The "q=-1" phenomenon via homology concentration

A few years ago, Dennis Stanton asked for a topological explanation for the fact that the Gaussian polynomial evaluated at -1 yields the number of self-complementary partitions in a rectangle, and for related instances of Stembridge's q=-1 phenomenon. In joint work with John Shareshian and Dennis Stanton, we provide such an explanation by introducing chain complexes whose "face numbers" are the coefficients in the Gaussian polynomial and whose homology is concentrated in even dimensions. These complexes make sense in more generality, but do not always have such homology concentration.

As time permits, I will discuss a short, topological proof that these complexes are acyclic whenever they are odd dimensional and a Morse matching lemma that is the main ingredient in the aforementioned homology concentration result. I will also briefly discuss a related chain complex whose "face numbers" count partitions in a 3-dimensional box and whose homology is again concentrated in dimensions all of the same parity, with homology basis indexed by semistandard domino tableaux of rectangular shape.