nLab Specht module




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

For every partition (in other words Young diagram) λn\lambda \vdash n, the Specht module associated to λ\lambda is usually denoted S λS^{\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 MM of Sym(n)Sym(n) is of the form

MλnM λ M \cong \underset{\lambda \vdash n}{\bigoplus} M^{\lambda}

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

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


Textbook accounts:

See also

Last revised on February 5, 2023 at 08:48:46. See the history of this page for a list of all contributions to it.