MTH 786-Q (Spring 2018): PROBLEM ASSIGNMENTS

Problems should be solved primarily on your own. Some "reasonable" collaboration is permitted, but you shouldn't just obtain the solution from another source. Do not hand in a solution that you did not obtain on your own or by collaboration with another student in the course!

• Due Tuesday, January 23. Problems 1, 4, 7, 8.
  
  
• Due Thursday, February 1. Problems 10, 11, 15, 20, 24. For Problem 20(b) you will need some knowledge of enumerative combinatorics that can be found in EC1, Chapter 1.
  
  
• Due Tuesday, February 13. Problems 21, 28, 31, 49, 54. For #54 you may assume the (easy) result Σ_n≥0 C(2n,n)xⁿ = (1-4x)^-1/2, where C(2n,n) denotes a binomial coefficient (since I don't know how to write the usual binomial coefficient notation in html). See EC1, Exercise 8(a).
  
  
• Due Thursday, February 22. Problems 18, 57, 61, 78(a) (for a(m,n) only). For #78(a), all you need to know about c^λ_μν is its definition <s_λ,s_μs_ν>. Bonus. Problem 55. Also hand in the following computational problem: let A be the 3×3 matrix of all 1's. (a) Find the plane partition with at most three rows and columns associated with A as discussed in class. (b) Find the pair (P,Q) of tableaux obtained by applying the dual RSK algorithm to A.
  
  
• Due Tuesday, March 6. Problems 30(a), 53, 66, 67. Also do the following two (easy, I hope) problems.
  (A1) Let λ be a partition, identified with its diagram. Show that the number of ways to add a horizontal strip of size m to λ followed by adding a vertical strip of size n is equal to the number of ways of first adding a vertical strip of size n to λ followed by adding a horizontal strip of size m. (Minus 5 points for not finding a sufficiently elegant solution.)
  (A2) Let φ be the homomorphism from Λ to Λ defined by φ(h_i) = h_i+1 for all i≥1. For all partitions λ such that λ_n≥n (where n is the number of parts of λ) express φ(s_λ) as a linear combination of Schur functions.
  Hint for #30(a). Use the proof of Theorem 7.15.1.
  
  
• March 22: no class
  
  
• Due Tuesday, March 27. Problems 29(a), 46, 63, 84(a), 90(a). Hint for #46. Use RSK. Hint for #90(a). At some point use Exercise 7.82(a).
  Bonus. Let φ be the homomorphism from (A2) of the previous problem set (now corrected). Let f(m) denote the number of partitions λ of m for which φ(s_λ) = s_{λ1+1, λ2+1, ...}. Find a nice product formula for the generating function Σ_m≥0 f(m)x^m.
  
  
• Due Tuesday, April 10 (note postponement). Problems 73(a,d,e,f), 94(a), 98, 108. For #73(f), you need only give the first (easy) proof. The difficulty level of #108 is [2+]. Note Exercise 7.42 errata here. Remark for #73(f). You can use part (b). Another way is to use another Chapter 7 exercise. Hint for #98. In equation (7.20) on page 299, set the first t y-variables equal to one and the rest equal to zero. Differentiate with respect to t, set t=1, and consider only terms of degree n. Also consider the product s_n-kp_k.
  Also hand in: (A3) Let n≥3. Let A_n denote the set of all pairs (i,T), where i is an element of [n] and T is a 2-element subset of [n] not containing i. Thus #A_n = n(n-1)(n-2)/2. The symmetric group S_n acts on A_n in the obvious way by acting on each element separately. Find the decomposition of the character of this action into irreducible characters.
  
  
• Due Thursday, April 12. Hand in the following problem.
  (A4) Let S_n denote the set of all sequences a₁, a₂, ..., a_2n of n 1's and n -1's such that every partial sum is nonnegative. Given such a sequence α, define w(α) to be the sum of all indices i such that a_i=1 and a_i+1=-1. Find a simple product formula for the generating function Σ_α∈Sn q^w(α). (Show that this result is equivalent to a result involving Schur functions.)
  
  
• Thursday, April 19. IN-CLASS QUIZ. The quiz will be open book (EC2 only) and class material (returned homework, personal class notes). No outside material such as other books, internet connection, etc. No phones or computers.