---
tag: PWF62J9R1H
kind: concept
name: Young symmetrizer
sources:
  - wikipedia: en:1585226@r1346297036
    locator: Young symmetrizer
    note: adopted
    mention: wikipedia://en/1585226
    sections: [Summary]
same_as:
  wikidata: Q8058560
  wikipedia: Young symmetrizer
inventory: [WVT65328KH]
---

## Summary

In mathematics, a **Young symmetrizer** is an element of the group algebra of the [[symmetric group|0515A6D2Y1]] $S_n$ whose natural action on tensor products $V^{\otimes n}$ of a complex vector space $V$ has as image an irreducible representation of the group of invertible linear transformations $GL(V)$. All irreducible representations of $GL(V)$ are thus obtained. It is constructed from the action of $S_n$ on the vector space $V^{\otimes n}$ 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 positive characteristic (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.