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3 typed "really," meant "recently"

Alexei Oblomkov really recently told me about the beautiful theorem of Kerov and Vershik, which says that "almost all Young diagrams look the same." More precisely: take a random irreducible representation of S_n (in Plancherel measure, which assigns a probability of (dim chi)^2 / (n!) to an irrep chi) and draw its Young diagram, normalized to have fixed area. Rotate the diagram 45 degrees and place the vertex at the origin of the Cartesian plane, so that it lies above the graph of y = |x|. Then there is a fixed curve, with equation

$$y = \left\{\begin{array}{ll} |x|, & |x| > 2 \\ \frac 2\pi(x \arcsin\frac x2 + \sqrt{4-x^2}), & |x| \leq 2 \end{array}\right.$$

such that the normalized Young diagram is very close to the curve with probability very close to 1.

My question: what asymptotic statements about irreducible characters of S_n can be "read off" the Kerov-Vershik theorem? In a sense, this isn't a question about Kerov-Virshik at all, but a question about which interesting statistics of irreducible characters can be read off the shape of the Young diagram.

There are some tautological answers: for instance, if f(chi) is the height of the first column of chi, then I guess Kerov-Virshik shows that

f(chi) / sqrt(n)

is very probably very close to 2 as n gets large (if I did this computation right -- in any event it concentrates around a fixed value.) But I don't really have in mind any representation-theoretic meaning for f(chi).

2 latexify the function

Alexei Oblomkov really told me about the beautiful theorem of Kerov and Vershik, which says that "almost all Young diagrams look the same." More precisely: take a random irreducible representation of S_n (in Plancherel measure, which assigns a probability of (dim chi)^2 / (n!) to an irrep chi) and draw its Young diagram, normalized to have fixed area. Rotate the diagram 45 degrees and place the vertex at the origin of the Cartesian plane, so that it lies above the graph of y = |x|. Then there is a fixed curve, with equation

$$y = \left\{\begin{array}{ll} |x| x|, & |x| > 2 y = (2/pi)(x arcsin x/2 \\ \frac 2\pi(x \arcsin\frac x2 + sqrt(4-x^2)) \sqrt{4-x^2}), & |x| <= \leq 2 \end{array}\right.$$

such that the normalized Young diagram is very close to the curve with probability very close to 1.

My question: what asymptotic statements about irreducible characters of S_n can be "read off" the Kerov-Vershik theorem? In a sense, this isn't a question about Kerov-Virshik at all, but a question about which interesting statistics of irreducible characters can be read off the shape of the Young diagram.

There are some tautological answers: for instance, if f(chi) is the height of the first column of chi, then I guess Kerov-Virshik shows that

f(chi) / sqrt(n)

is very probably very close to 2 as n gets large (if I did this computation right -- in any event it concentrates around a fixed value.) But I don't really have in mind any representation-theoretic meaning for f(chi).

1

# Statistics of irreps of S_n that can be read off the Young diagram, and consequences of Kerov-Vershik

Alexei Oblomkov really told me about the beautiful theorem of Kerov and Vershik, which says that "almost all Young diagrams look the same." More precisely: take a random irreducible representation of S_n (in Plancherel measure, which assigns a probability of (dim chi)^2 / (n!) to an irrep chi) and draw its Young diagram, normalized to have fixed area. Rotate the diagram 45 degrees and place the vertex at the origin of the Cartesian plane, so that it lies above the graph of y = |x|. Then there is a fixed curve, with equation

y = |x| |x| > 2 y = (2/pi)(x arcsin x/2 + sqrt(4-x^2)) |x| <= 2

such that the normalized Young diagram is very close to the curve with probability very close to 1.

My question: what asymptotic statements about irreducible characters of S_n can be "read off" the Kerov-Vershik theorem? In a sense, this isn't a question about Kerov-Virshik at all, but a question about which interesting statistics of irreducible characters can be read off the shape of the Young diagram.

There are some tautological answers: for instance, if f(chi) is the height of the first column of chi, then I guess Kerov-Virshik shows that

f(chi) / sqrt(n)

is very probably very close to 2 as n gets large (if I did this computation right -- in any event it concentrates around a fixed value.) But I don't really have in mind any representation-theoretic meaning for f(chi).