geq 1, \qquad 1 \geq \mu_2 \geq 0,\]

and hence are

\[(2,1),\quad (2,0),\quad (1,1),\quad (1,0).\]

Iterating this interlacing rule along the chain $U(d)\supset U(d-1)\supset\cdots\supset U(1)$ produces triangular Gelfand-Tsetlin patterns. Each pattern records the successive highest weights, and the admissible patterns index the corresponding Gelfand-Tsetlin basis. Because the branching is multiplicity-free and the final $U(1)$ representations are one-dimensional, these patterns label the basis vectors uniquely, up to normalization and phase.

SU(2) case

The simplest example comes from SU(2), a subgroup of U(2) that differs from it only by phase. A U(2) element may have any determinant on the complex unit circle; multiplying the element by a suitable unimodular scalar produces an SU(2) element with unit determinant. The Lie algebra su(2) is generated by

\[[J_z, J_\pm] = \pm J_\pm, \qquad [J_+, J_-] = 2 J_z.\]

The states $|j, m\rangle$, with $m = -j, -j+1, \ldots, j$, form a basis for the spin-$j$ irrep. They satisfy $J_z |j,m\rangle = m |j,m\rangle$, while $J_\pm$ act as ladder operators. These states describe spin and, for $j\in\mathbb{N}$, angular momentum.

The same algebraic structure also describes two bosonic modes with creation operators $a^\dagger_1$ and $a^\dagger_2$. Define

\[J_z = \tfrac{1}{2}(a^\dagger_1 a_1 - a^\dagger_2 a_2), \qquad J_+ = a^\dagger_1 a_2, \qquad J_- = a^\dagger_2 a_1.\]

These operators satisfy the commutation relations above. Here, $J_z$ measures the occupation difference between the modes, while $J_\pm$ move particles between them. A state with $n_1$ particles in mode 1 and $n_2$ in mode 2 has

\[j = \tfrac{1}{2}(n_1 + n_2), \qquad m = \tfrac{1}{2}(n_1 - n_2).\]

With these identifications, the Fock state $|n_1, n_2\rangle$ is mathematically equivalent to the spin state $|j, m\rangle$.

For SU(2), a GT pattern is an inverted triangle of the form

\[\left|\begin{array}{ccc} 2j&&0\\ &m+j& \end{array}\right\rangle.\]

The top row specifies the total spin $j$, while the single entry in the bottom row determines $m$. The next section develops the general structure. For now, consider a simple example:

julia> using GroupFunctions
julia> # Irrep [2,0] of SU(2): spin-1, dimension 3
       λ = [2, 0];
julia> basis = basis_states(λ);
julia> for b in basis
           print(b)
       end│ 2   0 ╲
│   0   ╱
│ 2   0 ╲
│   1   ╱
│ 2   0 ╲
│   2   ╱

SU(N): Multiple $J^{(l)}_z$ operators

For SU(N), there are $N-1$ commuting operators $J^{(l)}_z$, each with corresponding ladder operators $J^{(l)}_\pm$. Each triple $J^{(l)}_z, J^{(l)}_\pm$ satisfies the su(2) commutation relations. See Alex et al.^[2] for explicit definitions. Because these operators and states describe many different physical settings, we retain the abstract viewpoint here.

A GT pattern labels a basis vector by recording the integer partitions obtained through successive group restrictions. This vector is also a simultaneous eigenstate of all $J^{(l)}_z$, whose eigenvalues determine the p-weight defined below. In general, the p-weight fixes the state only when the irrep is totally symmetric or totally antisymmetric. The pattern forms a triangular array:

\[\left|\begin{array}{ccccccc} m_{1,N} & & m_{2,N} & & \cdots & & m_{N,N} \\ & m_{1,N-1} & & \cdots & & m_{N-1,N-1} & \\ & & \ddots & & ~ & & \\ & & m_{1,2} & & m_{2,2} & & \\ & & & m_{1,1} & & & \end{array}\right\rangle\]

The top row $(m_{1,N}, m_{2,N}, \ldots, m_{N,N})$ specifies the irrep and corresponds to a Young diagram, or integer partition. The remaining rows identify a state within that irrep by satisfying the betweenness condition:

\[m_{k, l+1} \geq m_{k, l} \geq m_{k+1, l+1},\]

julia> # Irrep [2,1,0] of SU(3): 8-dimensional
       λ = [2, 1, 0];
julia> basis = basis_states(λ);
julia> println("Dimension: ", length(basis))Dimension: 8
julia> # Show first few patterns
       for b in basis[1:3]
           display(b)
       end

The p-weight of a pattern is the sequence of differences between consecutive row sums: if $\sigma_l = \sum_k m_{k,l}$, then $w_l = \sigma_l - \sigma_{l-1}$. For certain irreps, the p-weight has a direct physical interpretation.

