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

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

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

$sYTableaux_n \overset{ q }{\longrightarrow} YDiagrams_n$is the Schur-Weyl measure.

]]>Added the result of using (12):

$p_{\lambda, i} \;=\; \tfrac{\left\vert ssYT_{\lambda}(N)\right\vert}{N^n} \,.$ ]]>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.

]]>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*.

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

Sent now. I have put you in cc.

]]>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 $S_n$ 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.

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

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

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

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

]]>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 $\mathfrak{gl}(t)$-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 $Rep(GL(t))$ for non-integral $t$, 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 $\mathbb{C} Mod$ 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.)

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

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

Just because I bumped into it, re #149 the Secret Blogging Seminar once took on $GL_t$ for non-integer $t$, here.

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

]]>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.)

]]>That’s very pleasing!

]]>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

$EigVals[e^{- \beta \cdot d_C}]_\lambda \;=\; e^{- \beta \cdot N} \underset { { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } } {\prod} \big( e^{\beta}+ j - i \big) \,.$ ]]>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.

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

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

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.

]]>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 $e^\beta$ 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?

]]>A related MO question mentioned in #106.

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

$\mathcal{D}^{pb}_n \overset{ perm }{\longrightarrow} Sym(n) \overset{ e^{ - \beta \cdot d_c(e,-) } }{\longrightarrow} \mathbb{C}$along the map $\mathcal{D}^{pb}_n \longrightarrow \mathcal{A}^{pb}_n$ from the set of horizontal chord diagrams to the algebra of horizontal chord diagrams.

The issue here is that the algebra $\mathcal{A}^{pb}_n$ is not just the linear span of the monoid $\mathcal{D}^{pb}_n$, 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 $\mathfrak{gl}(n)$-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 $\mathfrak{gl}(n)$ at non-integer values of $n$.

]]>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…

]]>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 $\mathfrak{gl}(n)$-weight systems on chord diagrams correspond to the Cayley distance kernel at log-integral inverse temperature $\beta = ln(n)$. 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 $\mathfrak{gl}(n)$-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.

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

]]>Found some spare time after all, and so:

I have added (here) the (quicker) proof of the (stronger, in fact optimal) lower bound $e^\beta \gt n - 1$ 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.

]]>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 $(x_1 = 1, \cdots, x_N = 1)$ into the Schur polynomial.

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

]]>