update website link

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

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

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.)

]]>a stub, for the moment just to record the bare definition for density matrices

]]>brief `category:people`

-entry for hyperlinking references at *holographic tensor network*

brief `category:people`

-entry for hyperlinking references at *holographic tensor network*

brief `category:people`

-entry for hyperlinking references at *holographic tensor network*

There was a lot of discussion about entropy, including this kind, a few years ago, summed up at John Baez’s nLab page.

]]>brief `category:people`

-entry for hyperlinking references at *Renyi entropy*

accent in title

]]>a stub for the moment, to make links work

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]]>A follow-up question.

]]>This MO question looks useful, especially the motivation. $\sigma \mapsto k^{#Cycles(\sigma)}$ is the character for the representation of $S_N$ on the $N$-fold product of a $k$-dimensional vector space.

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

The final comments are very close to what we want.

]]>You find a character table, such as this one for $S_6$. Pick a row, such as 411. Use character values at conjugacy classes as coefficients of $x^{#Cycles}$, multiplying by the number in each conjugacy class, which can also helpfully be obtained, such as here.

]]>I guess it’s just a typo coming from a change of mind between writing $S^n$ or $\underset{i \in [n]}{\prod} S$.

]]>Thanks, interesting. How did you “try out” that example, though? Could you explain how you computed that eigenvalue polynomial for $(4,1,1)$?

]]>How are we to read $\prod_{i:[n]} (-)_i \colon S^n \to S$? Am I being slow?

I am also confused. An n-ary operation on a set $S$ is just a function $S^n \to S$.

]]>Come to think of it, the hunch wasn’t supposed to be precise, just some factors without their constants.

To try one out: in $S_6$ the rep corresponding to $(4,1,1)$ gives for $\sum_{\sigma} \chi^{(4,1,1)}(\sigma) x^{#Cycles(\sigma)}$: $120 x + 40 x^2 - 120 x^3 + 40 x^4 -30 x^3 -90 x^4 +30 x^5 + 10x^6 = 10x(12 + 4x -15x^2 - 5x^3 + 3x^4 +x^5) = 10x( x +3)(x-2)(x+2)(x-1)(x+1)$.

Looks good with a factor. Perhaps it’s a Littlewood–Richardson rule thing.

]]>starting something, but nothing much here yet

]]>Let’s consider my hunch from above:

Just as the eigenvalues for $1$ and $sign$ are $\underoverset {k = 0} {n - 1} {\prod}\big(e^{\beta} + k \big)$ and $\underoverset {k = 0} {n - 1} {\prod}\big(e^{\beta} - k \big)$, the others are $\underoverset {k = -s} {n -s -1} {\prod}\big(e^{\beta} + k \big)$.

I guess in Lemma 4.12, we might as well use $1$ rather than $x$, and so have $n! s_{\lambda}(1, 1, \ldots, 1, 0, \ldots 0) = \sum_{\sigma} \chi^{(\lambda)}(\sigma) N^{#Cycles(\sigma)}$.

So we know that $n! s_{(n)}(1, 1, \ldots, 1, 0, \ldots 0) = \sum_{\sigma} N^{#Cycles(\sigma)} = \underoverset {k = 0} {n - 1} {\prod}\big(N + k \big)$ and $n! s_{1^n}(1, 1, \ldots, 1, 0, \ldots 0) = \sum_{\sigma} sgn(\sigma) N^{#Cycles(\sigma)} = \underoverset {k = 0} {n - 1} {\prod}\big(N - k \big)$.

There’s plenty of structure on the Schur polynomials, when considering them as decategorified Schur functors.

There must be things to consider about how to combine reps, e.g., a partition of $m$ and a partition of $n$ combine to a partition of $m + n$.

If my hunch is right then we’d be interested in how $\underoverset {k = -s} {n -s -1} {\prod}\big(N + k \big) = \frac{1}{N} \underoverset {k = -s} {0} {\prod}\big(N + k \big) \cdot \underoverset{k = 0}{n -s -1}{\prod}\big(N + k \big) = \frac{1}{N} s_{1^{(s+1)}}(1, 1, \ldots, 1, 0, \ldots 0) \cdot s_{(n-s-1)}(1, 1, \ldots, 1, 0, \ldots 0)\cdot (s+1)!\cdot (n-s-1)!$.

So for some $\lambda \in S_n$, $n! \cdot s_{\lambda}(1, 1, \ldots, 1, 0, \ldots 0)= \frac{1}{N} s_{1^{(s+1)}}(1, 1, \ldots, 1, 0, \ldots 0) \cdot s_{(n-s-1)}(1, 1, \ldots, 1, 0, \ldots 0)\cdot (s+1)!\cdot (n-s-1)!$.

]]>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.

]]>starting a new list of $n$Lab entries, to be used as floating context menu for all entries related to “quantum systems”

Maybe this needs to be broken up further. Currently the list has these “subsections”:

quantum probability (states and observables)

Here the section “quantum probability” has its own contents-page *states and observables – content* and so maybe the other subsections should eventually get their own page, too

Thanks, true. And similarly the list of examples at *polygon* should not say “square” but, maybe, quadrilateral or 4-gon or whatever.