Plethystic exponential

In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions, as a concise relation between the generating series for elementary, complete and power sums homogeneous symmetric polynomials in many variables. Its name comes from the operation called plethysm, defined in the context of so-called lambda rings.

In combinatorics, the plethystic exponential is a generating function for many well studied sequences of integers, polynomials or power series, such as the number of integer partitions. It is also an important technique in the enumerative combinatorics of unlabelled graphs, and many other combinatorial objects.

In geometry and topology, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariant of its symmetric products.

Definition, main properties and basic examples

Let $R[[x]]$ be a ring of formal power series in the variable $x$, with coefficients in a commutative ring $R$. Denote by

$R^{0}[[x]]\subset R[[x]]$

be the ideal of power series without constant term. Then, given $f(x)\in R^{0}[[x]]$ its plethystic exponential, denoted $PE[f]$ is given by

$PE[f](x)=\exp \left(\sum _{k=1}^{\infty }{\frac {f(x^{k})}{k}}\right)$

where $\exp(\cdot )$ is the usual exponential function. It is readily verified that (writing simply $PE[f]$ when the variable is understood):

${\begin{aligned}[ll]PE[0]&=1\\PE[f+g]&=PE[f]PE[g]\\PE[-f]&=PE[f]^{-1}\end{aligned}}$

Some basic examples are:

${\begin{aligned}[ll]PE[x^{n}]&={\frac {1}{1-x^{n}}},n\in \mathbb {N} \\PE\left[{\frac {x}{1-x}}\right]&=1+\sum _{n\geq 1}p(n)x^{n}\end{aligned}}$

In this last example, $p(n)$ is number of partitions of $n\in \mathbb {N}$.

Relation with symmetric functions

Working with variables $x_{1},x_{2},\cdots x_{n}$, denote by $h_{k}$ the complete homogeneous symmetric polynomial, that is the sum of all monomials of degree k in the variables $x_{i}$, and by $e_{k}$ the elementary symmetric polynomials. Then, the $h_{k}$ and the $e_{k}$ are related to the power sum polynomials: $p_{k}=x_{1}^{k}+\cdots +x_{n}^{k}$ by Newton's identities, that can succinctly be written, using pletyhistic exponentials, as:

$\sum _{n=0}^{\infty }h_{n}\,t^{n}=PE[p_{1}\,t]=PE[x_{1}t+\cdots +x_{n}t]$

$\sum _{n=0}^{\infty }(-1)^{n}e_{n}\,t^{n}=PE[-p_{1}\,t]=PE[-x_{1}t-\cdots -x_{n}t]$

Macdonald's formula for symmetric products

Let X be a finite CW complex, of dimension d, with Poincaré polynomial$$P_{X}(t)=\sum _{k=0}^{d}b_{k}(X)\,t^{k}$$where $b_{k}(X)$ is its kth Betti number. Then the Poincaré polynomial of the nth symmetric product of X, denoted $Sym^{n}(X)$, is obtained from the series expansion:$$PE[P_{X}(-t)\,x]=\prod _{k=0}^{d}\left(1-t^{k}x\right)^{(-1)^{k+1}b_{k}(X)}=\sum _{n\geq 0}P_{Sym^{n}(X)}(-t)\,x^{n}$$

The plethystic programme in mathematical physics

In a series of articles, a group of mathematical physicists, including Bo Feng, Amihay Hanany and Yang-Hui He, proposed a programme for systematically counting single and multi-trace gauge invariant operators of supersymmetric gauge theories. In the case of quiver gauge theories of D-branes probing Calabi-Yau singularities, this count is codified in the plethystic exponential of the Hilbert series of the singularity.