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Duality Interface

This page specifies the required function signatures, mathematical properties, and error conditions for the two duality functions of the Yuzuha Protocol. These functions formalise how representations and CG tensors behave under arrow reversal, and underpin the Frobenius-Schur phase bookkeeping needed for physically consistent dualisation in tensor network contractions.

fs_phase

Signature

fs_phase(rep: RepLabel) → float

Semantics

The Frobenius-Schur (FS) phase of a representation \(r\) is defined by the behaviour of the invariant metric \(g^{(r)}\) under self-composition:

\[\left(g^{(r)}\right)^2 = \text{FS}(r)\cdot I\]

This is well-defined for any group whose irreducible representations are self-dual (i.e. every irrep \(r\) is isomorphic to its dual \(\bar{r}\)), since in that case \(g^{(r)}\) is an invertible scalar-multiple of the identity when squared. The value \(\text{FS}(r) \in \{+1, -1\}\) distinguishes real representations (\(\text{FS} = +1\), metric symmetric) from pseudo-real ones (\(\text{FS} = -1\), metric antisymmetric). The SU(2)-specific formula is given in Arrow Conventions — Frobenius-Schur Indicator.

Required Properties

  1. Values: The return value is always \(+1.0\) or \(-1.0\).
  2. Bosonic representations: If \(g^{(r)}\) is symmetric (\(g^\top = g\)), then \(\text{FS}(r) = +1\).
  3. Fermionic representations: If \(g^{(r)}\) is antisymmetric (\(g^\top = -g\)), then \(\text{FS}(r) = -1\).
  4. Identity representation: \(\text{FS}(\mathbf{1}) = +1\).
  5. Bond inversion: Flipping the arrows of a contracted edge pair with representation \(r\) multiplies the X-symbol by \(\text{FS}(r)\).

SU(2) Reference

For SU(2) with \(\text{rep} = \text{Spin}(n)\) (i.e. \(j = n/2\)):

\[\text{FS}(j) = (-1)^{2j} = \begin{cases} +1 & j \in \mathbb{Z} \\ -1 & j \in \mathbb{Z} + \tfrac{1}{2} \end{cases}\]
import yuzuha

yuzuha.fs_phase_for_spin(yuzuha.Spin(1))   # j=1/2 → -1.0
yuzuha.fs_phase_for_spin(yuzuha.Spin(2))   # j=1   → +1.0

Error Conditions

Condition Exception
rep is not a valid representation label ValueError

compute_conjugate

Signature

compute_conjugate(spec: Spec) → (float, Spec)

Semantics

The conjugate of a spec \(S\) is the spec \(\bar{S}\) obtained by reversing all edge directions while keeping the same representation labels. The function also returns a scalar conjugate phase \(\phi \in \{+1, -1\}\).

The phase encodes the discrepancy between the actual arrow configuration of \(S\) and the implementation's chosen canonical conjugate pattern — a fixed reference direction assignment used to define the normalised dual tensor. Different implementations may adopt different canonical patterns; some may always return \(\phi = +1\) by absorbing the sign into the basis convention. The SU(2)-specific canonical pattern and phase formula are documented in Arrow Conventions — Canonical Conjugate Pattern.

Required Properties

  1. Spec structure: The returned \(\bar{S}\) has the same representation labels as \(S\) on each edge, with all directions flipped. The internal representations (alphas) are preserved unchanged.
  2. Phase values: \(\phi \in \{+1.0, -1.0\}\).

  3. Double conjugation: Applying compute_conjugate twice recovers the original spec and a net phase of \(+1\):

    \[\phi(\bar{S}) \cdot \phi(S) = +1 \qquad \text{and} \qquad \overline{\bar{S}} = S\]

  4. Self-consistency: \(\phi\) must be consistent with fs_phase in the sense that any bond-inversion phase accumulated by the arrow reversal equals the product of \(\text{FS}(r_k)\) over the affected edges.

SU(2) Reference

import yuzuha

jhalf = yuzuha.Spin(1)   # j = 1/2
j1    = yuzuha.Spin(2)   # j = 1

spec = yuzuha.CGSpec.from_edges([
    yuzuha.Edge.incoming(jhalf),
    yuzuha.Edge.incoming(jhalf),
    yuzuha.Edge.outgoing(j1),
])

phase, conj_spec = yuzuha.compute_conjugate(spec)
# phase: +1.0 or -1.0
# conj_spec: CGSpec with all edge directions flipped

Error Conditions

Condition Exception
The conjugated edge configuration admits no valid coupling ValueError

Relationship Between the Two Functions

compute_conjugate is built on top of fs_phase: the accumulated phase it returns is the product of fs_phase(rep) over the deviating edges. Implementations may expose fs_phase as a standalone utility so that callers can compute bond-inversion phases for individual edges without constructing a full conjugated spec.

Conformance Test Summary

Test Checks
FS phase values fs_phase(rep) ∈ {+1.0, -1.0} for all valid representations
FS phase identity fs_phase(identity_rep) == +1.0
Double conjugation spec compute_conjugate(compute_conjugate(spec)[1])[1] == spec
Double conjugation phase compute_conjugate(spec)[0] * compute_conjugate(compute_conjugate(spec)[1])[0] == +1.0
Phase consistency Returned phase consistent with fs_phase over arrow-reversed edges