# nLab Schur-Weyl measure

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

## Applications

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

The Schur-Weyl measure is probability distribution on the set of Young diagrams of given number of boxes, closely related to expressions appearing in Schur-Weyl duality.

## Definition

For $n \in \mathbb{N}_+$, write:

• $YDiagrams_n$ for the set of partitions of $n$, hence of Young diagrams with $n$ boxes;

• for $\lambda \in YDiagrams_n$ write:

• $sYTableaux_\lambda$ for the set of standard Young tableaux of shape $\lambda$;

• for $N \in \mathbb{N}_+$ write

• $ssYTableaux_\lambda(N)$ for the set of semistandard Young tableaux of shape $\lambda$ and with labels $\leq N$.

Then the Schur-Weyl measure for $n,N \in \mathbb{N}$ is

$\array{ YDiagrams_N & \overset{ p^{SW} }{\longrightarrow} & [0,1] \\ \lambda &\mapsto& \frac { \left\vert sYTableaux_\lambda \right\vert \cdot \left\vert ssYTableaux_\lambda(N) \right\vert } {N^n} }$

(e.g. Petrov 19, slide 76)

In the special case that $N = n$, and writing

this is equal to (see at hook length formula):

$p^{SW}(\lambda) \;=\; \frac { dim\big( S^{(\lambda)} \big) \cdot dim\big( V^{(\lambda)} \big) } {N^n} \,.$

## Properties

### Entropy

With respect to the Schur-Weyl measure on $Part(n)$ and in the limit of large $n = c N^2 \to \infty$, the logarithm of the Schur-Weyl probability is almost surely approximately constant (i.e. independent of $\lambda$) on the value

$- ln p^{SW} \;\sim\; \sqrt{n}\cdot H_c$

for some $H_c \in \mathbb{R}$, in that for all $\epsilon \in \mathbb{R}_+$ we have

$\underset{n = c N^2 \to \infty}{\lim} p^{SW} \left( \left\{ \lambda \in Part(n) \;\left\vert\; \tfrac{-1}{\sqrt{n}} ln p^{SW}(\lambda) - H_c \;\lt\; \epsilon \right. \right\} \right) \;=\; 1 \,.$

### Relation to Cayley distance kernel

The Schur-Weyl measure is the pushforward measure of the probability distribution on pure states encoded by the “Cayley state” (here): the quantum state that is given by the Cayley distance kernel – see there for details.

Review: