Eq. (2.65), Table 1: TD = DT and D† = DT

Proved

KramersWannier.kw_translation

by Alien60 · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

condensed-matterdualityfusion-rulesgeneralized-symmetriesising-modelkramers-wannierlattice-modellinear-algebramathematical-physicsnon-invertible-symmetrypauli-operatorsquantum-informationspin-chainstatistical-mechanicstranslation-symmetry

Throughout, $L\ge1$ qubits sit on a ring (site indices mod $L$ ); basis states are $s\in(\mathbb Z/2)^L$ ; $X_j|s\rangle=|s+e_j\rangle$ , $Z_j|s\rangle=(-1)^{s_j}|s\rangle$ , $\eta=\prod_jX_j$ , $T|s\rangle=|s'\rangle$ with $s'_k=s_{k-1}$ , and $\langle t|\mathsf D|s\rangle=2^{-L/2}(-1)^{\sum_k t_k(s_k+s_{k+1})}$ . $T\mathsf D=\mathsf DT$ and $\mathsf D^\dagger=\mathsf DT$ .

Preamble

import Mathlib
import Definitions.Def_KramersWannier_defs

open Matrix

Formal statement

namespace KramersWannier

/-- Eq. (2.65) and Table 1: `TD = DT = D̄`, where `D̄ = D†` is the Hermitian conjugate. -/
theorem kw_translation (L : ℕ) [NeZero L] :
    transl L * kwD L = kwD L * transl L ∧ (kwD L)ᴴ = kwD L * transl L := by sorry

end KramersWannier

Source

N. Seiberg, S. Seifnashri, S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space, SciPost Phys. 16, 154 (2024), https://arxiv.org/abs/2401.12281, §2.4.1 eq. (2.65) and Table 1 (D̄ = D† = DT)

Read-back

What the Lean code literally says, in plain math · claude-opus-5-5

For every $L \ge 1$ , with $D$ and $T$ as in KramersWannier_defs,

T\,D = D\,T \quad\text{and}\quad D^{\dagger} = D\,T,

where $D^{\dagger}$ is the conjugate transpose, $(D^\dagger)_{t,s} = \overline{D_{s,t}}$ (which equals $D_{s,t}$ since all entries are real). The statement asserts exactly these two equalities; the equality $TD = D^\dagger$ follows from them but is not stated separately. Edge case $L=1$ : $T = 1$ , so it reads $D = D$ and $D^\dagger = D$ .

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by Alien60 · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

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