# Origin of terms "flag", "flag manifold", "flag variety"?

These terms have become common in Lie theory and related algebraic geometry and combinatorics, as seen in many questions posted on MO, but it's unclear to me where they first came into use. Probably the concrete notion of (complete) flag of subspaces $$0 \subset V_1 \subset \dots \subset V_n =V$$ with $$\dim V_k = k$$ in an $$n$$-dimensional vector space $$V$$ occurs very early in the literature (Grassmann?), though I'm not sure whether a specific national flag (drapeau, Flagge, ... ) is supposed to come to mind. Those wanting to visualize the simplest case may find the picture here useful (or not).

At some point in the development of Lie groups and their homogeneous spaces, the notion of flag manifold got attached to the set of all complete flags in $$\mathbb{C}^n$$ or $$\mathbb{R}^n$$: this is realized as the quotient of the corresponding general linear group $$G$$ by the isotropy subgroup $$B$$ of a standard flag (say the group of upper triangular nonsingular matrices). In the 1950s such connected maximal solvable subgroups became known as Borel subgroups, while the notion of flag manifold came to mean the quotient $$G/B$$ for an arbitrary connected reductive Lie group $$G$$ and a Borel subgroup $$B$$. In the setting of real compact Lie groups, the analogue takes the form $$G/T$$ for a maximal torus $$T$$: this is apparently the earliest version of the flag manifold.

The work of Borel and Chevalley led to parallel developments for reductive algebraic groups over fields of arbitrary characteristic, with algebraic geometry replacing differential geometry and the term flag variety becoming common.

While the people I've mentioned certainly deserve most of the credit for recognizing the essential role of flag manifolds or varieties in the study of geometry and representation theory associated to reductive groups, I'm still left with some uncertainty:

What are the earliest sources in the literature for these terms?

ADDED: The original reason for using the word "flag" in this context is a minor though interesting part of my question; but as the comments here indicate there is some variation in the folklore. In old literature on traditional projective geometry a "flag" is sometimes defined as a pair consisting of a point and a line through it, but that doesn't help much with the etymology. Is there more than folklore?

Following the answer by Charles, I've looked further at the thesis of Ehresmann and the May 1951 Bourbaki talk by Borel (which like other expository talks is not included in his collected papers). I see more clearly how the notion of flag variety or flag manifold evolved from the older and still somewhat mysterious use of the term flag (Flagge, Fahne, ...) in projective geometry, for instance to refer to an incident point-line pair. Near the end of Borel's talk he notes: Cela permet en particulier de montrer que les "varietes de drapeaux" complexes considerees par EHRESMANN dans sa these sont sans torsion .... His use of quotation marks suggests to me that this might actually be the first time that label was applied to what later became known as the flag variety $$G/B$$ of a reductive Lie or algebraic group. (I wish I had thought to ask him at the right time.)

• During a Lie theory class, the professor (Siddhartha Sahi, I believe?) identified Israel Gelfand as the person who coined the term 'flag'. This could have been Rutgers-based favoritism, though. Mar 13, 2011 at 15:15
• The points of the flag variety are supposed to actually look like flags. A flag that you see in front of a high school (see abcteach.com/free/f/flag2bnw.jpg) typically has a roughly 0-dimensional ball at the top, with a roughly 1-dimensional pole coming out of it and sticking into the ground, with a roughly 2-dimensional cloth sharing a dimension with the pole, with a roughly 3-dimensional space surrounding the cloth, all in a roughly 4-dimensional space-time. Mar 13, 2011 at 17:15
• @Anton, that is an interesting way of seeing it. For me it was always obvious why flags are called flags, but my visualization is completely different :) Mar 13, 2011 at 19:36
• @Mariano: What is your visualization? I always assumed the one I posted is what everybody else had in mind. Mar 13, 2011 at 20:08
• Surely the terminology was imported from the study of polyhedra/polytopes? Mar 14, 2011 at 16:36

Armand Borel's Bourbaki Seminar 121 Groupes algébriques is from 1955, and uses "drapeau" (page 7). (It's online at archive.numdam.org.) This may not be the earliest occurrence, but there is a good reason for attention to the full flag variety in this context (the theory of Borel subgroups).

The concept traces back some way, to Ehresmann's thesis in the 1930s; Kolchin's work (the Lie-Kolchin theorem) uses the non-intrinsic language of upper triangular form. Hodge & Pedoe talks about Schubert spaces in general, which would be natural in the enumerative geometry tradition, for which full flags is just one of the cases.

