Lectures: Tuesdays 11:15-12:45 VinH 364 and Thursdays 11:15-12:05 AMundH 104 (Note schedule change)
The main parts of this course will be on symmetric functions and Schubert polynomials. Within this framework I'll touch on a number of aspects of combinatorial theory. Here's a note I wrote about Young tableaux that is some relevant light reading that will hopefully give you a sense of what the class is about.
I'll put up ( very unpolished! READ-AT-YOUR-OWN-RISK) latexed notes on the lecture after each class. These are just intended to give you a reminder of what we talked about. However, as I plan to eventually polish these up, comments and corrections are welcome!
Class 1: Lecture notes 1: motivations, symmetric function rings, partitions and orderings, monomial and elementary symmetric functions, fundamental theorem of symmetric functions. We also discussed moving the time to Tues 11:15-12:45 and Thurs 11:15-12:05.
Class 2: Lecture notes 2: homogeneous and power sum symmetric polynomials. The involution "omega", specializations.
Class 3: Lecture notes 3: Stirling numbers, bilinear form on Lambda, Schur polynomials, and their symmetry.
Class 4: Lecture notes 4: Finished up on symmetry of the Schur polynomials. Kostka numbers, some bijections with SYT, orthonormality of Schur polynomials. ``The operator approach''. I'll finish up this lecture next class. (Errata: Prop 1 should say K_{lambda,lambda}=1.)
Class 5: Finished up on "Lecture notes 4". Discussed simplicial complexes, shellability, Reisner-Stanley rings and correspondence, Grobner bases, h-vectors of shellable complexes in terms of new faces. Explained the ball of semistandard Young tableaux, set-valued tableaux. Ended by defining the Grothendieck polynomials G_{\lambda} and the principle that Schur polynomial theorems should have Grothendieck analogues.
Class 6: Lecture notes 5: Talked about RSK and Schensted. Also mentioned the question of whether hook lengths determine a shape (up to conjugation).
Class 7: Finished up lecture 5 (note: I updated the file also).
Class 8: Lecture notes 6: Classical definition of Schur polynomials; the Jacobi Trudi identity. Stated the Littlewood-Richardson rule. Professor Calin Chindris will be Thursday's guest lecturer.
Class 9: Guest lecturer Professor Calin Chindris on the saturation conjecture/theorem and its implications.
Class 10: (Lecture 6 continued; file above updated)
Class 11: Finished lecture 6: the jeu de taquin formulation of the Littlewood-Richardson rule. Problems on quantum cohomology of Grassmannians.
Class 12: Our next three classes (or so) we'll study lecture 7 .
Class 13: Finished the proof of the first and second fundamental theorems of jdt by ending with the infusion involution.
Class 14: Wrap up on lecture 7; also the hook-length formula, see lecture 8 .
Classes 15,16,17: Guest lecturer Professor Drew Armstrong will discuss the combinatorics of Coxeter groups.
Class 18: I'll start on representation theory of the symmetric group. See lecture 9 . We spoke about character theory and proved the Freobenius character map theorem and the Murnaghan-Nakayama rule.
Class 19: Continued lecture 9 (and updated the file). I'll speak a bit more about representations of S_n; applications to statistics.
Class 20: Finished lecture 9 (file updated once more).
Class 21: We'll begin on Schubert polynomials. See lecture 10 .
Class 22: Continued on Schubert polynomials: proof of the pipe dream formula.
Class 23: We'll continue on lecture 10 (file updated).
Class 24: Transition and marching; see lecture 11 .
Class 25: More on transition and marching.
Class 26: Schubert polynomials and plane partitions; Grothendieck polynomials.
Class 27: See lecture 12.
Class 28: Finished Lecture 12.