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In mathematics, plethytic exponential is a certain operator defined on (formal) power series which, like the usual exponential, translates addition into multiplication. In broad terms, if the input series has information on a certain object, the coefficients of its plethytic exponential usually provide important information on symmetric products of that object.
In complex analysis, the plethystic exponential is related to Weierstrass product expansions of entire functions.
In combinatorics, the plethystic exponential is a generating function for many well studied sequences of integers, polynomials or power series.
In geometry and topology, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariants of its symmetric products.
Definition, main properties and basic examples
[edit]Let be a ring of formal power series in the variable , with coefficients in a commutative ring . Denote by
![{\displaystyle R[[x]]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/5aacb21250f99f332a57763ecb7f92e3637ade19)
![{\displaystyle R^{0}[[x]]\subset R[[x]]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6d422050ca0756f711e41d5ff8af2b7635e0a6aa)
be the ideal of power series without constant term. Then, given its plethystic exponential, denoted is given by
![{\displaystyle f(x)\in R^{0}[[x]]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/12c3bdc8d8c75ad3e7ec8370c8d11d39d430d6da)
![{\displaystyle PE[f]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6a511e135362702c9fdf9d9d23ae76154f18a7fc)
=\exp \left(\sum _{k=1}^{\infty }{\frac {f(x^{k})}{k}}\right)}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d03991b1aacbeba9c1ff57ed1ecb8eeed419752c)
where is the usual exponential function. It is readily verified that:
![{\displaystyle {\begin{aligned}[ll]PE[0]&=1\\PE[f+g]&=PE[f]PE[g]\\PE[-f]&=PE[f]^{-1}\end{aligned}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b40dad0a97b050dc5c511d49e2dde764e5e9c356)
Some basic examples are (writing just when the variable is understood):
![{\displaystyle {\begin{aligned}[ll]PE[x]&={\frac {1}{1-x}}\\PE[x^{n}]&={\frac {1}{1-x^{n}}}\\PE\left({\frac {x}{1-x}}\right)&=1+\sum _{n\geq 1}p(n)x^{n}\end{aligned}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d43aa01716f07c8e22b4094ffc4bb292b01f99ea)
In this last example, is number of partitions of .
The plethystic programme in Mathematical-Physics
[edit]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 multitrace 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.
References
[edit]- Feng, Bo; Hanany, Amihay; He, Yang-Hui (2007), "Counting gauge invariants: the plethystic program", Journal of High Energy Physics, 2007, doi:10.1088/1126-6708/2007/03/090, MR 0010594
- Littlewood, D. E. (1944), "Invariant theory, tensors and group characters", Philosophical Transactions of the Royal Society A, 239: 305–365, doi:10.1098/rsta.1944.0001, JSTOR 91389, MR 0010594