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 [math]\displaystyle{ \Lambda_R(x_1,x_2,\ldots) }[/math] is generated as an R-algebra by the power sum symmetric functions

[math]\displaystyle{ p_k=x_1^k+x_2^k+x_3^k+\cdots. }[/math]

For any symmetric function [math]\displaystyle{ f }[/math] and any formal sum of monomials [math]\displaystyle{ A=a_1+a_2+\cdots }[/math], the plethystic substitution f[A] is the formal series obtained by making the substitutions

[math]\displaystyle{ p_k \longrightarrow a_1^k+a_2^k+a_3^k+\cdots }[/math]

in the decomposition of [math]\displaystyle{ f }[/math] as a polynomial in the p_k's.

Examples

If [math]\displaystyle{ X }[/math] denotes the formal sum [math]\displaystyle{ X=x_1+x_2+\cdots }[/math], then [math]\displaystyle{ f[X]=f(x_1,x_2,\ldots) }[/math].

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

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

Plethystic substitution can also be used to change the number of variables: if [math]\displaystyle{ X=x_1+x_2+\cdots,x_n }[/math], then [math]\displaystyle{ f[X]=f(x_1,\ldots,x_n) }[/math] is the corresponding symmetric function in the ring [math]\displaystyle{ \Lambda_R(x_1,\ldots,x_n) }[/math] of symmetric functions in n variables.

Several other common substitutions are listed below. In all of the following examples, [math]\displaystyle{ X=x_1+x_2+\cdots }[/math] and [math]\displaystyle{ Y=y_1+y_2+\cdots }[/math] are formal sums.

• If [math]\displaystyle{ f }[/math] is a homogeneous symmetric function of degree [math]\displaystyle{ d }[/math], then

  [math]\displaystyle{ f[tX]=t^d f(x_1,x_2,\ldots) }[/math]

• If [math]\displaystyle{ f }[/math] is a homogeneous symmetric function of degree [math]\displaystyle{ d }[/math], then

  [math]\displaystyle{ f[-X]=(-1)^d \omega f(x_1,x_2,\ldots) }[/math],

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

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