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 $\Lambda _{R}(x_{1},x_{2},\ldots )$ is generated as an R-algebra by the power sum symmetric functions

p_{k}=x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+\cdots .

For any symmetric function $f$ and any formal sum of monomials $A=a_{1}+a_{2}+\cdots$ , the plethystic substitution f[A] is the formal series obtained by making the substitutions

p_{k}\longrightarrow a_{1}^{k}+a_{2}^{k}+a_{3}^{k}+\cdots

in the decomposition of $f$ as a polynomial in the p_k's.

Examples

If $X$ denotes the formal sum $X=x_{1}+x_{2}+\cdots$ , then $f[X]=f(x_{1},x_{2},\ldots )$ .

One can write $1/(1-t)$ to denote the formal sum $1+t+t^{2}+t^{3}+\cdots$ , and so the plethystic substitution $f[1/(1-t)]$ is simply the result of setting $x_{i}=t^{i-1}$ for each i. That is,

f\left[{\frac {1}{1-t}}\right]=f(1,t,t^{2},t^{3},\ldots )

.

Plethystic substitution can also be used to change the number of variables: if $X=x_{1}+x_{2}+\cdots ,x_{n}$ , then $f[X]=f(x_{1},\ldots ,x_{n})$ is the corresponding symmetric function in the ring $\Lambda _{R}(x_{1},\ldots ,x_{n})$ of symmetric functions in n variables.

Several other common substitutions are listed below. In all of the following examples, $X=x_{1}+x_{2}+\cdots$ and $Y=y_{1}+y_{2}+\cdots$ are formal sums.

If $f$ is a homogeneous symmetric function of degree $d$ , then
$f[tX]=t^{d}f(x_{1},x_{2},\ldots )$
If $f$ is a homogeneous symmetric function of degree $d$ , then
$f[-X]=(-1)^{d}\omega f(x_{1},x_{2},\ldots )$ ,

where

\omega

is the well-known involution on symmetric functions that sends a Schur function

s_{\lambda }

to the conjugate Schur function

s_{\lambda ^{\ast }}

.

The substitution $S:f\mapsto f[-X]$ is the antipode for the Hopf algebra structure on the Ring of symmetric functions.
$p_{n}[X+Y]=p_{n}[X]+p_{n}[Y]$
The map $\Delta :f\mapsto f[X+Y]$ is the coproduct for the Hopf algebra structure on the ring of symmetric functions.
$h_{n}\left[X(1-t)\right]$ is the alternating Frobenius series for the exterior algebra of the defining representation of the symmetric group, where $h_{n}$ denotes the complete homogeneous symmetric function of degree $n$ .
$h_{n}\left[X/(1-t)\right]$ is the Frobenius series for the symmetric algebra of the defining representation of the symmetric group.