# Plethystic exponential > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Plethystic_exponential > Markdown URL: https://mediated.wiki/source/Plethystic_exponential.md > Source: https://en.wikipedia.org/wiki/Plethystic_exponential > Source revision: 1343759689 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematics](/source/Mathematics), the **plethystic exponential** is a certain [operator](/source/Operator_(mathematics)) defined on (formal) [power series](/source/Power_series) which, like the usual [exponential function](/source/Exponential_function), translates addition into multiplication. This exponential operator appears naturally in the theory of [symmetric functions](/source/Symmetric_function), as a concise relation between the [generating series](/source/Generating_function) for [elementary](/source/Elementary_symmetric_polynomial), [complete](/source/Complete_homogeneous_symmetric_polynomial) and [power sums](/source/Power_sum_symmetric_polynomial) homogeneous symmetric polynomials in many variables. Its name comes from the operation called [plethysm](/source/Plethysm), defined in the context of so-called [lambda rings](/source/Lambda_ring). In [combinatorics](/source/Combinatorics), the plethystic exponential is a [generating function](/source/Generating_function) for many well studied sequences of [integers](/source/Integer), [polynomials](/source/Polynomial) or power series, such as the number of integer [partitions](/source/List_of_partition_topics). It is also an important technique in the [enumerative combinatorics](/source/Enumerative_combinatorics) of unlabelled [graphs](/source/Graph_(discrete_mathematics)), and many other combinatorial objects.[1][2] In [geometry](/source/Geometry) and [topology](/source/Topology), the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariant of its symmetric products.[3] The inverse operator of the plethystic exponential is the [plethystic logarithm](/source/Plethystic_logarithm). ## 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]] the ideal consisting of power series without constant term. Then, given f(x)\in R^0[[x]], its plethystic exponential \operatorname{PE}[f] is given by - \operatorname{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 \operatorname{PE}[f] when the variable is understood): - \begin{align}[ll] \operatorname{PE}[0] & = 1\\[5pt] \operatorname{PE}[f+g] & = \operatorname{PE}[f] \operatorname{PE}[g]\\[5pt] \operatorname{PE}[-f] & = \operatorname{PE}[f]^{-1} \end{align} Some basic examples are: - \begin{align}[ll] \operatorname{PE}[x^n] & = \frac1{1-x^n}, n \in \mathbb{N} \\[5pt] \operatorname{PE}\left[ \frac x{1-x} \right] & = 1+\sum_{n\geq1}p(n)x^n \end{align} In this last example, p(n) is number of partitions of n\in\mathbb{N}. The plethystic exponential can be also defined for power series rings in many variables. ## Product-sum formula 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 f(x)=\sum_{k=1}^{\infty} a_k x^k denotes a formal power series with real coefficients a_k, then it is not difficult to show that: \operatorname{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](/source/Integer_partition) and to the [cycle index](/source/Cycle_index) of the [symmetric group](/source/Symmetric_group).[4] ## Relation with symmetric functions Working with variables x_1, x_2, \ldots, x_n, denote by h_k the [complete homogeneous symmetric polynomial](/source/Complete_homogeneous_symmetric_polynomial), that is the sum of all [monomials](/source/Monomial) of degree *k* in the variables x_i, and by e_k the [elementary symmetric polynomials](/source/Elementary_symmetric_polynomial). 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](/source/Newton's_identities), that can succinctly be written, using plethystic exponentials, as: - \sum_{n=0}^\infty h_n \,t^n = \operatorname{PE}[p_1 \,t] = \operatorname{PE}[x_1 t + \cdots + x_n t] - \sum_{n=0}^\infty (-1)^n e_n \,t^n = \operatorname{PE}[- p_1 \,t] = \operatorname{PE}[-x_1 t - \cdots - x_n t] ## Macdonald's formula for symmetric products Let *X* be a finite [CW complex](/source/CW_complex), of dimension *d*, with [Poincaré polynomial](/source/Poincar%C3%A9_polynomial) P_X (t) = \sum_{k=0}^d b_k(X) \, t^k where b_k(X) is its *k*th [Betti number](/source/Betti_number). Then the Poincaré polynomial of the *n*th symmetric product of *X*, denoted \operatorname{Sym}^n (X), is obtained from the series expansion: \operatorname{PE}[P_X(-t)\,x] = \prod_{k=0}^d (1-t^k x)^{(-1)^{k+1}b_k(X)} = \sum_{n\geq 0} P_{\operatorname{Sym}^n(X)}(-t) \, x^n ## Plethystic logarithm The inverse of the plethystic exponential is the plethystic logarithm. It is defined in the multivariate case as follows: if f(x_1, \ldots, x_n) is a formal power series[a] with constant term 1, then the plethystic logarithm \operatorname{PL}[f] is defined by \operatorname{PL}[f](t_1, t_2, \ldots, t_n) = \sum_{k=1}^\infty \frac{\mu(k)}{k} \ln(f(t_1^k, t_2^k, \ldots, t_n^k)), where \mu(k) is the [Möbius function](/source/M%C3%B6bius_function), defined by[5] \mu(k) = \begin{cases} 1 & \text{if } k = 1, \\ (-1)^n & \text{if } k \text{ is the product of } n \text{ distinct primes}, \\ 0 & \text{otherwise}, \end{cases} and ln is the [natural logarithm](/source/Natural_logarithm).[6] ## The plethystic programme in physics In a series of articles, a group of theoretical physicists, including Bo Feng, Amihay Hanany and [Yang-Hui He](/source/Yang-Hui_He), proposed a programme for systematically counting single and multi-trace gauge invariant operators of [supersymmetric gauge theories](/source/Supersymmetric_gauge_theory).[7] In the case of quiver gauge theories of [D-branes](/source/D-brane) probing [Calabi–Yau](/source/Calabi%E2%80%93Yau) singularities, this count is codified in the plethystic exponential of the [Hilbert series](/source/Hilbert_series) of the singularity. ## Notes 1. Or a function of n complex arguments. ## References 1. Pólya, G. & Read, R. C. (1987). [*Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds*](http://link.springer.com/10.1007/978-1-4612-4664-0). New York, NY: Springer New York. [doi:10.1007/978-1-4612-4664-0](https://doi.org/10.1007/978-1-4612-4664-0). ISBN 978-1-4612-9105-3. 1. Harary, Frank (1955-02-01). ["The number of linear, directed, rooted, and connected graphs"](http://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1955-0068198-2). *Transactions of the American Mathematical Society*. **78** (2): 445–463. [doi:10.1090/S0002-9947-1955-0068198-2](https://doi.org/10.1090/S0002-9947-1955-0068198-2). [ISSN 0002-9947](https://www.worldcat.org/issn/0002-9947) 1. Macdonald, I. G. (1962). ["The Poincare Polynomial of a Symmetric Product"](https://www.cambridge.org/core/product/identifier/S0305004100040573/type/journal_article). *[Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society)*. **58** (4): 563–568. [Bibcode:1962PCPS...58..563M](https://ui.adsabs.harvard.edu/abs/1962PCPS...58..563M). [doi:10.1017/S0305004100040573](https://doi.org/10.1017/S0305004100040573). [ISSN 0305-0041](https://www.worldcat.org/issn/0305-0041). [S2CID 121316624](https://api.semanticscholar.org/CorpusID:121316624) 1. Florentino, Carlos (2021-10-07). ["Plethystic Exponential Calculus and Characteristic Polynomials of Permutations"](https://www.dmlett.com/archive/v8/DML22_v8_pp22-29..pdf). *Discrete Mathematics Letters*. **8**: 22–29. [arXiv:2105.13049](https://arxiv.org/abs/2105.13049). [doi:10.47443/dml.2021.094](https://doi.org/10.47443/dml.2021.094). [ISSN 2664-2557](https://www.worldcat.org/issn/2664-2557). [S2CID 237451072](https://api.semanticscholar.org/CorpusID:237451072) 1. Abramowitz, Milton & Stegun, Irene A. (1972 [1964]). *Handbook of mathematical functions: with formulas, graphs and mathematical tables [conference under the auspices of the National science foundation and the Massachusetts institute of technology]*. Dover books on advanced mathematics. New York: Dover. p. 826. ISBN 978-0-486-61272-0. 1. Benvenuiti, Sergio; Feng, Bo; Hanany, Amihay; He, Yang-Hui (2006). "Counting BPS Operators in Gauge Theories". *Journal of High Energy Physics*. [arXiv:hep-th/0608050v2](https://arxiv.org/abs/hep-th/0608050v2). [doi:10.1088/1126-6708/2007/11/050](https://doi.org/10.1088/1126-6708/2007/11/050) 1. Feng, Bo; Hanany, Amihay; He, Yang-Hui (2007-03-20). ["Counting gauge invariants: the plethystic program"](http://stacks.iop.org/1126-6708/2007/i=03/a=090?key=crossref.bcaed087696ada7ddb3caa309da4f9f7). *Journal of High Energy Physics*. **2007** (3): 090. [arXiv:hep-th/0701063](https://arxiv.org/abs/hep-th/0701063). [Bibcode:2007JHEP...03..090F](https://ui.adsabs.harvard.edu/abs/2007JHEP...03..090F). [doi:10.1088/1126-6708/2007/03/090](https://doi.org/10.1088/1126-6708/2007/03/090). [ISSN 1029-8479](https://www.worldcat.org/issn/1029-8479). [S2CID 1908174](https://api.semanticscholar.org/CorpusID:1908174) --- Adapted from the Wikipedia article [Plethystic exponential](https://en.wikipedia.org/wiki/Plethystic_exponential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Plethystic_exponential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.