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    • CommentRowNumber101.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 6th 2021

    This MO question looks useful, especially the motivation. σk #Cycles(σ)\sigma \mapsto k^{#Cycles(\sigma)} is the character for the representation of S NS_N on the NN-fold product of a kk-dimensional vector space.

    So the quantity σ:S Nχ(σ)k #Cycles(σ)\sum_{\sigma: S_N} \chi(\sigma) k^{#Cycles(\sigma)} is counting the multiplicity of irreducible χ\chi in this character.

    The final comments are very close to what we want.

    • CommentRowNumber102.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 6th 2021

    A follow-up question.

    • CommentRowNumber103.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2021
    • (edited May 6th 2021)

    Interesting. I need to mull over this. But I suspect you’ll have it figured out before I am done mulling.

    \,

    Meanwhile, I was thinking about what the other eigenvalues would do for us:

    Now that we have dealt with the smallest eigenvalue, maybe next there is something of interest in the largest eigenvalue.

    It seems to be that the largest eigenvalue of the Cayley distance kernel should bound the largest eigenvalue of the corresponding state on chord diagrams, regarded as a density matrix.

    (Namely the latter is the maximum of ψ|ρ|ψ\langle \psi \vert \rho \vert \psi \rangle over unit norm elements ψ\psi in a Hilbert space on which horizontal chord diagrams are operators, which is equivalently the maximum of Tr(PP *ρ)Tr( P P^\ast \rho) over projectors P 2=P𝒜 pbP^2 = P \,\in\, \mathcal{A}^{pb}, which equals the quadratic value of the Cayley distance kernel on perm(P)perm(P) .)

    But the logarithm of the inverse of the largest eigenvalue of a density matrix is its min-entropy, a quantity of concrete interest. (Came to think of this from reading BPSW 18, Sec. 2.3.)

    • CommentRowNumber104.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    Actually, under the usual GNS construction (here), the largest eigenvalue of the Cayley distance kernel not just bounds but is equal to the the largest eigenvalue of the density matrix of the corresponding quantum state, hence the log of its inverse equals the min-entropy.

    • CommentRowNumber105.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    Ah, of course much more is true:

    Under the GNS construction, the eigenvalues EV λEV_\lambda of the Cayley distance kernel are exactly the weights of the density matrix of our corresponding state on chord diagrams. Hence they express the von Neumann entropy of the corresponding quantum state as

    S=λEV λln(EV λ). S \;=\; - \underset{ \lambda }{\sum} EV_\lambda \cdot ln(EV_\lambda) \,.

    This is tautological once one thinks about it, but it is also made explicit in section II.B of

    So this tells us the “meaning” of the eigenvalues of the CD kernel as we translate back to quantum states on hor. chord diagrams.

    • CommentRowNumber106.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    (more precisely: it’s that sum but with extra multiplicities, coming from the dimension of the kernel of perm:𝒜 N pb(Sym(N))perm : \mathcal{A}^{pb}_N \to \mathbb{C}(Sym(N)))

    • CommentRowNumber107.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2021
    • (edited May 7th 2021)

    So it seems that the MO question is telling us that for four of the irreducible characters my hunch is right. Their multiplicity in the character σk #Cycles(σ)\sigma \mapsto k^{#Cycles(\sigma)} associated to the obvious representation of S NS_N on N k\otimes^N \mathbb{C}^k is of the form A(MN)A \binom{M}{N}, as we had for the trivial and sign characters and also for the character of the standard (N1)(N-1)-dimensional rep.

    The ’motivation’ part of the question is the key passage.

    • CommentRowNumber108.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2021

    Spare minute, so I did as in #98 for the character of the rep 3,3 and it gives: 5x 2(x1)(x+1) 2(x+2)5 x^2(x-1)(x+1)^2(x+2). Not quite my hunch, but again roots are small integers.

    • CommentRowNumber109.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    I see.

    Meanwhile I was wondering if we might be able to say something about the entropy, just knowing that the eigenvalues come from that character formula, not necessarily knowing the concrete polynomial.

    One gets a lot of interesting hits when searching for this combination of keywords (e.g. “Quantum Information and the Representation Theory of theSymmetric Group” pdf) but not sure yet what one can say.

