Plethystic substitution

Plethystic substitution is a shorthand notation for a common kind of substitution in the Ring of [symmetric functions|algebra of symmetric functions] and that of symmetric polynomials. It is essentially basic substitution of variables, but allows for a change in the number of variables used.

Definition

The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions is generated as an R-algebra by the power sum symmetric functions
For any symmetric function and any formal sum of monomials, the plethystic substitution f is the formal series obtained by making the substitutions
in the decomposition of as a polynomial in the p_k's.

Examples

If denotes the formal sum, then.
One can write to denote the formal sum, and so the plethystic substitution is simply the result of setting for each i. That is,
Plethystic substitution can also be used to change the number of variables: if, then is the corresponding symmetric function in the ring of symmetric functions in n variables.
Several other common substitutions are listed below. In all of the following examples, and are formal sums.

If is a homogeneous symmetric function of degree, then
:
If is a homogeneous symmetric function of degree, then
:,
The substitution is the antipode for the Hopf algebra structure on the Ring of symmetric functions.
The map is the coproduct for the Hopf algebra structure on the ring of symmetric functions.
is the alternating Frobenius series for the exterior algebra of the defining representation of the symmetric group, where denotes the complete homogeneous symmetric function of degree.
is the Frobenius series for the symmetric algebra of the defining representation of the symmetric group.