Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-14T11:44:46.522462Z
Paper Citation Record · LEDGER
Crystallographic splitting theorem for band representations and fragile topological photonic crystals
As of 16 August 2026, this Paper Citation Record lists 21 of 21 outbound references and 0 inbound Pith citation observations for arXiv:1908.08541.
A citation records a reference. It does not transfer a finding from one paper to another.
Coverage vector
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-16T06:30:59.297886+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
Reference resolution
21 of 21 outbound references displayed
- verified exact1
- verified fuzzy15
- unresolved5
- parse uncertain0
- malformed identifier0
- metadata mismatch0
External citation measurements
No source-named external measurement is stored.
Outbound references
Observation 744eeb78-55b4-4046-a6db-ce3a72a6c989 · outbound
This paper cites an unresolved cited work.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Unresolved cited work
Reference 1
- Resolution
- unresolved
- raw_fallback, observed 2026-08-14T11:44:46.883126Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 7f75ddef-69dc-4e8b-97d6-e271510b686d · outbound
This paper cites Since Pj is invariant under H [cf.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Since Pj is invariant under H [cf
Reference 2
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.917440Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 9ac977a2-cdfc-4e76-a66d-b12089544068 · outbound
This paper cites an unresolved cited work.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Unresolved cited work
Reference 3
- Resolution
- unresolved
- raw_fallback, observed 2026-08-14T11:44:46.900856Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 79407668-2bd9-4ced-81e3-d410a4177ba5 · outbound
This paper cites V B that the splitting P = ⊕N j=1Px j into bands of the projected position operator [cf.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals V B that the splitting P = ⊕N j=1Px j into bands of the projected position operator [cf
Reference 4
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.865727Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation fe54adab-e5e0-48c6-93db-f644414697cf · outbound
This paper cites The reduced real-space coordinates of the two pairs of orbitals are (0 , 0, 0), in an orthog- onal basis of Bravais lattice vectors.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals The reduced real-space coordinates of the two pairs of orbitals are (0 , 0, 0), in an orthog- onal basis of Bravais lattice vectors
Reference 5
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.845058Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 24b8fb8d-4af4-4189-bcc6-901702596ebf · outbound
This paper cites an unresolved cited work.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Unresolved cited work
Reference 6
- Resolution
- unresolved
- raw_fallback, observed 2026-08-14T11:44:46.826119Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation ad717717-be14-421f-a95b-bd8bf3ae9b4a · outbound
This paper cites A finite group G is solvable if there exists a series of normal groups, i.e., C1 =G0◁G 1◁G 2...◁G k =G (F1) for a k ≥ 1, such that Gj+1/Gj is abelian for all j = 1,...,k − 1.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals A finite group G is solvable if there exists a series of normal groups, i.e., C1 =G0◁G 1◁G 2...◁G k =G (F1) for a k ≥ 1, such that Gj+1/Gj is abelian for all j = 1,...,k − 1
Reference 7
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.806368Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 935a0810-90d1-439e-8c01-e7b30112bd4f · outbound
This paper cites an unresolved cited work.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Unresolved cited work
Reference 8
- Resolution
- unresolved
- raw_fallback, observed 2026-08-14T11:44:46.789692Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 172697d5-8716-4a3e-9ced-6873cac6e3ae · outbound
This paper cites The 11 point groups constructed in this way are S2, C2h, C3i, C4h, C6h, D2h, D3d, D4h, D6h, Th and Oh.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals The 11 point groups constructed in this way are S2, C2h, C3i, C4h, C6h, D2h, D3d, D4h, D6h, Th and Oh
Reference 9
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.773811Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 4a99fad9-9344-46cf-a8b4-c19737a60ad0 · outbound
This paper cites Review of the semi-direct product.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Review of the semi-direct product
Reference 10
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.757454Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 18e3f031-7613-40bc-b10d-21cc6365ca31 · outbound
This paper cites This two-fold rotational symmetry generates C′′ 2.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals This two-fold rotational symmetry generates C′′ 2
Reference 11
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.741066Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 8744502c-0b68-496b-bb90-19e0217a4706 · outbound
This paper cites F 1 to prove that all 27 noncubic double point groups [class (2)] are monomial.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals F 1 to prove that all 27 noncubic double point groups [class (2)] are monomial
Reference 12
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.725752Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 2c99cca6-6757-454a-956e-543d688260bf · outbound
This paper cites F 4 b that allP in class (2)A are monomial; then, according to the Lemma for monomial direct-product groups in App.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals F 4 b that allP in class (2)A are monomial; then, according to the Lemma for monomial direct-product groups in App
Reference 13
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.710436Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 237aefc1-0385-4bb9-9d94-3ce2ea68dad2 · outbound
This paper cites Our proof relies on Wigner’s seminal result, 104 namely that all irreps ofPT =P× ZT 2 are induced from irreps of the crystallographic point group P.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Our proof relies on Wigner’s seminal result, 104 namely that all irreps ofPT =P× ZT 2 are induced from irreps of the crystallographic point group P
Reference 14
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.693022Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 399383a8-f46b-472d-9610-f1969f457de2 · outbound
This paper cites ˜T and ˜O are standard examples of non-monomial groups.188 Example of non-monomial irrep of double cubic point group ˜T.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals ˜T and ˜O are standard examples of non-monomial groups.188 Example of non-monomial irrep of double cubic point group ˜T
Reference 15
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.676261Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation e717ca84-c5ba-4ed5-8583-832754fdebdf · outbound
This paper cites an unresolved cited work.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Unresolved cited work
Reference 16
- Resolution
- unresolved
- raw_fallback, observed 2026-08-14T11:44:46.658490Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation ed83186e-e3b0-4d53-8c1f-e3a305188e2a · outbound
This paper cites Of the three remaining double cubic point groups, two have the direct-product form: ˜Th = ˜T× Zi 2 and ˜Oh = ˜O× Zi.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Of the three remaining double cubic point groups, two have the direct-product form: ˜Th = ˜T× Zi 2 and ˜Oh = ˜O× Zi
Reference 17
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.643190Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 85b91ad0-2251-45fd-a234-988bf675c1c7 · outbound
This paper cites F 4.) Since ˜O and ˜T are non-monomial, it follows that ˜Oh and ˜Th must also be non-monomial, according to the Lemma for monomial direct-product groups in App.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals F 4.) Since ˜O and ˜T are non-monomial, it follows that ˜Oh and ˜Th must also be non-monomial, according to the Lemma for monomial direct-product groups in App
Reference 18
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.627314Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 35b29194-213f-4990-bdc6-3ce43cefb05f · outbound
This paper cites A 1 from the perspective of band theory.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals A 1 from the perspective of band theory
Reference 19
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.608636Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation 7366a28e-bdae-4949-a34a-ae26e89b98b5 · outbound
This paper cites For simplicity, let us consider a rank-N BR(G, ϖ,D ).
Crystallographic splitting theorem for band representations and fragile topological photonic crystals For simplicity, let us consider a rank-N BR(G, ϖ,D )
Reference 20
- Resolution
- verified fuzzy
- raw_fallback, observed 2026-08-14T11:44:46.590764Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Observation f1c69a6b-29ae-4645-a39d-e614052c5b7c · outbound
This paper cites Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry.
Crystallographic splitting theorem for band representations and fragile topological photonic crystals Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
Reference 21
- Resolution
- verified exact
- local_arxiv, observed 2026-08-14T11:44:46.571819Z
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.
Pith citing papers
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