⧼exchistory⧽
4 exercise(s) shown, 0 hidden
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Write down a complete, simplified proof for the factorization
[[math]]
\xymatrix{C(S_{NM}^+)\ar[rr]^{\pi_L}\ar[rd]&&M_{NM}(\mathbb C)\\&C(S_M^+\wr_*G_H)\ar[ur]&}
[[/math]]
found above, for [math]L=H\otimes_QK[/math], in the scalar matrix case.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that the number of partial permutations is given by
[[math]]
|\widetilde{S}_N|=\sum_{k=0}^Nk!\binom{N}{k}^2
[[/math]]
that is, [math]1,2,7,34,209,\ldots\,[/math], and that we have the estimate
[[math]]
|\widetilde{S}_N|\simeq N!\sqrt{\frac{\exp(4\sqrt{N}-1)}{4\pi\sqrt{N}}}
[[/math]]
in the [math]N\to\infty[/math] limit.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that we have an isomorphism
[[math]]
C(\widetilde{S}_2^+)\simeq\left\{(x,y)\in C^*(D_\infty)\oplus C^*(D_\infty)\Big|\varepsilon(x)=\varepsilon(y)\right\}
[[/math]]
where [math]\varepsilon:C^*(D_\infty)\to\mathbb C1[/math] the usual counit map.
BBot
Apr 22'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Develop a theory of partial Hadamard matrices with noncommutative entries, and of the associated quantum permutation semigroups.