nLab Specht module

Idea

Specht modules are linear representations of the symmetric group $Sym(n)$ (for any $\in \mathbb{N}$), hence modules over the group ring $k(Sym(n))$, which are indexed by the partitions of $n$.

For every partition (in other words Young diagram) $\lambda \vdash n$, the Specht module associated to $\lambda$ is usually denoted $S^{\lambda}$.

In characteristic 0, they are irreducible and exhaust the isomorphism classes of irreps (e.g. Sagan 01, Thm. 2.4.6).

In this case, any finite dimensional representation $M$ of $Sym(n)$ is of the form

$$M \cong \underset{\lambda \vdash n}{\bigoplus} M^{\lambda}$$

where $M^{\lambda}$ is the isotypic component of $M$ associated to $\lambda$ which is of the form $M^{\lambda} \cong (S^{\lambda})^{\oplus^{m_{\lambda}}}$, with $m_{\lambda}$ called the multiplicity of $S^{\lambda}$ in $M$.

Over a field of positive characteristic $p$, where $p \mid n!$, the Specht modules are not irreducible, but every irreducible module does appear as the cosocle of a Specht module.

Related concepts

• representation theory of the symmetric group

References

Textbook accounts:

• Bruce Sagan, Section 2.3 in: The symmetric group, Springer 2001 (doi:10.1007/978-1-4757-6804-6, pdf)

• Jean-Louis Loday, Bruno Vallette, Annex A in: Algebraic Operads, Grundlehren der mathematischen Wissenschaften 346, Springer 2012 (ISBN 978-3-642-30362-3, pdf)

See also

• Wikipedia, Specht module