Signed Chromatic Quasisymmetric Invariant

• The signed chromatic quasisymmetric invariant is defined for directed signed graphs, extending traditional chromatic symmetric functions by incorporating a proper coloring weighted by an ascent statistic.
• It organizes colorings via signed-graphic hyperplane arrangements, linking proper colorings with chamber decompositions and refined reciprocity theorems.
• The invariant takes values in the algebra SQSym, offering a framework that connects signed graph theory with quasisymmetric functions through explicit fundamental expansions.

Searching arXiv for papers on signed chromatic quasisymmetric invariants and closely related work. The signed chromatic quasisymmetric invariant is a chromatic generating function for a directed signed graph that records proper colorings together with an ascent statistic derived from Zaslavsky’s compatibility condition for signed orientations. For a directed signed graph $\vec{\Sigma}=(V,E,\sigma,\tau)$, it is defined by

$$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$

where the variables are indexed by $\mathbb{Z}$, $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$, and $\mathrm{asc}(\kappa)$ counts the edges on which the coloring is not compatible with the chosen signed orientation. The construction extends Stanley’s chromatic symmetric function, the Shareshian–Wachs–Ellzey chromatic quasisymmetric function, and the earlier chromatic signed-symmetric function of a signed graph; its natural target is the algebra $SQSym$ of signed quasisymmetric functions (Aval et al., 27 Aug 2025).

1. Definition on directed signed graphs

A signed graph is $\Sigma=(V,E,\sigma)$, where $\sigma:E\to\{+,-\}$. In the directed setting, each edge is equipped with a bidirection $\tau$ satisfying

$\sigma(e)=-\tau(u,e)\tau(v,e)$

for $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$0, with $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$1. A proper coloring is a map $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$2 such that

$$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$3

for every edge $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$4. Thus positive edges impose $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$5, while negative edges impose $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$6. This is the signed analogue of the ordinary proper-coloring condition, but with the color set expanded from positive integers to all integers (Aval et al., 27 Aug 2025).

The ascent statistic is defined from Zaslavsky’s compatibility relation. For a coloring $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$7 and an oriented signed edge $$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$8, compatibility means

$$X_{\vec{\Sigma}}(x;t)=\sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)}x^\kappa,$$9

An ascent is an edge on which this inequality is violated, equivalently one satisfying

$\mathbb{Z}$0

The invariant $\mathbb{Z}$1 is obtained by weighting each proper coloring by $\mathbb{Z}$2. When $\mathbb{Z}$3, the statistic is forgotten and one recovers the previously studied signed chromatic symmetric function of Kuroda–Tsujie and related work (Aval et al., 27 Aug 2025).

2. Hyperplane arrangements and chamber decompositions

The invariant is organized by the signed-graphic hyperplane arrangement associated with $\mathbb{Z}$4. For a signed graph on vertices $\mathbb{Z}$5, the arrangement is

$\mathbb{Z}$6

Proper colorings correspond to integer points outside this arrangement, and acyclic orientations in the sense of Zaslavsky correspond to chambers of $\mathbb{Z}$7. The signed chromatic quasisymmetric invariant can therefore be regrouped chamberwise as

$\mathbb{Z}$8

where $\mathbb{Z}$9 is constant on each chamber (Aval et al., 27 Aug 2025).

This chamber description refines the earlier arrangement-theoretic definition of the chromatic signed-symmetric function

$x^\kappa=\prod_{v\in V}x_{\kappa(v)}$0

for a signed graph $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$1. In that earlier theory, the arrangement chambers and their closures also support a reciprocity theorem

$x^\kappa=\prod_{v\in V}x_{\kappa(v)}$2

with $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$3 defined by summing over integer points in chamber closures. This extends Stanley’s reciprocity from ordinary graphic arrangements to signed-graphic arrangements (Kuroda et al., 2021).

3. The target algebra $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$4

The signed chromatic quasisymmetric invariant takes values in $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$5, where $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$6 is the algebra of signed quasisymmetric functions. A formal power series $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$7, with variables $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$8, is signed quasisymmetric if it has bounded degree and if its coefficients depend only on the pattern of exponents on $x^\kappa=\prod_{v\in V}x_{\kappa(v)}$9, the positive-index variables, and the negative-index variables, not on the specific indices. Concretely, for any increasing positive index sequences $\mathrm{asc}(\kappa)$0 and $\mathrm{asc}(\kappa)$1, any bicomposition $\mathrm{asc}(\kappa)$2, and $\mathrm{asc}(\kappa)$3,

$\mathrm{asc}(\kappa)$4

This is the signed analogue of ordinary quasisymmetry (Aval et al., 27 Aug 2025).

The monomial basis is indexed by pairs $\mathrm{asc}(\kappa)$5, where $\mathrm{asc}(\kappa)$6 is a bicomposition, that is, a pair of integer vectors $\mathrm{asc}(\kappa)$7 with no column equal to $\mathrm{asc}(\kappa)$8. The corresponding basis element is

$\mathrm{asc}(\kappa)$9

Multiplication is given by a quasi-shuffle product on bicompositions,

$SQSym$0

and the coproduct is defined by deconcatenation. The algebra is graded, and its Hilbert series is

$SQSym$1

The space is invariant under the action of the signed symmetric group, and the paper explicitly presents it as a type $SQSym$2 extension of $SQSym$3 (Aval et al., 27 Aug 2025).

