Paper Citation Record · LEDGER

Crystallographic splitting theorem for band representations and fragile topological photonic crystals

As of 21 August 2026, this Paper Citation Record lists 21 of 21 outbound references and 0 inbound Pith citation observations for arXiv:1908.08541.

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pith.paper-citation-record.v1
1908.08541 v2

Coverage vector

measured 21 of 21 reference resolution

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Source: paper_references, paper_reference_links, observed 2026-08-14T11:44:46.522462Z

measured 21 of 21 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-21T06:32:19.484+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

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measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

21 of 21 outbound references displayed

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External citation measurements

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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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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

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No event found in the named queried sources as of 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+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 2026-08-21T06:32:19.484+00:00.

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