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With David Corfield and Hisham Sati we are finalizing an article:
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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.
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Comments are welcome. If you do take a look, please grab the latest version of the file from behind the above link.
This just made it into Dror Bar-Natan’s Pensieve:
Example 3.11
respectivey
We close with briefly explaining
’by’ sounds better than ’with’
Thanks, fixed now locally. I’ll upload after adding a pointer to Jucys’ theorem which we have re-derived…
A few days back, Carlo Collari has posted a followup:
claiming that the statement (that $\mathfrak{gl}_2$ weight systems are quantum states) generalizes from the fundamental representation to all “irreducible Young diagram representations”.
The idea of the proof is to observe:
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);
that the Young symmetrisers are star-selfadjoint and square to a multiple of themselves (Remark 16 and last lines on p. 11);
that precomposition with such operators preserves states (Lemma 25).
Don’t know if it’s a useful generalization, but our kernel is viewed in the context of a family on p.4 of
Thanks for highlighting. I have recorded the reference here.
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