    • CommentRowNumber110.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    By the way, we know the sum over all eigenvalues (with multiplicity), from Schur orthogonality:

    λPart(n)(χ (λ)(e)) 2EigVals[e βd C] λ =λPart(n)(χ (λ)(e)) 2(1χ (λ)(e)σSym(n)e β(|Cycles(σ)|n)χ (λ)(σ)) =σSym(n)e β(|Cycles(σ)|n)λPart(n)χ (λ)(e)χ (λ)(σ)=δ e,σn! =n! \begin{aligned} & \underset{ \lambda \in Part(n) }{\sum} \big( \chi^{(\lambda)}(e) \big)^2 \cdot EigVals[e^{- \beta \cdot d_C}]_\lambda \\ & \;=\; \underset{ \lambda \in Part(n) }{\sum} \big( \chi^{(\lambda)}(e) \big)^2 \cdot \left( \frac{ 1 }{ \chi^{(\lambda)}(e) } \underset{\sigma \in Sym(n)}{\sum} e^{ \beta \cdot (\left\vert Cycles(\sigma) \right\vert - n ) } \cdot \chi^{(\lambda)}(\sigma) \right) \\ & \;=\; \underset{ \sigma \in Sym(n) }{\sum} e^{ \beta \cdot (\left\vert Cycles(\sigma) \right\vert - n ) } \underset{ = \delta_{e,\sigma} n! }{ \underbrace{ \underset{ \lambda \in Part(n) }{\sum} \chi^{(\lambda)}(e) \cdot \chi^{(\lambda)}(\sigma) } } \\ & \;=\; n! \end{aligned}

    I came to think of this when trying to understand how many irreps the GNS constructed Hilbert space for the weight systems w (𝔤𝔩(n),n)w_{(\mathfrak{gl}(n), \mathbf{n})} has. Unless I am mixed up, it must have number of irreps with eigenvalue EV λEV_\lambda such that the resulting weighted sum of eigenvalues is unity.

    So it can’t have (χ (λ)(e)) 2(\chi^{(\lambda)}(e))^2 irreps at λ\lambda, since the above expression is not unity.

    But maybe it has χ (λ)(e)\chi^{(\lambda)}(e) irreps: Can we see any further simplification of

    λPart(n)χ (λ)(e)EigVals[e βd C] λ=λPart(n)σSym(n)e β(|Cycles(σ)|n)χ (λ)(σ) \underset{ \lambda \in Part(n) }{\sum} \chi^{(\lambda)}(e) \cdot EigVals[e^{- \beta \cdot d_C}]_\lambda \;=\; \underset{ \lambda \in Part(n) }{\sum} \underset{\sigma \in Sym(n)}{\sum} e^{ \beta \cdot (\left\vert Cycles(\sigma) \right\vert - n ) } \cdot \chi^{(\lambda)}(\sigma)

    ?

    • CommentRowNumber111.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2021

    Since for S 3S_3 eigenvalues are quadratic polynomials in e βe^{-\beta}, I can’t see that another linear combination than the first one you tried will give you a constant.

    • CommentRowNumber112.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    added to the list of examples a graphics (here) of lowest eigenvalue of Cayley distance kernel on Sym(6)Sym(6), which just appeared in

    diff, v45, current

    • CommentRowNumber113.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2021

    Another confirmation of the positive after n1n-1 conjecture.

    Do you know what software he uses?

    • CommentRowNumber114.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    That’s surely a Mathematica notebook.

    • CommentRowNumber115.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021
    • (edited May 7th 2021)

    Which means that if you can get hold of a Mathematica installation, you should be able to immediately load CayleyDistanceKernel.nb and play with the parameters.

    • CommentRowNumber116.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    re: #110

    Wait, this may come out as unity, after all:

    As we extract the density matrix corresponding to the Cayley distance kernel, we must use the other inner product which is the canonical Cartesian inner product on the n !n_!-dimensional vector space (Sym(n))\mathbb{C}(Sym(n)) with respect to its canonical basis.

    This means that for instance the unit-vector version of the homogeneous distribution eigenvector is 1n!(1) σSym(n)\tfrac{1}{\sqrt{n!}}\big( 1\big)_{\sigma \in Sym(n)}. and hence the corresponding entry in the density matrix is not EV[e βd C] (n)EV[e^{- \beta \cdot d_C}]_{(n)}, but 1n!EV[e βd C] (n) \tfrac{1}{n!} EV[e^{- \beta \cdot d_C}]_{(n)}.

    If the same scaling by 1/n!1/\sqrt{n!} holds for the other eigenvectors, too, then the sum n!n! in #110 becomes unity, after all.

