Mathematics Colloquia and Seminars

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In this talk, we present the Kromatic symmetric function, a K-theoretic analogue of X_G formed by assigning a set of colors to each vertex such that adjacent vertices share no color in common. Our main result is that the Kromatic symmetric function of a claw-free incomparability graph has a positive expansion into the basis of Grothendieck functions, themselves a K-theoretic analogue of Schur functions. Furthermore, the Grothendieck coefficients enumerate certain tableaux that contain both poset elements and positive integers in a way that combines ideas of Gasharov (1996) and Lenart (2000). Thus, Gasharov's classic result lifts to this function, strongly implying that there should be a shared geometric interpretation for these functions. On the other hand, we show that all natural generalizations of the Stanley-Stembridge conjecture to the Kromatic symmetric function are fundamentally and unfixably false, suggesting that perhaps this more general conjecture is not topological in nature.