Updated information on "Positivity problems and conjectures in algebraic combinatorics"

• A solution to Problem 2 for convex polytopes was given by Kalle Karu, with a strengthening by Paul Bressler and Valery A. Lunts.
  
  
• Some progress on Problem 4 is due to Victor Reiner and his colloborators Naichung Conan Leung, Dennis Stanton, and Volkmar Welker. The paper with Welker is also related to Problem 20.
  
  
• The conjecture following Problem 9 is false as stated. Pavlo Pylyavskyy pointed out that if λ ≥ λ' then it may not be the case that |Π_λ| ≤ |Π_λ'|, the smallest example being λ = (6,2,2,1,1) and λ' = (5,3,1,1,1,1). The conjecture remains open if modified to state that φ_λ has full rank when λ ≥ λ', though this may very well be false. The original conjecture remains open in the case that would imply the Foulkes plethysm conjecture, viz., λ has m parts equal to n, where n ≥ m. However, a counterexample in that case was announced by J. Müller and Max Neunhoeffer, but details have not yet appeared. Update: see the addendum in the paper "Some computations regarding Foulkes' conjecture" by J. Müller and Max Neunhoeffer, Experimental Mathematics 14 (2005), No. 3, 277-283.
  
  
• A solution to the conjecture of Sundaram mentioned shortly after Problem 9 was found by Ben Joseph in April, 2000.
  
  
• A solution to Problem 13 (the nonnegativity of the coefficients of the (q,t)-Kostka polynomials) has been given by Mark Haiman.
  
  
• Problem 23 was solved affirmatively by Maria Chudnowsky and Paul Seymour in 2004.
  
  
• Problem 5 has a negative answer. I don't remember the reference.
  
  
• A negative answer to Problem 6 was found by Art Duval, Caroline Klivans, and Jeremy Martin.
  
  
• A positive answer to Problem 7 was found by Kalle Karu.
  
  
• A negative answer was found by Petter Brändén for general (i.e., not necessarily natural) labelled posets. Subsequently John Stembridge gave a negative answer for naturally labelled posets.
  
  
• Problem 24 was given a positive answer by Petter Brändén, James Haglund, Mirkó Visontai, and David Wagner.
  
  
• Problem 25: update forthcoming.