# Notes on flag varieties and Grassmannians for beginners

Can you suggest books or lecture notes (for beginners) covering basic material about flag varieties and Grassmannians (of reductive groups), with emphasis on the usual flag variety, i.e. flag variety of $GL(n, \mathbb{C})$. Topics like: classical Plücker embedding and its generalizations for other reductive groups. Line bundles on flag varieties and Borel-Weil theorem. Cohomology of flag varieties. Definition and basic properties of Schubert cells and varieties. Also connections with symplectic geometry (i.e. coadjoint orbits) would also be nice.

• For the combinatorial theory: perhaps Fulton's "Young Tableaux" Sep 21 '15 at 0:05
• You seem to be asking quite a bit for one reference. The Grassmannian section in Griffiths-Harris is a fast introduction to the $A_n$-type Grassmannian. The paper by Bernstein-Gelfand-Gelfand gives a description of the cohomology ring that applies to all types. Sep 21 '15 at 1:02
• Two books: 1) Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, and 2) Hiller, Geometry of Coxeter groups. Sep 21 '15 at 1:46
• I am not sure to what extent this qualifies as "(for beginners)", but the first sections of the book by Baston and Eastwood on the Penrose transform cover most of the material you mention. Sep 21 '15 at 7:55
• Thanks all for the helpful references. I already knew some of them but definitely the comments and replies helped.
– Kiu
Sep 26 '15 at 15:00