math-communicationnotationsoft-questiontag-removed
The Dynkin diagrams of type ADE are ubiquitous in mathematics as solutions of various classification problems. The diagram E6 is usually drawn by five dots in a row with a sixth dot above the third, see for example here. There would be many other ways to draw the diagram E6, for example the sixth dot below the five dots, or just a capital
E.
Is there a reason for drawing that diagram in that particular shape beside or is that just a confirmed habit?
The harmonic oscillator is a fundamental example in both classical and quantum mechanics.
It is not known why Dirichlet denoted his functions with an $L$. Perhaps he chose $L$ for Legendre (I am not serious). The reason may be alphabetical. Just before $L$-functions are introduced in his 1837 paper on primes in arithmetic progression (Math. Werke vol. 1, 313--342), there are certain functions $G$ and $H$, and the letters $I, J$, and $K$ may not have seemed appropriate labels for a function.
While $L(s,\chi)$ and $L(\chi,s)$ are common notations for the $L$-function of a character $\chi$, neither decorated notation is due to Dirichlet; he simply wrote different $L$-functions as $L_0, L_1, L_2,\dots$.
Update (Jan. 12, 2016): I learned a few days ago from Ellen Eischen that the Kubota Tractor Corporation has a model called the "(compact) Standard L-Series," and today I saw a Kubota L-series go past my department building. Here is a photo I took.
If you're looking for a modern reinterpretation of what the L stands for in L-series, the webpage https://www.kubota.com/product/tlbseries.aspx gives the answer, and it's not Langlands: L means Loader or Landscaper.
Best Answer
The question as stated is not really helpful, but it's worth pointing out that the ADE and other graphs/diagrams evolved over a couple of decades in different countries. The graphs, which encode at first the Coxeter data for finite (or more general) reflection groups, go back at least to Coxeter's 1934 paper. In Witt's 1941 paper these now familiar graphs reappear when the reflection groups are unified with the classification of root systems for semisimple Lie algebras over $\mathbb{C}$. Here the vertices correspond to simple roots, not just simple reflections, so the notation has to incorporate some length information. Dynkin's fundamental 1952 papers used the resulting "Dynkin diagrams" with extra labels 0, 1, 2 at vertices to classify efficiently the nilpotent orbits. Along the way a number of different choices were made about adding edges and arrows or making some vertices darker to distinguish lengths. Bourbaki's 1968 treatise is by now the easiest standard source to follow.
The graph itself is drawn in a typographically convenient way and can vary as Scott notes (note especially Coxeter's type E pictures). An even more arbitrary convention governs the numbering of vertices or simple roots. Here again the evolved version in Bourbaki is well established, though some authors like Carter depart a bit from that numbering. The history of ideas is quite interesting, but at the end of the day a convention is just a convention.