Edit: A further data point is Chern's paper On the Characteristic Classes of Complex Sphere Bundles and Algebraic Varieties (1953), which relies on Ehresmann's work to some extent. The word "flag" is absent (though used by Chern discussing it in his Selected Papers).

• Thanks for going back farther into the literature. The long 1934 Annals of Mathematics paper by Ehresmann based on his thesis work (influenced by E. Cartan and Lefschetz) essentially studies flag manifolds as homogeneous spaces, even without using the term "drapeau". The word seems to have become common by the time of Borel's Bourbaki seminar, but I haven't been able to track it down earlier in his own work. French origins seem most likely. Mar 16, 2011 at 13:49
• P.S. I went back to Borel's first Bourbaki talk (no. 45) in 1951, Cohomologie des espaces homogenes, for an even earlier use of "drapeau". Mar 17, 2011 at 18:10

I think the concept may date back to René De Saussure (1868-1943). He was interested in the Euclidean geometry of 3-dimensional space and used the term "géometrie des feuillets". I think this may have been his doctoral work. The work was criticized by Eduard Study because de Saussure failed to reference Study! Despite teaching Math at the University of Geneva de Saussure seems to be more famous for his work on Esperanto.

From his papers:

$$\hspace{3.3em}$$

$$\hspace{3.3em}$$

$$\hspace{3.3em}$$

• de Saussure with two "s" instead of two "c". Nov 22, 2018 at 0:08
• Isn't 'feuillette' 'leaf' rather than 'flag'? Nov 22, 2018 at 1:52
• This seems to be the correct answer. See screenshots added from (1906, 1909, 1910), where flag appears first in Esperanto, then French. As Borel (1951; 1953, p. 202) apparently wrote drapeau before Ehresmann (1955: Oeuvres I, p. 473) one might guess that although not a pupil of de Saussure (obit), he was exporting Swiss terminology. Nov 22, 2018 at 3:01
• I should add that de Saussure’s flageto appeared first in his Internacia Scienca Revuo (1908, p. 167), and consists of affine rather than vector subspaces. Among the few who quoted him are Bricard (1909), Cartan-Fano (1915/1955, §23), Coolidge (1940, p. 263). Nov 23, 2018 at 21:41

@Jim, I always asked myself the same question. You say that the "notion of flag variety or flag manifold evolved from the older and still somewhat mysterious use of the term flag in projective geometry, for instance to refer to an incident point-line pair." Once I saw a drawing in an expository article that solved that mistery for me: picture a point in the projective plane as the corresponding line in space, and a projective line as the corresponding plane in space, then an incident point-line becomes a line waving a plane in space, like a flag :)

One way to visualize the meaning of the word, "flag" is to think of those pennant-like tapering flags (cloth part only), starting at 0 width, to higher intermediary widths, to width n. Somewhat like various dimensions comprising a mathematical flag. Maybe the french word means this particular type of flag.

• Looking at the Wikipedia page of "drapeau" one sees some images; I am not sure this corresponds to what you mean.
– user9072
Jan 20, 2015 at 11:50
• Interesting that you came up with the same idea as my comment a day before your post. The idea must be in the air, I suppose. Jan 22, 2015 at 6:09
• View the Wiki article on pennant and note flags for naval ships. See etymology of pennant from perhaps pennon etymonline.com/index.php?term=pennon&allowed_in_frame=0 Jan 22, 2015 at 6:17

Regarding visualization, I see it as those flags which are all sewed together and become a very long line when we want and take them out to the wind.

Borel (1951) definitely drew on an existing tradition. Besides de Saussure’s work unearthed in J.M. Selig’s answer, he could have learned « flag » from:

• van der Waerden (1936, pp. 15, 30) (they likely met in Summer 1951):

... die projektiven Invarianten von drei „Flaggen“ zu finden. Eine Flagge besteht aus einem Punkt, einer Geraden durch diesen Punkt, einer Ebene durch diese Gerade usw.

... Nennen wir die Figur, die aus einem Punkt $$x$$, einer mit $$x$$ inzidenten Geraden $$p$$ und aus einer mit $$p$$ inzidenten Ebene $$u$$ besteht, eine Flagge $$\{x,p,u\}$$, so erhalten wir ...

• Freudenthal (1949, p. 22):

Deuxième exemple. — Les drapeaux du $$\mathrm P_3$$, c’est-à-dire les triples constitués d’un point, d’une droite et d’un plan, tous les trois en position unie.