TITLE: What are the most important open problems in algebraic combinatorics?
QUESTION [8 upvotes]: I have seen the paper of Stanley http://math.mit.edu/~rstan/pubs/pubfiles/116.pdf
but it is quite old and many of the problems are solved. I would like to know the two or three biggest open problems in algebraic combinatorics.

REPLY [2 votes]: One of the oldest standing open problems in algebraic combinatorics is Foulkes' conjecture; for some history and nice reformulations of the problem, see
On Foulkes' conjecture 
by William F. Doran IV in Journal of Pure and Applied Algebra (August 1998), 130 (1), pg. 85-98.