    Need to check this, but not tonight. But that’s exciting, seems to suggest that Schur orthogonality on the geometric-group-theory side is normalization of density matrices on the quantum-states-on-horizontal-chord-diagrams side.

    • CommentRowNumber117.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2021

    Right, it has to be a matter of the scaling factor. As hinted at in #111, no other linear combination up to a scaling factor is going to give a constant. And you’ve provided the rationale for the factor.

    • CommentRowNumber118.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2021

    Here’s a relevant paper

    • Alexander Gnedin, Vadim Gorin, Sergei Kerov, Block characters of the symmetric groups, (arXiv:1108.5044)

    So σk #Cycles(σ)\sigma \mapsto k^{#Cycles(\sigma)} is what they call a block character (Prop. 2.1). Prop 2.4 gives its decomposition into irreducibles via Schur-Weyl duality.

    • CommentRowNumber119.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    Maybe to think about where we are headed:

    I was trying to get a handle on the von Neumann entropy of the fundamental weight systems w (𝔤𝔩(n),n)w_{(\mathfrak{gl}(n),\mathbf{n})}, regarded as quantum states, based on the intuition that these ought to be a convex combination of pure states weighted by (some rescaling of) the eigenvalues of the Cayley distance kernel.

    If that intuition is right, it would give meaning to the (your) effort of getting more insight into the eigenvalues.

    But I am still not sure how to realize that intuition. (Following the previous comments, this morning I wrote out (here) the dual basis to our eigenvectors, using Schur orthogonality – but not sure where this really leads.)

    I guess one way to say it is that we expect there to be hermitian projectors

    P λ,μ𝒜 N pb,P λ,μ *=P λ,μ,P λ,μP λ,μ=P λ,μ P_{\lambda, \mu} \;\in\; \mathcal{A}^{pb}_N \,,\;\;\;\;\;\; P_{\lambda, \mu}^\ast = P_{\lambda, \mu} \,,\;\;\;\;\; P_{\lambda, \mu} \cdot P_{\lambda,\mu} = P_{\lambda, \mu}

    in the algebra of horizontal chord diagrams (nn and NN now as in our article, opposite to as in the nnLab entry, λ\lambda ranging over partitions of NN and μ\mu denoting a multiplicity index that possibly ranges between 1 and (χ λ(e)) 2(\chi^{\lambda}(e))^2), and that

    w (𝔤𝔩(n),n)()=λ,μw (𝔤𝔩(n),n)(P λ,μ) w_{(\mathfrak{gl}(n), \mathbf{n})}(-) \;=\; \underset{\lambda, \mu}{\sum} w_{(\mathfrak{gl}(n), \mathbf{n})}(- \cdot P_{\lambda,\mu})

    with

    w (𝔤𝔩(n),n)(P)EigVals[e ln(n)d c] λ w_{(\mathfrak{gl}(n), \mathbf{n})}(P) \;\propto\; EigVals[e^{- \ln(n) d_c}]_{\lambda}

    But if that’s the case, it seems to be less obvious than yesterday I felt it ought to be.

    (NB, these formulas are meant to be impressionistic, am just brainstorming here.)

    • CommentRowNumber120.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021

    Sorry, I didn’t meant to distract from what you were saying in #118.

    So this formula Prop. 2.4 implies, again with Schur orthogonality, that the λ\lambdath eigenvector is proportional to the number of semistandard Young tableaux of shape λPart(n)\lambda \in Part(n) with entries N\leq N.

    That’s great, I’ll make a note of that in the entry now…

    Hm, this should in particular explain the zero-eigenspace. But does it? Maybe I should write it out first…

    • CommentRowNumber121.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021

    Probably they mean in 2.4 that kk is the largest entry that actually appears in the Young tableau?

    And this would nicely explain our zero-eigenvectors.

    • CommentRowNumber122.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2021

    So if you have a column in the Young diagram of length greater than NN, the factor is clearly 00.

    • CommentRowNumber123.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    That’s the only barrier, no? Otherwise just fill the first row with 1s, second row with 2s, etc. So it’s just a question of the length of the first (longest) column.

    • CommentRowNumber124.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021

    Oh, I was thinking the wrong way. Right. Okay, just a moment…

    • CommentRowNumber125.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    Re #121, I don’t think so. It’s that you can use numbers from 11 to kk.

    The Young tableau of one column corresponding to the sign character, can only be filled if the NN of Sym(N)Sym(N) is at most kk. Perhaps we’ll need a v.2 for the paper. This material seems rather to the point.

