Chromatic symmetric functions via the group algebra of $S_n$

Abstract

We prove some Schur positivity results for the chromatic symmetric function $X_G$ of a (hyper)graph $G$, using connections to the group algebra of the symmetric group. The first such connection works for (hyper)forests $F$: we describe the Schur coefficients of $X_F$ in terms of eigenvalues of a product of Hermitian idempotents in the group algebra, one factor for each edge (a more general formula of similar shape holds for all chordal graphs). Our main application of this technique is to prove a conjecture of Taylor on the Schur positivity of certain $X_F$, which implies Schur positivity of the formal group laws associated to various combinatorial generating functions. We also introduce the pointed chromatic symmetric function $X_{G,v}$ associated to a rooted graph $(G,v)$. We prove that if $X_{G,v}$ and $X_{H,w}$ are positive in the generalized Schur basis of Strahov, then the chromatic symmetric function of the wedge sum of $(G,v)$ and $(H,w)$ is Schur positive.

Figures (1)

• Figure 1: The trees with at most 10 vertices whose chromatic operator is positive semistable, not including the paths.

Theorems & Definitions (86)

• Definition 1.1: stanley-chromatic
• Definition 1.2
• Lemma 1.3: cf. Lemma \ref{['lem:trace']}
• Theorem 1.4: cf. Theorem \ref{['thm:chromatic-operator']}
• Example 1.5
• Example 1.6
• Lemma 2.1: denes
• proof
• Corollary 2.2
• Theorem 2.3: stanley-chromatic, Theorem 2.5