Plethystic substitution

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

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

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

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

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

• If ${\displaystyle f}$ is a homogeneous symmetric function of degree ${\displaystyle d}$, then

  ${\displaystyle f[tX]=t^{d}f(x_{1},x_{2},\ldots )}$

• If ${\displaystyle f}$ is a homogeneous symmetric function of degree ${\displaystyle d}$, then

  ${\displaystyle f[-X]=(-1)^{d}\omega f(x_{1},x_{2},\ldots )}$,

where ${\displaystyle \omega }$ is the well-known involution on symmetric functions that sends a Schur function ${\displaystyle s_{\lambda }}$ to the conjugate Schur function ${\displaystyle s_{\lambda ^{\ast }}}$.

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

• Combinatorics, Symmetric Functions, and Hilbert Schemes (Haiman, 2002)

• M. Haiman, Combinatorics, Symmetric Functions, and Hilbert Schemes, Current Developments in Mathematics 2002, no. 1 (2002), pp. 39–111.