    • CommentRowNumber126.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021

    Yes, thanks, looks like we overlapped.

    Meanwhile I have typed out the conclusion, starting here and culminating in this new Proposition.

    • CommentRowNumber127.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    Now I have extracted the conclusion on positivity as a stand-alone corollary (here).

    Yes, this makes for a shortcut in a v2. I’ll work it into the file.

    I should say that I will be (rather: already should be) busy with working on a research-center proposal that we want to hand in by end of the month. This will absorb us for a fair bit in the next two weeks. Maybe we could revise and submit after that.

    • CommentRowNumber128.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021

    I have ended up updating the file already, after all. Now Prop. 3.15

    • CommentRowNumber129.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    Thanks for doing that. I’m going to be pretty busy too over the next few weeks.

    Out of interest, is it that the Schur-Weyl aspect of this is relevant to the larger picture? I mean the idea of Sym(N)Sym(N) acting on N( k)\otimes^N (\mathbb{C}^k) not just as a means to calculate positivity.

    • CommentRowNumber130.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    I was thinking about this, but not sure yet.

    But knowing that the eigenvalues are counting Young tableaux is fascinating. There is a big story of combinatorics of Young tableaux controlling 𝒩=2\mathcal{N} = 2 SYM, originating around arXiv:hep-th/0306211 (I’ll try to dig out references.)

    In any case, it seems to bring us closer to computing the entropy of our states. It would be fascinating if we could compute its scaling with NN. (For 5-branes there is a famous conjecture that the entropy should scale as N 3N^3.)

    Maybe it makes sense to just assume/guess for the moment that the finite probability distribution in question is that of the kernel eigenvalues with multiplicities, divided by n!n!, and see if with that assumption we can compute the entropy of this distribution, as a function of NN.

    Because, even if that guess will not match the entropy of our quantum states on chord diagrams, it still seems like the natural probability distribution on the geometric group theory side, and surely must be relevant for something.

    So assuming that, we find ourselves computing logarithms of numbers of semistandard Young tableaux. Given that Gnedin et al. talk about random Young tableaux, the answer might even be hidden in their article somewhere.

    • CommentRowNumber131.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2021

    By the way, with the new formula for the eigenvalues, we know the largest eigenvalue at β=ln(N)\beta = ln(N) now:

    It must be that of the homogeneous distribution, because its Young diagram λ=(n)\lambda = (n) imposes the least constraints on its colorings to an ssYT, hence gives the largest possible numerator, while the dimension of the corresponding irrep gives the smallest possible denominator.

    So then we also know the min-entropy.

    But I am not sure yet if the result is saying anything interesting.

    • CommentRowNumber132.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 8th 2021
    • (edited May 8th 2021)

    I’m probably only going to have time for brief bouts of thought for recovery of sanity while a pile of marking falls my way over the next fortnight. Perhaps then a concrete problem to think about would be good. So the quantity to calculate is as in #105, corrected by #106?

    Probably better than random skimming of the literature, though I did notice the intriguing

    • BPS states, conserved charges and centres of symmetric group algebras, (arXiv:1911.11649).

    But maybe the literature is rather large, as you suggest in #130.

    • CommentRowNumber133.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2021

    Yes, that’s what I meant in #130: While I am still not sure about the multiplicities in #106, maybe it makes sense to assume for the time being that the probability distribution must be that given by the eigenvalues with their (χ (λ)) 2(\chi^{(\lambda)})^2-multiplicity and normalized by 1/n!1/n!.

    Regarding the literature: Yes, it’s huge. But that’s a nice one you found there, had not seen that. So let’s keep this on the back-burner.

    • CommentRowNumber134.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2021
    • (edited May 9th 2021)

    [ removed ]

    • CommentRowNumber135.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 12th 2021
    • (edited May 12th 2021)

    Just a brief query on entropy. To give a concrete example, in the case of Sym(3)Sym(3) acting on 3( 3)\otimes^3(\mathbb{C}^3), the numbers in ssYT 3(3)ssYT_3(3) are

    (3):10;(2,1):8;(1,1,1):1(3): 10; (2, 1): 8; (1,1,1): 1.

    Is it the quantity in #110 that’s of main interest, eigenvalues with multiplicity, (χ (λ)(e)) 2EigVals[e βd C] λ \big( \chi^{(\lambda)}(e) \big)^2 \cdot EigVals[e^{- \beta \cdot d_C}]_\lambda?

