Plethystic substitution


Plethystic substitution is a shorthand notation for a common kind of substitution in the 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 pk'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.
,
where is the well-known involution on symmetric functions that sends a Schur function to the conjugate Schur function.