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Atrium > Mathematics, Physics & Philosophy: "Fundamental weight systems are quantum states"

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2021
    • (edited May 6th 2021)

    With David Corfield and Hisham Sati we are finalizing an article:

    \,

    \,

    The title refers to a proof presented, that all fundamental gl(n)-weight systems on horizontal chord diagrams are states with respect to the star-involution of reversion of strands.

    But the bulk of the article rephrases this theorem and its proof as a special case of a more general statement in geometric group theory: characterizing the (non/semi-)positive definite phases of the Cayley distance kernel on the symmetric group.

    Finally, a last section recalls from Sati, Schreiber 2019c the original motivation and interpretation of this result: Under Hypothesis H it proves, from first principles, that a pair of coincident M5-branes (transversal on a pp-wave background, as in the BMN matrix model) indeed do form a (bound) state. This is of course generally expected, but there did not use to be a theory to derive this from first principles.

    \,

    Comments are welcome. If you do take a look, please grab the latest version of the file from behind the above link.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    This just made it into Dror Bar-Natan’s Pensieve:

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021

    Example 3.11

    • should be λ rows(λ)\lambda_{rows(\lambda)} at end of first bracket
    • in 1), should be “labels to fill its first column”
    • in 2) should be “any labelling of the first column”
    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021
    • (edited May 17th 2021)

    respectivey

    We close with briefly explaining

    ’by’ sounds better than ’with’

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    Thanks, fixed now locally. I’ll upload after adding a pointer to Jucys’ theorem which we have re-derived…

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2022
    • (edited Oct 18th 2022)

    A few days back, Carlo Collari has posted a followup:

    claiming that the statement (that 𝔤𝔩 2\mathfrak{gl}_2 weight systems are quantum states) generalizes from the fundamental representation to all “irreducible Young diagram representations”.

    The idea of the proof is to observe:

    1. that the weight systems for these more general representations are obtained by pre-composing the fundamental weight system (which we proved is a state) by the corresponding Young symmetriser (p. 9);

    2. that the Young symmetrisers are star-selfadjoint and square to a multiple of themselves (Remark 16 and last lines on p. 11);

    3. that precomposition with such operators preserves states (Lemma 25).

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 16th 2022

    Don’t know if it’s a useful generalization, but our kernel is viewed in the context of a family on p.4 of

    • Iskander Azangulov, Viacheslav Borovitskiy, Andrei Smolensky, On power sum kernels on symmetric groups (arXiv:2211.05650)
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2022

    Thanks for highlighting. I have recorded the reference here.

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