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.
The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions
1
x
2
, … )
is generated as an R-algebra by the power sum symmetric functions For any symmetric function
f
and any formal sum of monomials
=
a
, the plethystic substitution f[A] is the formal series obtained by making the substitutions in the decomposition of
as a polynomial in the pk's.
denotes the formal sum
One can write
to denote the formal sum
, and so the plethystic substitution
is simply the result of setting
That is, Plethystic substitution can also be used to change the number of variables: if
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,
are formal sums.