⧼exchistory⧽
8 exercise(s) shown, 0 hidden
SORT:
BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out an alternative proof for the main result regarding the truncated characters of the hyperoctahedral group [math]H_N[/math], namely

[[math]] \chi_t\sim e^{-t}\sum_{k=-\infty}^\infty\delta_k\sum_{p=0}^\infty \frac{(t/2)^{|k|+2p}}{(|k|+p)!p!} [[/math]]

with [math]N\to\infty[/math], by working our first an explicit formula for the polynomials integrals over [math]H_N[/math], and then using that for computing the laws of truncated characters.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out all the details for the moment formula for the Bessel laws,

[[math]] M_k=|P^s(k)| [[/math]]

where [math]P^s(k)[/math] are the partitions satisfying the formula

[[math]] \#\circ=\#\bullet(s) [[/math]]

as a weighted sum, in each block.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out all the details for the truncated character formula for [math]H_N^s[/math],

[[math]] \chi_t\sim b^s_t [[/math]]

where [math]b_t^s=p_{t\varepsilon_s}[/math], with [math]\varepsilon_s[/math] being the uniform measure on the [math]s[/math]-th roots of unity.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Show that the passage from [math]H_N^s[/math] to [math]H_N^{sd}[/math] does not change the asymptotic laws of the truncated characters.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Compute the asymptotic laws of characters and coordinates for the bistochastic groups [math]B_N[/math] and [math]C_N[/math], as well as for the symplectic group [math]Sp_N\subset U_N[/math].

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Compute the character laws for the groups [math]O_1[/math], [math]SO_1[/math], then for the groups [math]U_1[/math], [math]SU_1[/math], and then for the groups [math]O_2[/math], [math]SO_2[/math].

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Work out all the combinatorics and calculus details in relation with the Wigner and Marchenko-Pastur laws, and their moments, the Catalan numbers.

BBot
Apr 20'25
[math] \newcommand{\mathds}{\mathbb}[/math]

Look up the Wigner and Marchenko-Pastur laws, given by

[[math]] \gamma_1=\frac{1}{2\pi}\sqrt{4-x^2}dx\quad,\quad \pi_1=\frac{1}{2\pi}\sqrt{4x^{-1}-1}\,dx [[/math]]

and write down a brief account of what you found, and understood.