The symmetric irrep $\lambda = [N, 0, \ldots, 0]$, whose Young diagram has a single row, describes $N$ indistinguishable bosons in $d$ modes. Reversing the p-weight of this irrep gives the occupation numbers $(n_1, n_2, \ldots, n_d)$. Accordingly, the function occupation_number used below is defined as reverse ∘ pweight.

julia> # 2 photons in 3 modes
       λ = [2, 0, 0];
julia> basis = basis_states(λ);
julia> for b in basis
           println(b, " → occupation ", occupation_number(b)) # will output all Fock states of 3 modes with total number of particles = 2
       end│ 2   0   0 ╲
│   0   0    〉
│     0     ╱
 → occupation [0, 0, 2]
│ 2   0   0 ╲
│   1   0    〉
│     0     ╱
 → occupation [0, 1, 1]
│ 2   0   0 ╲
│   1   0    〉
│     1     ╱
 → occupation [1, 0, 1]
│ 2   0   0 ╲
│   2   0    〉
│     0     ╱
 → occupation [0, 2, 0]
│ 2   0   0 ╲
│   2   0    〉
│     1     ╱
 → occupation [1, 1, 0]
│ 2   0   0 ╲
│   2   0    〉
│     2     ╱
 → occupation [2, 0, 0]

Each Fock state corresponds to exactly one GT pattern.

The antisymmetric irrep $\lambda = [1, 1, \ldots, 1,0,0,\ldots,0]$, whose Young diagram has a single column, describes fermions. The Pauli exclusion principle restricts each occupation number to 0 or 1.

julia> # 2 fermions in 3 modes
       λ = [1, 1, 0];
julia> basis = basis_states(λ);
julia> println("Dimension: ", length(basis))  # Should be 3Dimension: 3
julia> for b in basis
           println(b, " → occupation ", occupation_number(b)) # will output all fermionic states -- in Fock space, permutations of [1,1,0]
       end│ 1   1   0 ╲
│   1   0    〉
│     0     ╱
 → occupation [0, 1, 1]
│ 1   1   0 ╲
│   1   0    〉
│     1     ╱
 → occupation [1, 0, 1]
│ 1   1   0 ╲
│   1   1    〉
│     1     ╱
 → occupation [1, 1, 0]

Mixed irreps such as $[2, 1]$ are neither fully symmetric nor fully antisymmetric. They arise physically when particles are partially distinguishable.

Consider three photons entering an interferometer. Two arrive simultaneously with identical spectral properties and can therefore interfere fully. Because the third arrives later, it cannot interfere with the earlier pair. The resulting quantum state has components in both the symmetric sector $[3,0,0]$ and the mixed-symmetry sector $[2,1,0]$. This decomposition appears in observable coincidence rates; see Section 3.2 of Amaro-Alcalá et al.^[3] for an explicit computation.

The simpler case of two photons with partial overlap likewise involves both $[2]$ (symmetric, contributing via the permanent) and $[1,1]$ (antisymmetric, contributing via the determinant); see Section 2.1 of the above paper.

julia> # Mixed irrep [2,1,0]: dimension 8
       λ = [2, 1, 0];
julia> basis = basis_states(λ);
julia> for b in basis
           println(occupation_number(b))
       end[0, 1, 2]
[1, 0, 2]
[1, 1, 1]
[0, 2, 1]
[1, 1, 1]
[2, 0, 1]
[1, 2, 0]
[2, 1, 0]

For mixed irreps, multiple GT patterns can share the same p-weight. This inner multiplicity shows that occupation numbers alone do not uniquely specify a state when particles have mixed exchange symmetry.

Once GT patterns have labeled the basis states, we can ask how their transition amplitudes depend on the unitary transformation $U$. For bosons, the answer involves permanents; for fermions, determinants; and for mixed symmetry, a generalization called the immanant.