    As a quick check

    λPart(n)(χ (λ)(e)) 2EigVals[e βd C] λ=λPart(n)(χ (λ)(e)) 2n!N nχ (λ)(e)|ssYT λn(N)|\underset{ \lambda \in Part(n) }{\sum} \big( \chi^{(\lambda)}(e) \big)^2 \cdot EigVals[e^{- \beta \cdot d_C}]_\lambda = \underset{ \lambda \in Part(n) }{\sum}\big(\chi^{(\lambda)}(e) \big)^2 \cdot \tfrac{n!}{N^n \cdot \chi^{(\lambda)}(e)} \left\vert ssYT_{\lambda \vdash n}(N)\right\vert.

    And this is

    λPart(n)χ (λ)(e)n!N n|ssYT λn(N)|=3!3 3(10+28+1)=3! \underset{ \lambda \in Part(n) }{\sum}\chi^{(\lambda)}(e) \cdot \tfrac{n!}{N^n} \left\vert ssYT_{\lambda \vdash n}(N)\right\vert = \frac{3!}{3^3}(10 + 2 \cdot 8 + 1) = 3!,

    as expected. So the probability distribution here is (1027,1627,127)\big(\frac{10}{27}, \frac{16}{27}, \frac{1}{27}\big).

    • CommentRowNumber136.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2021
    • (edited May 12th 2021)

    I still don’t know, but I suspect that each eigenvector should appear in the probability distribution with multiplicity the dimension of the corresponding irrep.

    Let me recall the reasoning:

    \,

    We are trying to write the linear form

    w (𝔤𝔩(n),n):𝒜 n pb w_{(\mathfrak{gl}(n), \mathbf{n})} \;\colon\; \mathcal{A}^{pb}_n \longrightarrow \mathbb{C}

    as a convex combination with maximally many summands (which then are the pure states, normalized by their probability in the mixture).

    But we know that this actually factors through the algebra homomorphism to the group algebra

    w (𝔤𝔩(N),N):𝒜 pbperm(Sym(n))[e][e ln(N)d C][], w_{(\mathfrak{gl}(N), \mathbf{N})} \;\colon\; \mathcal{A}^{pb} \overset{perm}{\longrightarrow} \mathbb{C}(Sym(n)) \overset{ [\mathrm{e}] \cdot [e^{- ln(N) \cdot d_C}] \cdot [-] }{\longrightarrow} \mathbb{C} \,,

    (where cdot on the right denotes matrix multiplication).

    This suggests that – as a warmup exercise – we regard the group algebra [Sym(n)]\mathbb{C}[Sym(n)] as our algebra of observables, and ask instead for maximal convex combinations of the linear form

    [Sym(n)][e][e ln(N)d C][]. \mathbb{C}[Sym(n)] \overset{ [\mathrm{e}] \cdot [e^{- ln(N) \cdot d_C}] \cdot [-] }{\longrightarrow} \mathbb{C} \,.

    Here we know that [Sym(n)]\mathbb{C}[Sym(n)], as a module over itself (as which it is the regular representation), decomposes as a direct sum of irreps S (λ)S^{(\lambda)} with multiplicity their dimension.

    Writing P i (λ):[Sym(n)][Sym(n)]P^{(\lambda)}_i \;\colon\; \mathbb{C}[Sym(n)] \to \mathbb{C}[Sym(n)] for the projector onto the iith copy of the λ\lambdath irrep S (λ)S^{(\lambda)}, I suppose that the above linear form becomes equal to the convex combination

    [e][e ln(N)d C][]=λ,i λEigVals[e ln(N)d C] λ([e][P i λ (λ)]). [\mathrm{e}] \cdot [e^{- ln(N) \cdot d_C}] \cdot [-] \;\; = \;\; \underset{\lambda, i_\lambda}{\sum} EigVals[e^{- ln(N) \cdot d_C}]_\lambda \cdot \left( [\mathrm{e}] \cdot [P^{(\lambda)}_{i_\lambda} -] \right) \,.

    So regarded as a mixed state on the star-algebra [Sym(n)]\mathbb{C}[Sym(n)], this corresponds to the probability distribution on the set of pairs (λ,i λ)(\lambda, i_\lambda) with probability of the λ\lambdath diagram being EigVals[e ln(N)d C] λEigVals[e^{- ln(N) \cdot d_C}]_\lambda.

    If this is right and with fingers crossed (or better: with some insight that I keep missing) this result might transfer to the actual state on the actual algebra 𝒜 pb\mathcal{A}^{pb} that we are interested in.

