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 pk'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.