Schubert Calculus Learning Seminar

This seminar is organized by Alexander Caviedes, Cesar Ceballos, Peter Crooks and Iva Halacheva. We are following Fulton's book "Young-Tableaux". 


Location: Fields Institute
Tuesdays 11:00am. Fall 2013
Wednesdays 2:00pm. Winter 2014 
Fridays 11:00 am. Summer 2014




Schedule 2013-2014


September 24 Cesar Ceballos Bumping and Sliding : Operations on Young Tableaux. Schensted-bumping operation. Schutzerberg-sliding algorithm. [F, Int, Ch1]
October 1 Alexander Caviedes Words and Elementary transformations: Words. Knuth Transformations. Schur Polynomials. [F, Sect 2.1, 2.2] Alex' Notes
October 8 Alexander Caviedes Schur Polynomials: Schur Polynomials. Pieri's Formula. Kostka Numbers. [F, Sect 2.2] Alex' Notes
October 15 Iva Halacheva Robinson-Schensted-Knuth Correspondence I: RSK correspondence. [F, Sect 4.1]
October 22 Iva Halacheva Robinson-Schensted-Knuth Correspondence II: Applications of the RSK correspondence. Symmetry Theorem. Matrix Ball construction. [F, Sect 4.2, 4.3]
October 29 Cesar Ceballos The Littlewood-Richardson Rule: Yamanouchi word. Littlewood-Richardson Tableaux. Littlewood-Richardson Rule. [F, Ch 5]
November 5 Alexander Caviedes Symmetric Polynomials: Complete, Elementary, Monomial, Newton and Schur symmetric Polynomials. Jacobi-Trudi Formula. The ring of symmetric functions. [F, Ch 6] Alex' Notes
November 12 Peter Crooks The action of the symmetric group on tableaux. Specht Modules I. [F, Sect 7.1, 7.2]
November 19 Peter Crooks Representations of the symmetric group. Specht Modules II. Irreducible representations of the symmetric group. [F, Sect 7.1, 7.2] Peter's Notes
November 26 Iva Halacheva The ring of representations and symmetric functions I. [F, Sect 7.3]
December 10 Iva Halacheva The ring of representations and symmetric functions II. [F, Sect 7.3]
December 17 Cesar Ceballos A straightening algorithm. [F, Sect 7.4] Cesar's Notes
January 13* Alexander Caviedes Schur Modules. [F, Sect 8.1 ]
January 22 Alexander Caviedes, Peter Crooks, Iva Halacheva Exercise Section I: Hook length formula [F, Sect 4.3, 7.3]. Character of Mlambda [F, Sect 7.2]. Consequences of Litlewood-Richardson's rule [F, Chapter 5]
January 29 Alexander Caviedes Schur Modules II, Irreducible representations of GLm(C)
February 5 Cesar Ceballos, Peter Crooks, Iva Halacheva Exercise Section II
February 12 Peter Crooks Gl(E) representations from Sn representations I. Weyl functor [F, Sect 8.1, 8.2]
February 26 Peter Crooks Gl(E) representations from Sn representations II. Character of Elambda [F, Sect 8.3, 8.4]
March 7* Iva Halacheva Plucker embeddings I. Projective embeddings of flag varieties [F, Sect 9.1]
March 12 Iva Halacheva Plucker embeddings II. Projective embedding of flag varieties [F, Sect 9.1]
March 19 Iva Halacheva Plucker embeddings III. Homogeneous coordinate ring of the flag variety. Geometric Invariant theory [F, Sect 9.1, 9.2]
March 26 Peter Crooks Representations of Gl(E) and line bundles I. [F, Sect 9.3]
April 2 Peter Crooks Representations of Gl(E) and line bundles II [F, Sect 9.3]
April 9 Peter Crooks Representations of Gl(E) and line bundles III [F, Sect 9.3]
April 16 Cesar Ceballos Schubert Calculus on Grassmannians. Schubert cells and varieties. Duality Theorem [F, Sect 9.4]
April 30 Cesar Ceballos Schubert Calculus on Grassmannians. Schubert cells and varieties. Pieri's Formula [F, Sect 9.4] Cesar's Notes
May 7 Alexander Caviedes Fixed points of torus actions. Schubert cells in flag manifolds [F, Sect 10.1, 10.2]
May 14 Alexander Caviedes Schubert varieties. Dual Schubert cells and dual Schubert varieties in flag manifolds. Duality Theorem [F, Sect 10.2]
July 11 Alexander Caviedes Chern Classes of line bundles. Leray Hirsch Theorem [F, Sect 10.2, Appendix B1]
July 18 Alexander Caviedes Cohomology ring of flag manifolds [F, Sect 10.2, Appendix B1]
July 25 Peter Crooks Relations among Schubert varieties I [F, Sect 10.3]
August 1* Peter Crooks Relations among Schubert varieties II [F, Sect 10.3]
August 8 Peter Crooks Relations among Schubert varieties III [F, Sect 10.3]
August 8 Cesar Ceballos Schubert Polynomials [F, Sect 10.4]
August 15 Cesar Ceballos Bruhat Order [F, Sect 10.5]
August 15 Iva Halacheva Applications to the Grassmannian [F, Sect 10.6]


Bibliography
[F] Young-Tableaux. William Fulton.