On K-theoretic polynomials and the chromatic symmetric function

Abstract

This thesis explores various problems related to polynomials from combinatorial K-theory and/or to the chromatic symmetric function. We prove four main results:

1. Two alternating sum formulas involving K-theoretic polynomials, conjectured by Monical, Pechenik, and Searles (2021). 2. The fact that certain properties of a graph can be recovered from its Kromatic symmetric function. 3. A power sum expansion for the Kromatic symmetric function, which we show has integer coefficients. 4. Formulas for certain pieces of the chromatic symmetric homology for star graphs.

The organization is as follows:

In Chapter 1, we introduce background on symmetric functions and K-theory. The basic idea of K-theory is to deform cohomology rings by introducing an extra parameter β, often set to -1.

In Chapter 2, we prove an alternating sum conjecture of Monical, Pechenik, and Searles (2021) concerning four different sets of K-theoretic polynomials: the Lascoux atoms, the kaons, the quasiLascoux polynomials, and the glide polynomials. Monical, Pechenik, and Searles (2021) proved that the coefficients in the kaon expansion of the Lascoux atoms and the coefficients in the glide expansion of the quasiLascoux polynomials are monomials in β. We prove their conjecture that for β = -1, the sum of the coefficients in each of these expansions is always either 0 or 1.

In Chapter 3, we give background on the chromatic symmetric function (introduced by Stanley (1995)), and its K-analogue, the Kromatic symmetric function (introduced by Crew, Pechenik and Spirkl (2023)). A proper coloring of a graph is a way of assigning a color to each vertex such that adjacent vertices always receive different colors. The chromatic symmetric function is computed by assigning a variable to each color then summing the monomials for all proper colorings of G, where in each monomial, the exponent of each variable is the number of times that color is used. The Kromatic symmetric function is defined similarly except that the monomials correspond to proper set colorings, meaning each vertex is assigned a nonempty set of colors such that adjacent vertices have non-overlapping color sets.

In Chapter 4, we study a question posed by Crew, Pechenik, and Spirkl (2023) about how much information about a graph can be recovered from its Kromatic symmetric function. It is known to be a stronger graph invariant than the chromatic symmetric function, and we conjecture that it is in fact a complete invariant that distinguishes all graphs. As evidence toward this conjecture, we prove that the number of copies in G of certain induced subgraphs can be recovered from its Kromatic symmetric function.

In Chapter 5, we give a formula for the expansion of the Kromatic symmetric function using a K-analogue for the power sum basis of the ring of symmetric functions. We show that this expansion has all integer coefficients, as conjectured by Crew, Pechenik, and Spirkl (2023).

In Chapter 6, we study the chromatic symmetric homology, which is a categorification of the chromatic symmetric function introduced by Sazdanovic and Yip (2018). We compute the multiplicity of certain Specht modules in in the case where G is a star graph, confirming and extending some conjectures of Chandler, Sazdanovic, Stella, and Yip (2023).