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November 3, 2016

Linear Algebraic Groups (Part 5)

Posted by John Baez

Now let’s look at projective geometry from a Kleinian viewpoint. We’ll take the most obvious types of figures — points, lines, planes, and so on — and see which subgroups of GL(n)\mathrm{GL}(n) they correspond to. This leads us to the concept of ‘maximal parabolic subgroup’, which we’ll later generalize to other linear algebraic groups.

We’ll also get ready to count points in Grassmannians over finite fields. For that, we need the qq-deformed version of binomial coefficients.

  • Lecture 5 (Oct. 6) - Projective geometry from a Kleinian perspective. The Grassmannians Gr(n,j)\mathrm{Gr}(n,j) as spaces of points, lines, planes, etc. in projective geometry. The Grassmannians as quotients of the general linear group by the maximal parabolic subgroups P n,jP_{n,j}. Claim: the cardinality of Gr(n,j)\mathrm{Gr}(n,j) over the finite field 𝔽 q\mathbb{F}_q is the qq-binomial coefficient (nj) q\binom{n}{j}_q. The mysterious sense in which set theory is linear algebra over the ‘field with one element’.
Posted at November 3, 2016 1:35 AM UTC

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