nLab representation theory of the symmetric group

Idea

The representation theory of the symmetric groups.

Properties

Irreducible representations

Proposition

In characteristic zero, the irreducible representations of the symmetric group are, up to isomorphism, given by the Specht modules labeled by partitions $\lambda \in Part(n)$.

(e.g. Sagan 01, Thm. 2.4.6).

Dimension of irreps and hook length

Over the complex numbers:

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

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

(e.g. Sagan, Thm. 2.6.5)

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

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

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

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

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

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

Examples

Example

(standard representation)
What is sometimes called the standard representation of the symmetric group $Sym_n$ is the restriction of the $n$-dimensional permutation representation on $k^n$ (by permutation of the canonical basis vectors) to the $(n-1)$-dimensional subspace where the sum of coefficients of these basis vectors is zero.

This is an irreducible representation whose corresponding partition (according to Prop. ) is $\big((n-1),1\big)$.

(cf. Kao Def. 2.5, Groupprops: Standard representation of the symmetric group)

Related concepts

• Specht module

• parastatistics

• representation theory of the general linear group

• representation theory of the special unitary group

• Schur-Weyl duality

• Jucys-Murphy element

• FI-representation

References

Monographs:

• G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer (1978) [doi:10.1007/BFb0067708]

• Gordon D. James, Adalbert Kerber: The Representation Theory of the Symmetric Group, Cambridge University Press (1984) [doi:10.1017/CBO9781107340732]

• Persi Diaconis, Chapter 7 of: Group Representations in Probability and Statistics, IMS Lecture Notes Monogr. Ser., 11: 198pp. (1988) (jstor:i397389, ISBN: 0940600145, pdf)

• William Fulton, Section 7 of: Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997 (doi:10.1017/CBO9780511626241)

• Bruce Sagan, The symmetric group, Springer 2001 (doi:10.1007/978-1-4757-6804-6, pdf)

See also:

• Wikipedia, Representation theory of the symmetric group

Notes:

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

• Daphne Kao: Representations of the Symmetric Group, VIGRE 2010 [pdf]

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

• Alexander Gnedin, Vadim Gorin, Sergei Kerov, Block characters of the symmetric groups, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 (arXiv:1108.5044, doi:10.1007/s10801-012-0394-9)

From the perspective of the seminormal representation:

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

• Anatoly Vershik, Andrei Okounkov, A New Approach to the Representation Theory of the Symmetric Groups,

  Part I: Selecta Mathematica, New Series 2, 581-605 (arXiv:math/0503040, doi:10.1007/BF02433451); Part II (incorporates Part I in revised and improved form): Russian version: Записки научных семинаров ПОМИ 307 (2004), 57–98 (Zapiski nauchnyh seminarov POMI 307 (2004), 57–98); English version: Journal of Mathematical Sciences 131 (2005), 5471–5494 (doi:10.1007/s10958-005-0421-7).

In relation to quantum information theory:

• Alonso Botero, Quantum Information and the Representation Theory of the Symmetric Group, Rev. colomb. mat. vol.50 no. 2 Bogotá July/Dec. 2016 (doi:10.15446/recolma.v50n2.62210, pdf)