nLab
Schur-Weyl measure

Contents

Context

Measure and probability theory

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

  • YDiagrams nYDiagrams_n for the set of partitions of nn, hence of Young diagrams with nn boxes;

  • for λYDiagrams n\lambda \in YDiagrams_n write:

    • sYTableaux λsYTableaux_\lambda for the set of standard Young tableaux of shape λ\lambda;

    • for N +N \in \mathbb{N}_+ write

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

YDiagrams N p SW [0,1] λ |sYTableaux λ||ssYTableaux λ(N)|N n \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=nN = n, and writing

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

p SW(λ)=dim(S (λ))dim(V (λ))N n. 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)Part(n) and in the limit of large n=cN 2n = c N^2 \to \infty, the logarithm of the Schur-Weyl probability is almost surely approximately constant (i.e. independent of λ\lambda) on the value

lnp SWnH c - ln p^{SW} \;\sim\; \sqrt{n}\cdot H_c

for some H cH_c \in \mathbb{R}, in that for all ϵ +\epsilon \in \mathbb{R}_+ we have

limn=cN 2p SW({λPart(n)|1nlnp SW(λ)H c<ϵ})=1. \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 \,.

(Mkrtchyan 14, Thm. 1.1)

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.

References

Review:

See also:

Last revised on June 3, 2021 at 15:14:48. See the history of this page for a list of all contributions to it.