Exercises

Apr 21'25

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Formulate and prove the classical De Finetti theorem, concerning sequences which are invariant under [math]S_\infty[/math], without using representation theory methods.

BBot

Apr 21'25

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Formulate and prove the free De Finetti theorem, concerning sequences which are invariant under [math](S_N^+)[/math], without using representation theory methods.

BBot

Apr 21'25

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Work out the full proof of the explicit formula for the Weingarten function for [math]S_N[/math], namely

[[math]] W_{kN}(\pi,\nu)=\sum_{\tau\leq\pi\wedge\nu}\mu(\tau,\pi)\mu(\tau,\nu)\frac{(N-|\tau|)!}{N!} [[/math]]

then of the main estimate for this function, namely

[[math]] W_{kN}(\pi,\nu)=N^{-|\pi\wedge\nu|}( \mu(\pi\wedge\nu,\pi)\mu(\pi\wedge\nu,\nu)+O(N^{-1})) [[/math]]

where [math]\mu[/math] is the Möbius function of [math]P(k)[/math].

BBot

Apr 21'25

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Work out estimates for the integrals of type

[[math]] \int_{S_N^+}v_{i_1j_1}v_{i_2j_2}v_{i_3j_3}v_{i_4j_4} [[/math]]

and then for the Weingarten function of [math]S_N^+[/math] at [math]k=4[/math].

BBot

Apr 21'25

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Prove directly that the function

[[math]] d(\pi,\nu)=\frac{|\pi|+|\nu|}{2}-|\pi\vee\nu| [[/math]]

is a distance on [math]P(k)[/math].