4. Fundamental expansions and structural properties

The fundamental family for $SQSym$4 is built from signed $SQSym$5-partitions on signed chains. These functions generalize the classical fundamental basis of $SQSym$6, and the paper extracts a basis using “minimal chains” together with an explicit bijection to bicompositions. In this basis the signed chromatic quasisymmetric invariant admits an explicit expansion: $SQSym$7 where $SQSym$8 is the signed symmetric group, $SQSym$9 is the fundamental function indexed by the descent set and sign word of $\Sigma=(V,E,\sigma)$0, and $\Sigma=(V,E,\sigma)$1 is the corresponding inversion statistic (Aval et al., 27 Aug 2025).

Several formal properties parallel the ordinary chromatic theory. The invariant is multiplicative on disjoint unions: $\Sigma=(V,E,\sigma)$2 It specializes to the signed chromatic symmetric function at $\Sigma=(V,E,\sigma)$3, and setting $\Sigma=(V,E,\sigma)$4 yields the zero-free version. However, the output is usually not signed symmetric: it generally lies in $\Sigma=(V,E,\sigma)$5, not in the algebra $\Sigma=(V,E,\sigma)$6 of signed symmetric functions, except for specific classes of graphs treated in the symmetry results of the paper (Aval et al., 27 Aug 2025).

5. Relation to earlier chromatic invariants

The new invariant sits at the intersection of two established lines of generalization. In the unsigned direction, Stanley’s chromatic symmetric function $\Sigma=(V,E,\sigma)$7 was refined by Shareshian, Wachs, and Ellzey to a chromatic quasisymmetric function for directed graphs, and the signed construction is designed to recover that theory when all edge signs are positive and the orientation is the standard unsigned one. In the signed direction, the specialization $\Sigma=(V,E,\sigma)$8 recovers the chromatic signed-symmetric function, whose ordinary-graph specialization is obtained by the projection $\Sigma=(V,E,\sigma)$9 sending $\sigma:E\to\{+,-\}$0 for $\sigma:E\to\{+,-\}$1 and retaining $\sigma:E\to\{+,-\}$2 for $\sigma:E\to\{+,-\}$3; for a positive simple graph $\sigma:E\to\{+,-\}$4, one has $\sigma:E\to\{+,-\}$5 (Aval et al., 27 Aug 2025, Kuroda et al., 2021).

A separate unsigned refinement is the $\sigma:E\to\{+,-\}$6-chromatic quasisymmetric function $\sigma:E\to\{+,-\}$7, defined by summing over $\sigma:E\to\{+,-\}$8-balanced colorings of a simple graph. It satisfies $\sigma:E\to\{+,-\}$9, is positive in the fundamental basis, and gives rise to a generalized chromatic polynomial $\tau$0 whose negative evaluations generalize Stanley’s theorem relating $\tau$1 to acyclic orientations (Humpert, 2010). This places the signed chromatic quasisymmetric invariant within a broader program in which chromatic generating functions are refined by orientation data, although the signed-graph setting uses signed edge constraints and signed quasisymmetry rather than $\tau$2-balancedness.

At the $\tau$3 level, the earlier signed-symmetric theory also investigated distinguishing power. The chromatic signed-symmetric function was shown to distinguish signed paths up to $\tau$4 vertices computationally, and it was proved that for signed paths indexed by compositions of length at most $\tau$5, or by unimodal compositions, equality of the functions implies isomorphism of the signed paths. The same work also showed that a signed tree is connected if and only if its chromatic signed-symmetric function is irreducible in the ring of signed-symmetric functions (Kuroda et al., 2021). A plausible implication is that analogous structural questions for the quasisymmetric refinement may become central in the signed setting.

6. Terminological boundaries and adjacent signed theories

The phrase “signed” appears in this area in more than one sense. In the signed chromatic quasisymmetric invariant, “signed” refers primarily to signed graphs, signed orientations, and the signed quasisymmetric target algebra $\tau$6. In a different line of work, “signed” refers to coefficient signs in basis expansions. For the chromatic quasisymmetric function $\tau$7 of a graph, a signed $\tau$8-expansion was proved for any natural unit interval graph: $\tau$9 and a sign-reversing involution then yields explicit positive $\sigma(e)=-\tau(u,e)\tau(v,e)$0-expansions for $\sigma(e)=-\tau(u,e)\tau(v,e)$1-chains and almost-$\sigma(e)=-\tau(u,e)\tau(v,e)$2-chains. The same paper gives a signed $\sigma(e)=-\tau(u,e)\tau(v,e)$3-expansion for arbitrary graphs using no-broken-circuit trees, with the claw graph exhibiting negative terms and thus failure of $\sigma(e)=-\tau(u,e)\tau(v,e)$4-positivity (Tom, 2023). This is a different notion of “signed” from the signed-graph framework, even though both are attached to chromatic quasisymmetric constructions.

A further structural boundary is supplied by kernel results for chromatic quasisymmetric maps. For graphs, the kernel of the noncommutative chromatic quasisymmetric map is spanned by modular relations, and in the commutative case one adds isomorphism relations. For hypergraphic polytopes, the kernel is generated by simple relations and generalized modular relations, and the image of the noncommutative map is the singleton commuting space $\sigma(e)=-\tau(u,e)\tau(v,e)$5 (Penaguiao, 2018). That work does not construct signed chromatic quasisymmetric invariants, but it states that any signed or “signed quasisymmetric” invariant that factors through the universal chromatic morphism must vanish on the same modular-type obstructions. This suggests that the signed chromatic quasisymmetric invariant is not only a new enumerator but also part of a tightly constrained universal chromatic framework.