In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group
whose natural action on tensor products
of a complex vector space
has as image an irreducible representation of the group of invertible linear transformations
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.
Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, and consider the action of
Define two permutation subgroups
of Sn as follows:[clarification needed] and Corresponding to these two subgroups, define two vectors in the group algebra
is the unit vector corresponding to g, and
is the sign of the permutation.
The product is the Young symmetrizer corresponding to the Young tableau λ.
Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer.
(If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.)
Let V be any vector space over the complex numbers.
Consider then the tensor product vector space
Let Sn act on this tensor product space by permuting the indices.
One then has a natural group algebra representation
, with the canonical Young tableau
the swap operator defined by
is where μ is the conjugate partition to λ.
are the symmetric and alternating tensor product spaces.
is an irreducible representation of Sn, called a Specht module.
We write for the irreducible representation.
for some rational number
In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra
Then one has If V is a complex vector space, then the images of
provides essentially all the finite-dimensional irreducible representations of GL(V).