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 ${\mathrm{\Lambda }}_R(x_1,x_2,\dots )$ 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

$p_k⟶a_1^k+a_2^k+a_3^k+\cdots$

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 $f[X]=f(x_1,x_2,\dots )$.

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,

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

Plethystic substitution can also be used to change the number of variables: if $X=x_1+x_2+\cdots +x_n$, then $f[X]=f(x_1,\dots ,x_n)$ is the corresponding symmetric function in the ring ${\mathrm{\Lambda }}_R(x_1,\dots ,x_n)$ 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

  $f[tX]=t^{d}f(x_1,x_2,\dots )$

• If $f$ is a homogeneous symmetric function of degree $d$, then

  $f[-X]=(-1{)}^d\omega f(x_1,x_2,\dots )$,

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 $\mathrm{\Delta }:f\mapsto f[X+Y]$ is the coproduct for the Hopf algebra structure on the ring of symmetric functions.
• $h_n[X(1-t)]$ 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 $n$.
• $h_n[X/(1-t)]$ is the Frobenius series for the symmetric algebra of the defining representation of the symmetric group.

External links

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

References

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