Abstract
Let $P$ be a poset, $\mathrm{inc}(P)$ its incomparability graph, and $X_{\mathrm{inc}(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in Adv. Math., 111 (1995) pp.166–194. Let $\omega$ be the standard involution on symmetric functions. We express coefficients of $X_{\mathrm{inc}(P)}$ and $\omega X_{\mathrm{inc}(P)}$ as character evaluations to obtain simple combinatorial interpretations of the power sum and monomial expansions of $\omega X_{\mathrm{inc}(P)}$ which hold for all posets $P$. Consequences include new combinatorial interpretations of the permanent, induced trivial character immanants, and power sum immanants of totally nonnegative matrices, and of the sum of elementary coefficients in the Shareshian-Wachs chromatic quasisymmetric function $X_{\mathrm{inc}(P),q}$ when $P$ is a unit interval order.
Highlights
The Frobenius isomorphism from the space Tn of symmetric group traces to the space Λn of homogeneous degree-n symmetric functions, Frob : Tn → Λn θ(w)pctype(w), (1)
One may use the inverse of the Frobenius isomorphism to study symmetric functions, such as Stanley’s chromatic symmetric functions XG [30], in terms of Snclass functions
We show that the expansion of any homogeneous degree-n symmetric function in any standard basis of Λn yields coefficients which are trace evaluations
Summary
The Frobenius isomorphism from the space Tn of symmetric group traces to the space Λn of homogeneous degree-n symmetric functions, Frob : Tn → Λn. We interpret “hook” irreducible character immanants in Theorem 27, induced trivial character immanants in Theorem 31, and power sum immanants in Theorem 32 These results include two new interpretations (55) – (56) of the permanent of a totally nonnegative matrix and play an important role in the evaluation of hyperoctahedral group characters at elements of the type-BC Kazhdan-Lusztig basis [29].
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