The orders of the exceptional Weyl groups

Question

Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?

Answer 1

The orders of the Weyl groups of root systems of type $E_6$, $E_7$, $E_8$ seem to appear in a 1894 Comptes Rendus paper [1] of Cartan, pp. 641-642. The paper [2] has the details and the orders appear on p.39, p. 48, p. 51.

[1] Cartan, É. Sur la réduction de la structure d’un groupe à sa forme canonique. C. R. 119, 639-642 (1894). zbMATH http://gallica.bnf.fr/ark:/12148/bpt6k30752.f639

[2] Cartan, É. Sur la réduction à sa forme canonique de la structure d’un groupe de transformations fini et continu. American J. 18, 1-61 (1896). zbMATH https://doi.org/10.2307/2369787

Answer 2

My guess: these orders were first computed by Coxeter in his thesis

The polytopes with regular-prismatic vertex figures, Philos. Transactions (A) 229, 329–425 (1930). ZBL56.1119.03,

$E_l$ as $(\mathrm{PA})_l$ on p. 384, $F_4$ as $\{3,4,3\}$ and $G_2$ as $\{6\}$ on p. 349. But one might say that he only identified them as Weyl groups in the concluding §3.6 (pp. 186–210) of the 1935 Weyl seminar notes, The structure and representation of continuous groups: see pp. 201 and 208–209 for the orders, and p. 185 for Weyl’s formula to compute them.

Note added: Coxeter’s later book Regular Polytopes devotes §11·9 to Weyl’s formula, and in historical section §11·x notes that the groups $[3,4,3]$ (of $F_4$), resp. $[3^{l-4,2,1}]$ (of $E_{l=6,7,8}$) were already, linearly represented and with orders, in Goursat (1889, pp. 86–88), resp. Burnside (1911, pp. 299–308) with notation $\{G_l,T\}$. Moreover Burnside identified

• (pp. 285, 302) his $\{G_6,T\}$ as the group of 27 lines on a cubic surface. That’s an implicit reference to Jordan (1870, nº441) or (1869, p. 658) where this group is defined and its order computed as 27·10·8·24 = 72·6! or indeed 51840.

• (pp. 285, 305) his $\{G_7,T\}$ modulo a normal $\mathbf Z_2$ as the group of 28 bitangents to a plane quartic. That’s an implicit reference to Weber (1883, p. 493) where this group is defined and its order computed as 4·9! or indeed one half of 2903040.

• (pp. 285, 308) his $\{G_8,T\}$ modulo a normal $\mathbf Z_2$ as Jordan’s first hypoabelian group on eight symbols. That’s an explicit reference to (1870, nº262) or (1869, p. 657) where this group is defined and its order computed in two ways (where $\newcommand\P{\mathscr P}$ $\P_n := 2^{2n-1} + 2^{n-1}$), both of which evaluate to 96·10! or indeed one half of 696729600: $$ \style{font-family:sans-serif}{\text{in 1870,}} \qquad \omega_4 = 2^0(\P_1-1)2^2(\P_2-1)2^4(\P_3-1)2^6(\P_4-1);\\ \style{font-family:sans-serif}{\text{in 1869,}} \qquad \mathrm O_4 = 2^1(2^2-1)2^3(2^4-1)2^5(2^6-1)2^7(2^8-1)/\P_4. $$

This I think supports @NoamD.Elkies’ notion that these groups (and orders!) had long been known and ready to be identified — with the subtlety of Cartan and Weyl’s slightly different definitions ($\mathscr G$ vs. $\mathscr G'$) mentioned in my other comment. Thus Cartan in 1896 cited the exact same three above-italicized groups, on pp. 43, 50, 54 of @testaccount’s reference [2]; and his whole notation, terminology (e.g. roots that “meet”), and switch from $\mathrm G$ to $\mathrm G_1$ (our $\mathscr G$ and $\mathscr G'$) in the $E_6$ case (p. 39) were clearly informed by knowing Jordan’s group of 27 lines.

Second note added: I. Dolgachev’s great book Classical Algebraic Geometry: a modern view ends Chap. 8|9 with a very relevant historical note (earlier detailed in his (1983, §§1 & 7)):

The Weyl group $\mathrm W(E_6)$ as the Galois group of 27 lines was first studied by C. Jordan (1870). Together with the group of 28 bitangents of a plane quartic isomorphic to $\mathrm W(E_7)$, it is discussed in many classical text-books in algebra (e.g. Weber). S. Kantor (1895) realized the Weyl group $\mathrm W(E_n)$, $n \leqslant 8$, as a group of linear transformations preserving a quadratic form of signature $(1, n)$ and a linear form. A. Coble (1916) was the first who showed that the group is generated by the permutations group and one additional involution. So we should credit him with the discovery of the Weyl groups as reflection groups. Apparently independently of Coble, this fact was rediscovered by P. Du Val (1936). We refer to Bourbaki for the history of Weyl groups, reflection groups and root systems. Note that the realization of the Weyl group as a reflection group in the theory of Lie algebras was obtained by H. Weyl in 1928, ten years later after Coble’s work.

Sure enough, looking these up finds the orders of $\mathrm W(E_6)$ and $\mathrm W(E_7)$ in Kantor (1895, p. 22), and those of $\mathrm W(E_6)$, $\mathrm W(E_7)/\mathbf Z_2$, $\mathrm W(E_8)/\mathbf Z_2$ in Coble (1916, p. 356). (On the other hand, it seems to me that what is attributed here to Coble was done by Burnside 5 years earlier...?)

Answer 3

Following up on Dave Benson's comment, the MacTutor Killing entry explains that, in correspondence with Engel, Killing found $G_2$ in May of 1887 and the others that fall. This is reported (including the dimensions, summarized on the last page) in the second part of a Mathematische Annalen series which carries a submission date of 2 February 1888.

Wilhelm Killing. Die Zusammensetzung der stetigen endlichen Transformationsgruppen, Zweiter Theil. Math. Ann. 33 (1889) 1–48.

Available at The European Digital Mathematics Library, reviewed by Engel in https://zbmath.org/20.0368.03.

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As Francois points out, this is not the intended parameter.