nLab Schur-Weyl measure

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

• $S^{(\lambda)} \in Rep(Sym(n))$ for the complex irreducible representation of the symmetric group which is labeled by $\lambda$ (the Specht module, see the representation theory of the symmetric group);

• $V^{(\lambda)} \in Rep(Sym(n))$ for the complex irreducible representation of the (special) general linear group which is labeled by $\lambda$ (see the representation theory of the general linear group),

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

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

(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.

Related concepts

• Plancherel measure

• asymptotic representation theory

• Robinson-Schensted-Knuth algorithm

References

Review:

• Leonid Petrov, slides 73-76 in: Random Matrices, lecture notes 2019 (pdf slides, pdf, webpage)

  (in the context of random matrix theory)

See also:

• Sevak Mkrtchyan, Entropy of Schur-Weyl Measures, Annales de l’I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 678-713. (arXiv:1107.1541, numdam:AIHPB_2014__50_2_678_0)

• Pierre-Loïc Méliot, Kerov’s central limit theorem for Schur-Weyl and Gelfand measures, DMTCS proc. AO, 2011, 669–680 (pdf)

• Pierre-Loïc Méliot, Kerov’s central limit theorem for Schur-Weyl measures of parameter 1/2 (arXiv:1009.4034)

• M. S. Boyko and N. I. Nessonov, Entropy of the Shift on $II_1$-representations of the Group $S(\infty)$ (pdf)