Seminar - Fall 2016

Here are the notes from a course on linear algebraic groups.

If you discover any errors in the notes please email me, and I'll add them to the list of errors.

John Simanyi took notes in LaTeX, and you can see this LaTeX files here.

Lecture 1 (Sept. 22) - The definition of a linear algebraic group. Examples: the general linear group \(\mathrm{GL}(n)\), the special linear group \(\mathrm{SL}(n)\), the orthogonal group \(\mathrm{O}(n)\), the special orthogonal group \(\mathrm{SO}(n)\), and the Euclidean group \(\mathrm{E}(n)\). The origin of groups in geometry: the parallel postulate and Euclidean versus non-Euclidean geometry. Elliptic and hyperbolic geometry.

Lecture 2 (Sept. 27) - The road to projective geometry. Treating Euclidean, elliptic and hyperbolic geometry on an equal footing: in each case the symmetry group is a linear algebraic group of 3 × 3 matrices over a field \(k\), points are certain 1d subspaces of \(k^3\), and lines are certain 2d subspaces of \(k^3\). In projective geometry we take the symmetry group to be all of \(\mathrm{GL}(3)\), take points to be all 1d subspaces of \(k^3\), and take lines to be all 2d subspaces of \(k^3\). It thus subsumes Euclidean, elliptic and hyperbolic geometry. In general we define projective \(n\)-space, \(k\mathrm{P}^n\), to be the set of 1d subspaces of \(k^{n+1}\).

Lecture 3 (Sept. 29) - The Schubert decomposition of \(k\mathrm{P}^n\) into Bruhat cells. Examples: the real projective line \(\mathbb{R}\mathrm{P}^1\), the complex projective plane \(\mathbb{C}\mathrm{P}^1\) and the real projective plane \(\mathbb{R}\mathrm{P}^2\). Projective geometry over finite fields: for any prime power \(q\), there is a field \(\mathbb{F}_q\) with \(q\) elements, and the cardinality of \(\mathbb{F}_q\mathrm{P}^n\) is the \(q\)-integer \([n]_q\). Abstract projective planes.

Lecture 4 (Oct. 4) - Pappus's hexagon theorem, and how it characterizes which abstract projective planes are of the form \(k\mathrm{P}^2\) for a field \(k\). Klein geometry and transitive group actions: each kind of highly symmetrical geometry corresponds to a group \(G\), and each type of geometrical figure in this geometry corresponds to a set on which \(G\) acts transitively. Transitive \(G\)-spaces all arise from subgroups of \(G\). Klein geometry studies invariant relations between transitive \(G\)-spaces.
Supplementary reading:
- Eric Moorhouse, Incidence Geometry.

Lecture 5 (Oct. 6) - Projective geometry from a Kleinian perspective. The Grassmannians \(\mathrm{Gr}(n,j)\) as spaces of points, lines, planes, etc. in projective geometry. The Grassmannians as quotients of the general linear group by the maximal parabolic subgroups \(P_{n,j}\). Claim: the cardinality of \(\mathrm{Gr}(n,j)\) over the finite field \(\mathbb{F}_q\) is the \(q\)-binomial coefficient \(\binom{n}{j}_q\). The mysterious sense in which set theory is linear algebra over the 'field with one element'.

Lecture 6 (Oct. 11) - Proof that the cardinality of \(\mathrm{Gr}(n,j)\) over \(\mathbb{F}_q\) is \(\binom{n}{j}_q\). The \(q\)-deformed version of Pascal's triangle. Bruhat cells for the Grassmanian. How to count the total number of Bruhat cells, which is just \(\binom{n}{j}\), and the number of cells of any given dimension.

Lecture 7 (Oct. 13) - Flags, and the the flag variety \(F(n_1, \dots, n_\ell, n)\), which consists of all chains of linear subspaces \( V_1 \subset V_2 \subset \cdots \subset V_\ell \subset k^n\). The flag variety as a quotients of the general linear group by parabolic subgroups, which are intersections of maximal parabolic subgroups. The complete flag variety \(F_n = F(1,2,\dots,n)\) as the quotient \(GL(n)/B(n)\) where the Borel subgroup \(B(n)\) is the group of invertible upper triangular matrices. The cardinality of the complete flag variety over \(\mathbb{F}_q\) is the \(q\)-factorial \([n]_q!\). When \(q = 1\) this reduces to the ordinary factorial, which counts 'set-theoretic flags'.

Lecture 8 (Oct. 18) - Group objects in various categories. What's the right category for linear algebraic groups? First try: algebraic sets. Over an algebraically complete field \(k\), Hilbert's Nullstellensatz says there's an order-reversing one-to-one correspondence between algebraic sets \(S \subseteq X\) in a finite-dimensional vector space \(X\) over \(k\) and radical ideals \(J \subseteq k[X]\) of the polynomial algebra \(k[X]\). The algebra \(k[S]\) of polynomials restricted to \(S\) is isomorphic to \(k[X]/J\). Problem: we'd like an 'intrinsic' approach that does not make use of the ambient space \(X\). Second try: affine algebras. For any algebraic set \(S\), the algebra \(k[S]\) is an affine algebra, meaning a finite-generated commutative algebra without nilpotents. Up to isomorphism, every affine algebra arises this way. Thus we can use affine algebras as a more intrinsic substitute for algebraic sets. Problem: all this works only over an algebraically complete field.
Supplementary reading:
- J. S. Milne, Algebraic Geometry.

Lecture 9 (Oct. 20) - The category \(\mathrm{AlgSet}_k\) of algebraic sets and regular functions versus the category \(\mathrm{AffAlg}_k\) of affine algebras and algebra homomorphisms. The duality between geometry and commutative algebra: when \(k\) is algebraically closed, \(\mathrm{AlgSet}_k \simeq \mathrm{AffAlg}_k^{\mathrm{op}}\). We can define the category of affine varieties over \(k\), \(\mathrm{AffVar}_k\), to be \(\mathrm{AffAlg}_k^{\mathrm{op}}\). Problems: 1) all this works only when \(k\) is algebraically closed, 2) it excludes infinite-dimensional spaces corresponding to algebras that aren't finitely generated, and 3) it excludes 'infinitesimal' spaces corresponding to algebras with nilpotents. Third try: affine schemes. Grothendieck simply defined the category of affine schemes over \(k\), \(\mathrm{AffSch}_k\), to be the opposite of the category of commutative algebras over \(k\).
Supplementary reading:
- Brian Osserman, Points and morphisms, from the classical and scheme-theoretic point of view.

Lecture 10 (Oct. 25) - Affine algebraic groups and affine algebraic group schemes.
Supplementary reading:
- J. S. Milne, Basic Theory of Affine Group Schemes (also called "AGS").
- J. S. Milne, Algebraic Groups (also called "iAG").
Lecture 11 (Oct. 27) - The big theorem on Bruhat decompositions for reductive algebraic groups.

Lecture 12 (Nov. 1) - Illustrating the big theorem on Bruhat decompositions: the case of \(\mathrm{GL}(3)\).

Lecture 13 (Nov. 3) - Illustrating the big theorem on Bruhat decompositions: the case of \(\mathrm{GL}(n)\). Permutahedra.