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representation theory of the symmetric group

Contents

Contents

Idea

The representation theory of the symmetric groups.

Properties

Irreducible representations

In characteristic zero, the irreducible representations of the symmetric group are, up to isomorphism, given by the Specht modules labeled by partitions λPart(n)\lambda \in Part(n) (e.g. Sagan 01, Thm. 2.4.6).

Dimension of irreps and hook length

Over the complex numbers:

The dimension of the irrep S (λ)S^{(\lambda)} (Specht module) equals the number of standard Young tableau of shape λ\lambda:

(1)dim(S (λ))=|sYTableaux λ| dim\big(S^{(\lambda)}\big) \;=\; \left\vert sYTableaux_\lambda \right\vert

(e.g. Sagan, Thm. 2.6.5)

Moreover, the number of standard Young tableaux of shape λ\lambda is given by the hook length formula

(2)|sYTableaux λ|=n!( 1irows(λ)1jλ ihook λ(i,j)) 1. \left\vert sYTableaux_\lambda \right\vert \;=\; n! \left( \prod_{ { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } } \ell hook_\lambda(i,j) \right)^{-1} \,.

This is due to Frame, Robinson & Thrall 54. Textbook accounts include Stanley 99, Cor. 7.21.6, Sagan 01 Thm. 3.10.2.

Combining (1) with (2) gives the hook length formula for the dimension of the Specht modules

dim(S (λ))=n!( 1irows(λ)1jλ ihook λ(i,j)) 1. dim\big(S^{(\lambda)}\big) \;=\; n! \left( \prod_{ { 1 \leq i \leq rows(\lambda) } \atop { 1 \leq j \leq \lambda_i } } \ell hook_\lambda(i,j) \right)^{-1} \,.

(e.g. James 78, Thm. 20.1)

hook length formulahook-content formula
number of standard Young tableauxnumber of semistandard Young tableaux
dimension of irreps of Sym(n)dimension of irreps of SL(n)

References

Textbook accounts:

See also:

Notes:

  • Yufei Zhao, Young Tableaux and the Representations of the Symmetric Group (pdf, pdf)

Discussion of characters for the symmetric group that depend only on Cayley distance from the neutral element (“block character”):

From the perspective of the seminormal representation:

On the representation theory of the symmetric group via the seminormal representation:

In relation to quantum information theory:

Last revised on May 19, 2021 at 01:22:39. See the history of this page for a list of all contributions to it.