    • CommentRowNumber137.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2021
    • (edited May 12th 2021)

    Probably I should simply keep going with this thought:

    It should follow that

    𝒜 n pbλi λperm 1(P i λ λ[Sym(n)]) \mathcal {A}^{pb}_n \; \coloneqq \; \underset{ \lambda \atop i_\lambda }{\oplus} perm^{-1} \big( P^{\lambda}_{i_\lambda} \mathbb{C}[Sym(n)] \big)

    is a decomposition into irreducible 𝒜 n pb\mathcal{A}^{pb}_n-modules, and that, writing 𝒫 i λ λ\mathcal{P}^{\lambda}_{i_\lambda} for their linear projectors, we have that

    w 𝔤𝔩(n),n=λi λEigVals[e ln(N)d C] λw 𝔤𝔩(n),n𝒫 i λ λ w_{ \mathfrak{gl}(n), \mathbf{n} } = \underset{ \lambda \atop i_\lambda }{\sum} EigVals[e^{ - ln(N) d_C } ]_\lambda \cdot w_{ \mathfrak{gl}(n), \mathbf{n} } \circ \mathcal{P}^{\lambda}_{i_\lambda}

    is our state expressed as a sum of pure states weighted by their probability.

    Seems obvious now. But last time I was thinking in this direction we couldn’t see that this is consistent with the normalization of the eigenvalues. Ah, but then we weren’t using Gnedin et al’s formula yet.

    • CommentRowNumber138.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2021
    • (edited May 12th 2021)

    So for the above to be correct it would have to be true that

    n!N nλPart(n)#ssYT λ(N)=1 \frac{n!}{N^n} \underset{\lambda \in Part(n)} {\sum} \# ssYT_\lambda(N) \; = \; 1

    Hm

    • CommentRowNumber139.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2021

    Ah, no, now I see the issue:

    The linear forms

    w 𝔤𝔩(n),n𝒫 i λ λ w_{ \mathfrak{gl}(n), \mathbf{n} } \circ \mathcal{P}^{\lambda}_{i_\lambda}

    are positive but not necessarily normalized anymore. The inverse of their normalization factor needs to be factored into the eigenvalues to get the actual probabilities!

    But need to quit now. Maybe tomorrow.

    • CommentRowNumber140.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 12th 2021

    The quantity totalling 1 here is λPart(n)χ (λ)(e)N n|ssYT λn(N)|=1 \underset{ \lambda \in Part(n) }{\sum} \tfrac{\chi^{(\lambda)}(e)}{N^n} \left\vert ssYT_{\lambda \vdash n}(N)\right\vert = 1.

    • CommentRowNumber141.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2021

    So what is that normalization factor

    [e]P i λ λ[e]=?? [e] \cdot P^\lambda_{i_\lambda} \cdot [e] = ??

    (i.e. the top left entry in the matrix representation of the projector onto the i λi_\lambdath copy of the λ\lambdath irrep in the regular rep)?

    It must be the quotient of the summands in #140 and #138, then (dim of irrep over n!n!). But need to prove this.

    • CommentRowNumber142.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021

    Coming back to the positivity proof:

    We don’t actually need that Proposition from Gnedin et al. to conclude that the eigenvalues of the Cayley distance kernel count semistandard Young tableaux: This follows readily by our original proof strategy via the Frobenius character formula (all closely related, of course, but still) simply by inserting the values (x 1=1,,x N=1)(x_1 = 1, \cdots, x_N = 1) into the Schur polynomial.

    I have now made this the first of two alternative proofs (the second via Gnedin et al.) of that proposition.

    diff, v54, current

    • CommentRowNumber143.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021
    • (edited May 16th 2021)

    Found some spare time after all, and so:

    I have added (here) the (quicker) proof of the (stronger, in fact optimal) lower bound e β>n1e^\beta \gt n - 1 for positivity that follows by using the hook-content formula (instead of the Gershgorin circle theorem).

    This was kindly pointed out to us by Abdelmalek Abdesselam.

    diff, v55, current

    • CommentRowNumber144.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 16th 2021

    Nice! So this proves what you conjectured earlier in the thread? Can you remind me what the implication is for the quantum states business?

    • CommentRowNumber145.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021

    For the quantum state property of the fundamental weight systems this has no further implication, which is why previously we went ahead without it. It’s just a nice addition that concludes the characterization of the positivitity of the Cayley distance.

