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Character tables, verified by the orthogonality theorem (#132)
Phase 1 of the MO correlation diagram roadmap. PointGroups already lists the irrep NAMES of 46 groups, which is all that is needed to label an orbital; deciding which irreps a set of orbitals spans needs the characters, which is what this adds -- for the thirteen groups the AH_n and ML_n scope calls for. A character table is a hand-entered grid of small integers, and a wrong sign in one cell gives a table that still looks like a character table while silently misassigning an irrep. So none of this was checked by eye: the great orthogonality theorem makes the data self-verifying, and tests/symmetry enforces the class counts against the group order, the squared dimensions against it, every row's norm, row orthogonality, column orthogonality, and that there are as many irreps as classes. 987 checks over thirteen groups. THE ONE THING ORTHOGONALITY CANNOT CATCH is two rows being swapped. A relabelling leaves every orthogonality relation intact -- the table stays perfectly self-consistent while two irreps wear each other's names. Confirmed rather than assumed: swapping Td's T1 and T2 passed the orthogonality checks cleanly. Closed by tying a label to something physical. The character of the 3-D Cartesian representation follows from the class NAMES alone -- 1+2cos for a proper rotation, -1+2cos for an improper one, 1 for a mirror, -3 for inversion -- so reducing it is an independent constraint on the labelling. It comes out T2 in Td, T1u in Oh, E'+A2'' in D3h, A1+B1+B2 in C2v, and so on for all thirteen, every one matching the standard tables. The swap is now caught. Every check was self-tested by introducing the error and watching it fail: a flipped sign, a row swap, a renamed irrep, a g/u mislabel. The irrep names are also checked against PointGroups for the same group, since the point of the labels is to be compared against the ones the codes themselves report in ORBSYM -- which the roadmap makes the oracle for the whole symmetry pipeline. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01EFMUNbw2UpRPmgsqviAowb
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‎data/client/config/CharacterTables‎

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# Character tables, for symmetry analysis of molecular orbitals.
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#
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# data/client/config/PointGroups already lists the IRREP NAMES of 46
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# groups, which is all that was needed to label an orbital. Deciding
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# which irreps a set of orbitals spans needs the characters themselves,
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# which is what this file adds.
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#
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# FORMAT. One block per group, blank-line separated:
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#
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# [GROUP] h=<order>
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# classes: <name> <name> ... the conjugacy classes, in order
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# counts: <n> <n> ... how many operations in each
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# <irrep>: <chi> <chi> ... one row per irreducible rep
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#
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# The irrep names must match PointGroups' spelling for the same group,
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# since that is what the codes' own ORBSYM labels are compared against.
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#
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# The "translations" line records what the Cartesian coordinates x, y, z
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# span in that group. It is there because orthogonality alone CANNOT
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# catch two rows being swapped -- a relabelling leaves every orthogonality
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# relation intact, so the table stays self-consistent while two irreps
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# have each other's names. The character of the 3-D Cartesian
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# representation is computable from the class names alone (1+2cos(theta)
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# for a proper rotation, -1+2cos(theta) for an improper one, 1 for a
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# mirror, -3 for inversion), so reducing it and comparing against this
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# line ties each label to something physical and pins the labelling too.
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#
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# THESE ARE VERIFIED MECHANICALLY, not by eye. The great orthogonality
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# theorem makes the data self-checking -- rows orthogonal, columns
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# orthogonal, sum of counts equal to h, sum of squared dimensions equal
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# to h, and every row's own norm equal to h. tests/symmetry enforces all
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# five, so a transcription slip in a hand-entered table cannot survive.
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# That is the only reason entering them by hand is acceptable.
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#
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# Only real characters appear here. The cyclic groups Cn with n > 2 have
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# complex characters for their E irreps; none of the groups below is one,
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# and a group that needs them must not be added without extending the
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# format rather than rounding them.
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[C1] h=1
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classes: E
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counts: 1
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A: 1
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translations: 3A
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[CS] h=2
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classes: E sh
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counts: 1 1
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A': 1 1
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A'': 1 -1
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translations: 2A' + A''
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[CI] h=2
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classes: E i
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counts: 1 1
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Ag: 1 1
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Au: 1 -1
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translations: 3Au
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[C2] h=2
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classes: E C2
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counts: 1 1
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A: 1 1
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B: 1 -1
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translations: A + 2B
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[C2V] h=4
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classes: E C2 sv_xz sv_yz
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counts: 1 1 1 1
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A1: 1 1 1 1
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A2: 1 1 -1 -1
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B1: 1 -1 1 -1
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B2: 1 -1 -1 1
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translations: A1 + B1 + B2
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[C3V] h=6
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classes: E C3 sv
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counts: 1 2 3
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A1: 1 1 1
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A2: 1 1 -1
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E: 2 -1 0
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translations: A1 + E
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[C4V] h=8
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classes: E C4 C2 sv sd
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counts: 1 2 1 2 2
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A1: 1 1 1 1 1
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A2: 1 1 1 -1 -1
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B1: 1 -1 1 1 -1
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B2: 1 -1 1 -1 1
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E: 2 0 -2 0 0
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translations: A1 + E
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[C2H] h=4
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classes: E C2 i sh
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counts: 1 1 1 1
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Ag: 1 1 1 1
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Bg: 1 -1 1 -1
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Au: 1 1 -1 -1
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Bu: 1 -1 -1 1
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translations: Au + 2Bu
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[D2H] h=8
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classes: E C2_z C2_y C2_x i s_xy s_xz s_yz
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counts: 1 1 1 1 1 1 1 1
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Ag: 1 1 1 1 1 1 1 1
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B1g: 1 1 -1 -1 1 1 -1 -1
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B2g: 1 -1 1 -1 1 -1 1 -1
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B3g: 1 -1 -1 1 1 -1 -1 1
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Au: 1 1 1 1 -1 -1 -1 -1
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B1u: 1 1 -1 -1 -1 -1 1 1
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B2u: 1 -1 1 -1 -1 1 -1 1
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B3u: 1 -1 -1 1 -1 1 1 -1
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translations: B1u + B2u + B3u
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[D3H] h=12
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classes: E C3 C2 sh S3 sv
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counts: 1 2 3 1 2 3
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A1': 1 1 1 1 1 1
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A2': 1 1 -1 1 1 -1
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E': 2 -1 0 2 -1 0
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A1'': 1 1 1 -1 -1 -1
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A2'': 1 1 -1 -1 -1 1
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E'': 2 -1 0 -2 1 0
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translations: E' + A2''
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[D4H] h=16
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classes: E C4 C2 C2' C2'' i S4 sh sv sd
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counts: 1 2 1 2 2 1 2 1 2 2
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A1g: 1 1 1 1 1 1 1 1 1 1
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A2g: 1 1 1 -1 -1 1 1 1 -1 -1
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B1g: 1 -1 1 1 -1 1 -1 1 1 -1
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B2g: 1 -1 1 -1 1 1 -1 1 -1 1
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Eg: 2 0 -2 0 0 2 0 -2 0 0
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A1u: 1 1 1 1 1 -1 -1 -1 -1 -1
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A2u: 1 1 1 -1 -1 -1 -1 -1 1 1
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B1u: 1 -1 1 1 -1 -1 1 -1 -1 1
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B2u: 1 -1 1 -1 1 -1 1 -1 1 -1
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Eu: 2 0 -2 0 0 -2 0 2 0 0
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translations: A2u + Eu
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[TD] h=24
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classes: E C3 C2 S4 sd
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counts: 1 8 3 6 6
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A1: 1 1 1 1 1
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A2: 1 1 1 -1 -1
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E: 2 -1 2 0 0
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T1: 3 0 -1 1 -1
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T2: 3 0 -1 -1 1
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translations: T2
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[OH] h=48
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classes: E C3 C2 C4 C2' i S4 S6 sh sd
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counts: 1 8 6 6 3 1 6 8 3 6
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A1g: 1 1 1 1 1 1 1 1 1 1
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A2g: 1 1 -1 -1 1 1 -1 1 1 -1
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Eg: 2 -1 0 0 2 2 0 -1 2 0
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T1g: 3 0 -1 1 -1 3 1 0 -1 -1
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T2g: 3 0 1 -1 -1 3 -1 0 -1 1
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A1u: 1 1 1 1 1 -1 -1 -1 -1 -1
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A2u: 1 1 -1 -1 1 -1 1 -1 -1 1
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Eu: 2 -1 0 0 2 -2 0 1 -2 0
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T1u: 3 0 -1 1 -1 -3 -1 0 1 1
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T2u: 3 0 1 -1 -1 -3 1 0 1 -1
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translations: T1u

