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I'm not sure whether my answer is conceptual in your sense, but here is a relatively short proof. First of all, your definition of $f$ suggests the notation $$s_p := \sum_{i=p}^n x_i.$$ Now consider t …

The key word here is "polarity". A polarity of a projective plane with point set $P$ and line set $L$ is a map $\pi$ from $P \cup L$ to itself mapping points to lines and lines to points, such that $\ …

There is a forthcoming book "Descent in buildings" by Bernhard Mühlherr, Holger Petersson and Richard Weiss, which should appear soon (published by Princeton University Press). See http://press.prince …

Dieudonné's "La Géometrie des Groupes Classiques" might be what you're looking for; it has a whole chapter on automorphisms and isomorphisms of the classical groups.

If $k$ is a commutative field of characteristic $p>0$, then the map $$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$ is a group homomorphi …

Have you tried Chapter 17, section 17.1 from Springer's book on algebraic groups? I believe that is as down-to-earth as it can get, and it is certainly rather detailed.

Tits' original lecture notes from 1974 (Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, 386, Springer-Verlag) can still serve as a very good introduction to the subject. …

Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO. Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\operat …

It's indeed a real pity that so many of Tits's papers are quite difficult to get. Luckily, there are some very concrete ideas by F. Buekenhout, B. Muehlherr, J.-P. Tignol and H. Van Maldeghem, to publ …

The most natural way to view your set of vectors is in the setting of projective geometry; the vector space $K^n$ (where $K$ is now an arbitrary field, so $K = \mathbb{C}$ for you) can be seen as a pr …

The map $T \colon \mathrm{PSL}_2(p) \to \operatorname{Sym}(\mathbb{F}_{2^n}) \colon f \mapsto T_f$ does not have its image in $\mathrm{GL}_n(2)$ for other Mersenne primes $p = 2^n - 1$, unlike the cas …

This is indeed known, and can be found, for instance, in the book "Algebraic combinatorics. I. Association schemes" by Bannai and Ito (1984), Section II.2, Example 2.1 (p. 53).

For a reference, you could use Theorem I.5.1 from C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, New York, 1954. (This also appears in volume 2 of his collected works.) The …