    The fundamental 𝔤𝔩(n)\mathfrak{gl}(n)-weight systems on chord diagrams correspond to the Cayley distance kernel at log-integral inverse temperature β=ln(n)\beta = ln(n). We already knew that at all these log-integral inverse temperatures the Cayley distance kernel is positive (semi-)definite, which means equivalently that all the fundamental 𝔤𝔩(n)\mathfrak{gl}(n)-weight systems are quanrtum states.

    But once one has translated the problem from weight systems to Cayley distance kernels this way, it is natural to keep going and fully analyze the kernels, even at temperatures that do not correspond to any weight systems (at least as far as we know). On that front we had previously only provided a loose lower bound for the positive phase at all sufficiently high inverse temperatures. The new argument provides the sharp lower bound.

    • CommentRowNumber146.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 16th 2021

    Oh, ok. But it seems to me suggestive that there is a kind of phase transition, from a “discrete spectrum” to a continuous one, so that deformations might be possible, In some sense. But maybe there really is nothing there between log-integral inverse temperature…

    • CommentRowNumber147.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021
    • (edited May 16th 2021)

    Yeah, one might ask whether the Cayley distance kernel at non-log-integral temperature could ever define a weight system, by descending the assignment

    𝒟 n pbpermSym(n)e βd c(e,) \mathcal{D}^{pb}_n \overset{ perm }{\longrightarrow} Sym(n) \overset{ e^{ - \beta \cdot d_c(e,-) } }{\longrightarrow} \mathbb{C}

    along the map 𝒟 n pb𝒜 n pb\mathcal{D}^{pb}_n \longrightarrow \mathcal{A}^{pb}_n from the set of horizontal chord diagrams to the algebra of horizontal chord diagrams.

    The issue here is that the algebra 𝒜 n pb\mathcal{A}^{pb}_n is not just the linear span of the monoid 𝒟 n pb\mathcal{D}^{pb}_n, but furthermore the quotient by the ideal generated by the 2T-relations and the 4T-relations.

    Now, these relations are secretly an incarnation of the Jacobi identity on Lie algebra objects. It is through this that 𝔤𝔩(n)\mathfrak{gl}(n)-weight systems are indeed weight systems.

    So if one is asking for weight systems corresponding to the Cayley distance kernel at non-log-integral inverse temperature, one is essentially asking for making sense of the Lie algebra 𝔤𝔩(n)\mathfrak{gl}(n) at non-integer values of nn.

    • CommentRowNumber148.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 16th 2021

    A related MO question mentioned in #106.

    • CommentRowNumber149.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021

    By the way, I have updated sec. 3.3 in the pdf here according to #143 . Please check it out.

    Namely I did find some time today, after all! Speaking of using precious time:

    The most promising next step to me still seems to be understanding the fundamental weight systems as mixed states given by a probability distribution on the set of Young diagrams.

    Of course, if anyone sees how to understand the Cayley kernel at non-integer e βe^\beta as a weight system, I’ll be interested, too. But at the moment that seems like far-fetched speculation – an impression not dispelled by that MO discussion, right?

    • CommentRowNumber150.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 16th 2021

    Well done for finding time. Will take a look in a marking break.

    Agreed for non-integer e βe^{\beta}.

    Do you mean to consider all distributions on the set of Young diagrams? That distribution I’m indicating in #140 after #135 seems central in the current context.

    • CommentRowNumber151.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021

    Yeah, I mean the distribution we have been talking about. I thought in #137 it finally clicked. With the factor in #141 this must give whatever formula it is that sums to unit.

    • CommentRowNumber152.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2021

    Following the announcement in another thread here I have added (here) statement and proof of the kind of formula that that you (David C.) have been conjecturing all along should exist. The right formula should be

    EigVals[e βd C] λ=e βN1irows(λ)1jλ i(e β+ji). EigVals[e^{- \beta \cdot d_C}]_\lambda \;=\; e^{- \beta \cdot N} \underset { { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } } {\prod} \big( e^{\beta}+ j - i \big) \,.

    diff, v57, current

    • CommentRowNumber153.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 16th 2021

    That’s very pleasing!

    • CommentRowNumber154.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2021
    • (edited May 17th 2021)

    Yes, this nicely concludes the analysis of the Cayley distance kernel. Case closed.

    (I have added it to the file as a Sec. 3.4.)

    • CommentRowNumber155.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021

    Just because I bumped into it, re #149 the Secret Blogging Seminar once took on GL tGL_t for non-integer tt, here.

