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 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 ''p''_k's.

Examples

If $X$ denotes the formal sum $X=x_1+x_2+\cdots$, then f 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-t)/math> is simply the result of setting $x_i=t^$ for each i. That is,

: f\left frac\rightf(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 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 Xt^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 $s_$ to the conjugate Schur function $s_{\lambda^\ast}$.
*The substitution S:f\mapsto f X/math> is the antipode for the Hopf algebra structure on the Ring of symmetric functions.
*p_n +Yp_n p_n /math>
*The map \Delta: f\mapsto f +Y/math> is the coproduct for the Hopf algebra structure on the ring of symmetric functions.
*h_n\left (1-t)\right/math> 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 /(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.

Combinatorics
Symmetric functions