Plethystic exponential

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

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

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

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

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

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

Some basic examples are:

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

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

The plethystic exponential can be also defined for power series rings in many variables.

The plethystic exponential can be used to provide innumerous product-sum identities. This is a consequence of a product formula for plethystic exponentials themselves. If ${\displaystyle f(x)=\sum _{k=1}^{\infty }a_{k}x^{k}}$ denotes a formal power series with real coefficients ${\displaystyle a_{k}}$, then it is not difficult to show that:

$${\displaystyle {\text{PE}}[f](x)=\prod _{k=1}^{\infty }(1-x^{k})^{-a_{k}}}$$

The analogous product expression also holds in the many variables case. One particularly interesting case is its relation to integer partitions and to the cycle index of the symmetric group.[4]

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

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

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

Let X be a finite CW complex, of dimension d, with Poincaré polynomial

$${\displaystyle P_{X}(t)=\sum _{k=0}^{d}b_{k}(X)\,t^{k}}$$

where ${\displaystyle b_{k}(X)}$ is its kth Betti number. Then the Poincaré polynomial of the nth symmetric product of X, denoted ${\displaystyle \operatorname {Sym} ^{n}(X)}$, is obtained from the series expansion:

$${\displaystyle {\text{PE}}[P_{X}(-t)\,x]=\prod _{k=0}^{d}\left(1-t^{k}x\right)^{(-1)^{k+1}b_{k}(X)}=\sum _{n\geq 0}P_{\operatorname {Sym} ^{n}(X)}(-t)\,x^{n}}$$

In a series of articles, a group of theoretical 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.[5] 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.