Chromatic symmetric functions from the modular law
Introduction
The chromatic polynomial can be characterized as the unique function that has the following three properties.1
- (A)
It satisfies the deletion-contraction recurrence, for every edge .
- (B)
It is multiplicative, .
- (C)
It has values at complete graphs given by .
The chromatic polynomial of a graph admits a symmetric function generalization introduced by Stanley in [31]. Given a graph G it is defined as where the sum runs through all proper colorings of the vertices and . A coloring κ is proper if whenever v and are adjacent. If Λ is the algebra of symmetric functions, it turns out that csf is a function from Graphs to Λ. This function is multiplicative and its values at complete graphs are given by (where is the elementary symmetric function of degree n). However, it does not satisfy the deletion-contraction recurrence, one simple reason being that the chromatic symmetric function is homogeneous of degree equal to the number of vertices of G.
In this paper we will restrict ourselves to indifference graphs, i.e., graphs whose set of vertices can be identified with and such that if is an edge with , then and are also edges for every k such that . This class of graphs, restrictive as it might look, ends up having deep relations with geometry and representation theory, see, for example, [8], [20] and [1].
Indifference graphs can be naturally associated with Hessenberg functions and Dyck paths (see Fig. 1A). A Hessenberg function is a non-decreasing function such that for every . The graph associated to h is the graph with vertex set and set of edges . All indifference graphs arise from Hessenberg functions. To each Hessenberg function there is an associated Dyck path, which is the unique path with north steps before the i-th east step. We usually denote a Hessenberg function by the n-tuple of its values or by the word in n (north step) and e (east step) corresponding to its associated Dyck path. We denote by the set of Dyck paths, which will be identified with the set of Hessenberg functions and with the set of indifference graphs. In the rest of the introduction by graph we will always mean an indifference graph.
When G is the graph associated to h, we can recover the chromatic polynomial of G by a differential operator in the Weyl algebra . The operator is obtained from h by replacing each east step with ∂ and each north step with x. For example, if , then . It is not hard to check that the chromatic polynomial of G satisfies the following equality
Moreover, with this interpretation, the deletion-contraction recurrence (when applied to edges of G that correspond to corners in h) is, essentially, the well known formula , which gives See Fig. 1B.
As pointed out before, the fact that Formula (1a) is not homogeneous is one of the reasons for the deletion-contraction recurrence not holding for the chromatic symmetric function. On the other hand, it is not hard to find homogeneous relations for ∂ and x. For example, one can simply consider and . Explicitly this gives In particular, if , and are graphs associated to Dyck paths , and such that and are obtained from by replacing a subpath nen with enn and nne, respectively, then (see Fig. 1C)
Similarly, the same holds if and are obtained from by replacing a subpath ene with een and nee, respectively.
One can actually replace Property (A) (the deletion-contraction recurrence) with the recurrence in Equation (1c), and these properties will still characterize the chromatic polynomial for indifference graphs. In other words, the restriction of χ to the set of indifference graphs is the unique function that has the following three properties.
- (A')
Whenever , , and are graphs associated to Dyck paths , , and such that and are obtained from by replacing a subpath nen with enn and nne, respectively, or by replacing a subpath ene with een and nee, respectively, then .
- (B)
It is multiplicative, .
- (C)
It has values at complete graphs given by .
Given a connected graph , which is not complete, we have that its associated Dyck path must end with for some positive integers . If we define and by replacing the subpath of with and , respectively, we have that both and are Dyck paths (because is connected) and we can apply Equation (1e). This process eventually ends, since b will increase at each step. Of course, the chromatic polynomial of indifference graphs can be easily computed directly, but the above recurrence is useful, for instance, if one wants to write the chromatic polynomial in certain bases as in [7].
The idea is to repeat this process for the chromatic symmetric function. Actually, we will work with the chromatic quasisymmetric function introduced by Shareshian and Wachs in [29]. For a graph G with set of vertices , the chromatic quasisymmetric function is defined as where the sum runs through all proper colorings of G and is the number of ascents of the coloring κ. When G is an indifference graph, we have that is actually symmetric, so we will think of as function , where is the algebra of symmetric functions with coefficients in .
With a little experimentation one can see that Equation (1c) does not hold in general for the chromatic symmetric function of indifference graphs. However, it still holds if we add some extra assumptions on , and , which are summarized in Definition 2.1. The purpose of this article is to determine when Equation (1e) lifts to the chromatic symmetric function (see Proposition 2.4). Moreover, we prove that there are enough of these liftings to fully characterize Stanley's chromatic symmetric function on indifference graphs as stated in the following theorem. As usual, we define and .
Theorem 1.1 The function is the unique function that has the following three properties. It satisfies the modular law, as in Definition 2.1. It is multiplicative, . It has values at complete graphs given by .
