Young symmetrizer

In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group ${\displaystyle S_{n}}$ whose natural action on tensor products ${\displaystyle V^{\otimes n}}$ of a complex vector space ${\displaystyle V}$ has as image an irreducible representation of the group of invertible linear transformations ${\displaystyle GL(V)}$. All irreducible representations of ${\displaystyle GL(V)}$ are thus obtained. It is constructed from the action of ${\displaystyle S_{n}}$ on the vector space ${\displaystyle V^{\otimes n}}$ by permutation of the different factors (or equivalently, from the permutation of the indices of the tensor components). A similar construction works over any field but in characteristic p (in particular over finite fields) the image need not be an irreducible representation. The Young symmetrizers also act on the vector space of functions on Young tableau and the resulting representations are called Specht modules which again construct all complex irreducible representations of the symmetric group while the analogous construction in prime characteristic need not be irreducible. The Young symmetrizer is named after British mathematician Alfred Young.