### Math 665: Combinatorial Theory II

Topic for Winter 2000: symmetric functions.

Course meets: MWF 2:10-3, Room 2866 East Hall.

Instructor: Sergey Fomin, 2858 East Hall, 764-6297, fomin@math.lsa.umich.edu

Course homepage: http://www.math.lsa.umich.edu/~fomin/665.html

Level: introductory graduate.

Prerequisites: none (for graduate students).

Student work expected: several problem sets.

Synopsis: Symmetric functions, and especially the Schur functions and various generalizations thereof, are ubiquitous in representation theory, Schubert calculus, combinatorial enumeration, and other fields. The course will take the purely combinatorial (and entirely elementary for the most part) approach, primarily driven by the intrinsic logic of the subject, rather than by its group-theoretic or algebro-geometric applications (these will also be discussed, time permitting). A lot of time will be devoted to Young tableaux, related combinatorial algorithms, and enumeration problems.

Text:

[EC2]
R.P.Stanley, Enumerative combinatorics, vol.2, Cambridge University Press, 1999.
We will cover Chapter 7 (including Appendix 1). See table of contents.

Reference texts:

[Fu]
W.Fulton, Young tableaux , Cambridge University Press, 1997.
[Ma]
I.G.Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, 1995.
[Sa]
B.E.Sagan, The symmetric group, Wadsworth and Brooks/Cole, 1991.
[EC1]
R.P.Stanley, Enumerative combinatorics, vol.1, 2nd edition, Cambridge University Press, 1997.
Problem Set 1
1. Prove Euler's pentagonal number theorem.
2. [EC2, Exercise 7.2(a,b)].
3. Show that the dominance order corresponds to containment of convex hulls of Sn-orbits.
4. Show that the dominance order describes adjacency of nilpotent orbits.
5. Prove Muirhead's inequalities [Ma, Exercise I.2.18].
6. [EC2, Exercise 7.3].
7. [EC2, Exercise 7.4].
8. [EC2, Exercise 7.9].
9. [EC2, Exercise 7.6].
10. [Ma, Exercise I.2.15]
11. Compute Riemann's zeta function at points 2, 4, 6,..., following [Ma, Exercise I.2.21]
12. Prove that the cardinality of a Grassmannian over Fq is given by a q-binomial coefficient.
13. Prove the q-binomial theorem.
Problem Set 2
1. [EC2, Exercise 7.12].
2. [EC2, Exercise 7.17(a)].
3. [EC2, Exercise 7.21].
4. [EC2, Exercise 7.24(d)].
5. [EC2, Exercise 7.59(d)] and [EC2, Exercise 7.59(e)] (first part).
6. Prove the analog of [EC2, Corollary 7.12.6] for the Young-Fibonacci lattice.
7. [EC2, Exercise 7.34].
8. [EC2, Exercise 7.37(d)] (difficult!).
9. [EC2, Exercise 7.39].
10. [EC2, Exercise 7.40].
11. Let L(A,B)={SSYT of shape A whose reading is a Yamanouchi word of skew shape B}.
Establish a bijection L(A,B)->L(B,A).
12. [EC2, Theorem A1.3.8].
13. [EC2, Theorem A1.3.9].
14. [EC2, Theorem A1.3.11] (difficult!).
Problem Set 3
1. Show that the Schur function for the staircase shape is a polynomial in odd power sums.
2. Prove E.H.Moore's presentation theorem for the symmetric group.
3. Solve [EC2, Exercise 6.19aa] using content words of skew shapes.
4. [Sa, Exercise 3.12.4]
5. Prove that the number of SYT of a given shape for which i is a descent does not depend on i.
6. [EC2, Exercise 7.101(a)] (difficult).
7. [EC2, Exercise 7.62].
8. [EC2, Exercise 7.73].
9. Prove that self-evacuating SYT are equinumerous to the domino tableaux of the same shape
(assuming it has an even number of boxes).
10. Prove that noncommutative Schur functions in plactic variables commute pairwise.