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.

The formal definition of plethystic substitution relies on the fact that the ring of symmetric functions ${\displaystyle \Lambda _{R}(x_{1},x_{2},\ldots )}$ is generated as an R-algebra by the power sum symmetric functions

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

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

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

in the decomposition of ${\displaystyle f}$ as a polynomial in the p_k's.

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.