‎tests/symmetry/README.md‎

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# tests/symmetry — are the character tables right?
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```
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tests/symmetry/run_tests.py check every table
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tests/symmetry/run_tests.py -v show each check
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```
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Exit 0 if every table is sound, 1 otherwise. No build, no display, no
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chemistry codes.
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## Why
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`data/client/config/CharacterTables` is a hand-entered grid of small
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integers — exactly the kind of data a transcription slip survives in. A
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wrong sign in one cell gives a table that still *looks* like a character
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table and silently misassigns one irrep, which would then propagate into
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every MO symmetry label built on it (#132).
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It does not have to be checked by eye. The great orthogonality theorem
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makes the data self-verifying.
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## What is checked
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| | |
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|---|---|
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| class counts sum to the group order | catches a missing or duplicated class |
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| squared dimensions sum to the group order | catches a wrong dimension |
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| every irrep's own norm equals the order | catches most single-cell errors |
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| distinct irreps are orthogonal | catches the rest |
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| distinct classes are orthogonal, normalised | independent of the row checks |
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| as many irreps as classes | catches a whole row left out |
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| **(x, y, z) reduces to the recorded irreps** | catches a row **swap** |
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## The one orthogonality cannot catch
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Swapping two whole rows leaves **every** orthogonality relation intact —
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it is a relabelling, so the table stays perfectly self-consistent while
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two irreps wear each other's names. That was confirmed, not assumed: the
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Td T₁/T₂ swap passed the orthogonality checks cleanly.
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The fix is to tie a label to something physical. The character of the
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3-D Cartesian representation follows from the class *names* alone —
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1+2cos θ for a proper rotation, −1+2cos θ for an improper one, 1 for a
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mirror, −3 for inversion — so reducing it and comparing against the
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`translations:` line is an independent constraint. In Td it must come
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out T₂, in Oh T₁ᵤ, in D3h E′+A₂″.
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Every check here was self-tested by introducing the error and watching
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it fail: a flipped sign, a row swap, a renamed irrep, a g/u mislabel.
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## Scope
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Thirteen groups: C1, Cs, Ci, C2, C2v, C3v, C4v, C2h, D2h, D3h, D4h, Td,
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Oh — enough for AH_n and ML_n, which is the scope #132 sets for the
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first MO diagram.
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Only **real** characters. The cyclic groups Cn with n > 2 have complex
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characters for their E irreps; none of the above is one, and a group
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that needs them must not be added without extending the format rather
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than rounding them.

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