    I got there via their post on Howe duality, which seems to keep cropping up.

    • CommentRowNumber156.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021

    They arrive at the formula you have in #152 (or close to it) in this comment.

    For a permutation ww, define c(w)c(w) to be the number of cycles in ww. Define an n!×n!n! \times n! matrix, with rows and columns indexed by S nS_n, whose (u,v)(u,v) entry is t c(u 1v)t^{c(u^{-1} v)}. Our goal is to compute the determinant of this matrix, and show that all the roots are integers between n-n and nn.

    • CommentRowNumber157.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2021
    • (edited May 17th 2021)

    Interesting, thanks for the pointer!

    What, though, is the statement that the “conjectured combinatorial proof” in that comment is a conjectured proof of? I have been scrolling upwards, but have trouble spotting this.

    On the general idea of using this to make sense of non-integral 𝔤𝔩(t)\mathfrak{gl}(t)-weight systems:

    The “only” way I know to get weight systems is from metric Lie algebra representations over Lie algebra objects internal to any tensor category.

    From that perspective, it’s not immediately clear if it would help to have a definition of Rep(GL(t))Rep(GL(t)) for non-integral tt, as they were discussing on the Secret Blogging Seminar according to the pointers above.

    Instead, what would seem to be needed is a modified definition of Mod\mathbb{C} Mod whose objects may have non-integral trace, and such that it is a symmetric monoidal categoty. Then metric Lie representation objects internal to that exotic symmetric monoidal category would give weight systems, and they would plausibly have a possible relation to the Cayley distance kernel and non-integral exponentiated inverse temperature.

    But that still feels like fishing in the dark (that may well just due to me, personally, being in the dark, of course).

    I am reminded that by Deligne’s theorem on tensor categories we know all sensible tensor categories: They are the representation categories of algebraic supergroups. That seems to mean that if we do want to connect to non-integral exponentiated temperature, we should look for algebraic supergroups whose modules have non-integral traces, in some sense. (That seems to be a different use of supergroups than they were discussing in the Secret Seminar, but who knows.)

    • CommentRowNumber158.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021

    I’m in the dark too. I guess Speyer is answering his own

    Are there any values of tt other than the nonnegative integers for which you get a nontrivial radical?

    This post refers back to one on Deligne looking for a Sym(t)Sym(t) for non-integer tt, here.

    There’s a section ’The Quotient by the Radical’.

    • CommentRowNumber159.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2021
    • (edited May 17th 2021)

    I find this hard to read. But then I am looking at this on the side while doing something else. Will try to have a closer look when I have more leisure.

    Meanwhile, the question D. Speyer was asking in that comment you pointed to: “Does anyone remember enough S nS_n representation theory to see why?” is exactly the question we have answered now: “Use the hook length formula with the hook content formulas a few times”. I’ll email him, maybe he is still interested.

    • CommentRowNumber160.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2021

    I’ll email him, maybe he is still interested.

    Sent now. I have put you in cc.

    • CommentRowNumber161.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    So let’s finally get serious about computing the probability distribution encoded by the Cayley distance kernel ,when regarded as a quantum state. I have started a new thread on this at Cayley mixture.

    • CommentRowNumber162.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2021
    • (edited May 19th 2021)

    So I have started a new section here – Properties – Cayley state on the group algebra – with an expanded version of the previous notes in the Sandbox (rather detailed, but still unpolished), following our discussion in the other thread “Cayley mixture”.

    Looks like we have the convex decomposition into pure states and the formula for the probability distribution on these. Now just to reduce that formula to something more recognizable.

    diff, v60, current

    • CommentRowNumber163.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 19th 2021

    Added the result of using (12):

    p λ,i=|ssYT λ(N)|N n. p_{\lambda, i} \;=\; \tfrac{\left\vert ssYT_{\lambda}(N)\right\vert}{N^n} \,.

    diff, v62, current

    • CommentRowNumber164.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2021

    Added the observation (here) that the pushforward of this probability distribution along

    sYTableaux nqYDiagrams n sYTableaux_n \overset{ q }{\longrightarrow} YDiagrams_n

    is the Schur-Weyl measure.

    diff, v63, current

    • CommentRowNumber165.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2021

    added explicit mentioning (here) of the Hartley entropy of the Cayley state as the log of the number of standard Young tableaux with nn boxes and N\leq N rows.

    (as per the discussion in the other thread, starting around here)

    diff, v65, current