Meetings: MWF 2:30-3:20 Vincent 364
Instructor: Chris Fraser (call me "Chris"), cfraser@umn.edu
Office hours: MW 10-11 am in Vincent 204, or by appointment.
Text: Richard Stanley's Enumerative Combinatorics Vol. 2. We will cover Chapter 7 and both appendices. Other useful references are Fulton's Young Tableaux, Macdonald's Symmetric Functions and Hall polyomials, Sagan's The symmetric group, Manivel's Symmetric functions, Schubert polynomials, and degeneracy loci.
Content: Combinatorial aspects of the theory of symmetric functions. Representation theory of the general linear and symmetric group. Schubert calculus.
Prereqs: Familiarity with linear algebra and abstract algebra.
Homeworks: Posted here.
Hw 1: solve 5 problems from the problem list by Feb. 21.
problem : 1 2 3 4 5 6 7 8 9 14
# of solves: 4 9 4 7 6 10 7 1 4 12
HW 2: solve 4 more problems from the problem list by April 1.
problem : 1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 20 23 24 25
# of solves: 4 9 4 7 8 12 8 2 6 2 1 13 1 7 8 8 2 1 1 7
HW 3: solve THREE more problems from the problem list by May 8.
Schedule of lectures:
1/22: Monomial, elementary, homogeneous, and power symmetric functions.
1/24: Partitions and their orderings.
1/27: Transition matrix e-->m (0-1 matrices with prescribed row & col sum).
1/29: Fundamental theorem of symmetric functions (e's are a basis). Transition matrix h --> m (ℕ -matrices with prescribed row & col sum) .
1/31: Involution ω . h's and p's are bases. Transition matrix p --> m (ordered set-partitions with prescribed block sums).
2/3: p's are eigenfunctions for ω with eigenvalue \pm 1.
2/5: Hall inner product.
2/7: Combinatorial definition of Schur Functions.
2/10: Schur functions are symmetric: Bender-Knuth involution. Kostka numbers.
2/12: Schur functions are orthonormal via RSK correspondence.
2/14: Gelfand Tsetlin patterns & RSK via toggles (Sam Hopkins guest lecturing).
2/17: Symmetry of RSK under inversion of permutations / transposition of ℕ -matrices.
2/19: ω acts on s's by transposition of Young diagrams.
2/21: Ratio of alternants formula for Schur polynomials.
2/24: Pieri rule: how to multiply h's times s's or e's times s's.
2/26: Multiplication by s_lam is adjoint to "skewing" by lam. Two definitions of Littlewood-Richardson numbers. Coproduct on the algebra of symmetric functions.
2/28: Jacobi-Trudi identity via Lindstrom-Gessel-Viennot.
3/2: Murnaghan-Nakayama rule.
3/4: Group representations: examples.
3/6: Group representations: subreps, irreps, isomorphisms of reps, definition of characters.
The subsequent lectures have tablet notes from the Zoom meetings.
3/18: Characters are an O.N.B. for the space of class functions; decomposing a rep into irreps is writing its character as a sum of irr characters. Notes and Kayla's notes.
3/20: Inisomorphic irreps have orthogonal characters. Example of decomposing a rep using characters. Notes.
3/23: Frobenius characteristic map of inner product spaces: CF(S_n) to Lambda^n . Irreducible characters of S_n. Notes.
3/25: Complete proof that \chi^\lambda(\bullet) are the characters. Induction product of class functions. Small examples of character tables. Notes.
3/27: Decomposing the regular representation and RSK. Specht modules. Notes.
3/30: Specht modules are the irreps. Notes.
4/1: SYT polytabloids are basis for Specht module; Branching rule for Sn; L-R #s describe induction product. Notes.
4/3: Rep theory of GLn: examples. Notes.
4/6: Rep theory of GLn: characters; main theorem. Notes.
4/8: Rep theory of GLn: more on main theorem; plethysm. Notes.
4/10: Schur functors. Notes.
4/13: Schur functors and matrix minors; SSYT basis. Notes.
4/15: Branching rule for GLk and Gelfand-Tsetlin basis; Schur-Weyl duality. Notes.
4/17: More Schur-Weyl duality. Notes.
4/20: Cauchy identity and GL-rep theory; Hook length formula. Notes.
4/22: Hook length formula. Notes.
4/24: Hook content formula. Notes.
4/27: Shape of a permutation under RSK. Knuth-equivalent permutations have the same insertion tableau. Notes.
4/29: Littlewood-Richardson rule(s). Notes.
5/1: Intro to Schubert calculus. Notes.
5/4: Shubert cells and varieties, cohomology ring of the Grassmannian is a quotient of Lambda Notes.