Connections between Hessenberg varieties, chromatic quasisymmetric functions, and q-series

Abstract

In many ways, the combinatorics of symmetric functions can help us understand how other mathematical objects behave. For example, the Schur functions encode information about symmetric group representations as well as intersection theory in the Grassmannian. In this dissertation, we investigate connections between chromatic symmetric functions and Hessenberg varieties, and how each one can elevate the understanding of the other. Stanley and Stembridge conjectured in 1993 that the chromatic symmetric functions for unit interval graphs expanded with positive coefficients in the basis of elementary symmetric functions. This conjecture has been proved directly for several families of graphs, and a recent full proof was proposed by Hikita in 2024. Geometrically, this corresponds to showing that the cohomology rings of Hessenberg varieties, acted on by the symmetric group, decompose into permutation modules. Again, this result has been proven for several families of Hessenberg varieties, but in general remains open. For the Hessenberg function h=(h(1),n,...,n), the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. In this dissertation, we define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function h' = ((n-1)n-m, nm), and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of P-tableaux, motivated by the work of Gasharov, and use P-tableaux to find a new formula for the Poincare polynomial of these Hessenberg varieties. Another open problem is to determine conditions for which the chromatic quasisymmetric function is symmetric. In 2024, Aliniaeifard et al. showed that if P is a path graph, then X_P(x;q) is symmetric if and only if the vertices of P are labeled in increasing or decreasing order, and if S is a star graph, then X_S(x;q) is not symmetric. In this dissertation, we extend this result, and show that if G is any tree, other than the path graph given above, then X_G(x;q) is not symmetric. We also construct a family of graphs called mixed mountain graphs, which are similar to unit interval graphs, and show that their chromatic quasisymmetric functions are symmetric.