Actually we prove a more general result. For an indifference graph G with vertex set , we denote by its transposed graph, that is, we relabel the vertices of G via .
Theorem 1.2 Let A be a -algebra and let be a function that satisfies the modular law, as in Definition 2.1. Then f is determined by its values at the disjoint ordered union of complete graphs and these values are independent of the order in which the union is taken. Moreover, we have that for every indifference graph G.
Our proof of Theorem 1.2 is constructive. We find an algorithm (Algorithm 2.8) based exclusively on the modular law. This algorithm was implemented in SAGE and is available upon request.
Every function that satisfies the modular law is intimately related with the chromatic symmetric function, as seen in Corollary 3.2. It would be interesting to find functions with combinatorial interpretations that satisfy the modular law. In [2] the authors define one such function enumerating increasing spanning forests and use it to sharpen the description of the e-coefficients of unicellular LLT polynomials conjectured in [4] and [16].
When G is an indifference graph whose complement is bipartite, we observe in Remark 2.12 that the algorithm reduces to a planar network. In this situation, we show that can be computed in terms of q-hit numbers. This is a q-analogue of the Stanley-Stembridge combinatorial formula [32, Theorem 4.3]. We recall that the partition associated to h is given by .
Theorem 1.3 Let h be an Hessenberg function whose associated indifference graph has bipartite complement and let λ be the partition associated to h. If , then where are the Garsia-Remmel q-deformation of hit numbers, that is, it enumerates weighted rook placements on board with exactly j rooks on the Young diagram of λ. Moreover, we have that is e-unimodal.
In the last section we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function. In the breakthrough work [25] it is proved that the chromatic polynomial of a graph is log-concave. This result was later generalized to matroids in [3]. We supply some evidence supporting the logarithmic concavity of the e-coefficients of for .
We point out that several analogues of deletion-contraction exist for the chromatic symmetric function (or some closely related symmetric functions). A non-commutative chromatic symmetric function is defined in [17] which satisfies a deletion-contraction recurrence. In [19] and [27], a modular law for the chromatic symmetric function is introduced for any graph. When restricted to indifference graphs it is the analogue of Equation (1c). In [26], this relation is found for the closely related unicellular LLT polynomial. A chromatic symmetric function for weighted graphs is defined in [12] which satisfies a deletion-contraction recurrence when one considers contractions of weighted graphs. Other linear relations in various settings can be found in [27], [24], [13], and [6].
It is also worth mentioning that, for unicellular LLT polynomials, the analogy with differential operators was made precise in [9]. They defined operators and which play the roles of x and ∂ in the discussion above, and proved that the unicellular LLT polynomial associated to a Dyck path h can be computed as where is the operator obtained from h by replacing each east step with and each north step with .
In the recent paper [6] it is demonstrated how to obtain the chromatic symmetric function from a similar set of relations (see [6, Corollary 6.16]). Actually, one of the relations used in [6] is contained in the modular law used here. The authors also consider a bounce relation on Schröeder paths that implies the modular law and other relations.
Acknowledgments
We thank the anonymous referees for the careful reading of the paper and constructive suggestions.
Section snippets
The algorithm
Our main goal in this section is to prove Theorem 1.2. We first need some notation. We denote the set of Dyck paths by and by the set of Dyck paths of size n. We will also think of as the set of Hessenberg functions via the identification between Dyck paths and Hessenberg functions. There is a (non-commutative) product on the set given by concatenation of Dyck paths, while on Hessenberg functions the product of with is the function given by
The chromatic quasisymmetric function
We begin by recalling that the chromatic quasisymmetric function does indeed satisfy the modular law. This is already well known in the literature (see for instance [19, Proposition 3.1], [26, Theorem 3.4], [24, Theorem 3.1] and [4, Corollary 20 and Proposition 23]). In particular, if we write we have that the functions also satisfy the modular law. The following result proves that every other function is actually a -linear combination of .
Rook placements and q-hit numbers
In this section, we prove a q-analogue of [32, Theorem 4.3]. We mention that rook placements also appear in relation with terms in the e-expansion of , see [5]. First we must recall the definition of the q-analogue of hit numbers introduced in [15].
Let be the board and let λ be a partition such that its Young diagram fits in , that is . To keep the notation consistent with the last sections, we will number the rows of from bottom to top. This means that the Young
Logarithmic concavity
In this section we discuss the logarithmic concavity of the coefficients of . We need a few definitions first. We say that a polynomial is log-concave with no internal zeros if it is unimodal and for every j. For simplicity, in what follows we will write log-concave instead of log-concave with no internal zeros. It is a well know fact that the product of two log-concave polynomials still is a log-concave polynomial (see [30]). On the other hand, the sum of two
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