Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

Kabluchko Theorem 1.2 gives exactly the number of arrangement chambers whose metric projection from a generic point lies in a $k$-face, with the number equal to the absolute $k$-th characteristic coefficient.

• Theorem 2.5Generic-point projection dimensions encode the characteristic-polynomial coefficients of every real arrangement.

Lofano–Paolini Corollary 5.14 independently counts chambers by projection-face codimension. The standard Poincaré/characteristic conversion gives the focal dimension-indexed formula.

• Theorem 2.5Generic-point projection dimensions encode the characteristic-polynomial coefficients of every real arrangement.

The target labels graphical-arrangement regions by acyclic orientations and converts the arrangement characteristic polynomial to the graph chromatic polynomial. Stanley Proposition 2.5 and Theorem 2.7 are the exact external inputs; the projection/source-component arguments themselves are internal.

• Theorem 4.4For a graphical arrangement and strongly separated positive vector, projection dimension equals source-component count.
• Theorem 5.2For a natural unit interval graph, projection dimension is right-to-left minima among permutations with only graph descents.
• Corollary 5.3Derives the factorized characteristic polynomial of a natural unit interval graphical arrangement.

The type-B projection calculation represents flats by sign-stable partitions and zero blocks. Reiner verifies that intersection-lattice model. The ordering of blocks into faces is an elementary internal extension, not a statement supplied by Reiner.

• Theorem 6.17Claims that type-B projection dimension equals signed source-component count.
• Lemma 6.24Supplies the canonical signed-source face and a prefix-containment property for every other v-face.

The signed-source coefficient interpretation counts arrangement regions by symmetric acyclic orientations. Zaslavsky Theorem 4.4 gives the region/orientation correspondence for signed graphs, after the target's symmetric graph is represented as a signed multigraph when both signs occur.

• Theorem 6.8Interprets signed chromatic coefficients by signed source components.

Theorem 6.8 combines the arrangement coefficient count with the equality between signed chromatic and signed-graphic arrangement characteristic polynomials. Zaslavsky Lemma 4.3 is that equality.

• Theorem 6.8Interprets signed chromatic coefficients by signed source components.

Kabluchko Proposition 3.1 imports the generic central-arrangement chamber-section count and the arbitrary-subspace comparison from KVZ Theorem 3.3 and Lemma 3.5. These yield the polar-cone multiplicity used throughout the proof of Theorem 1.6.

• Theorem 1.2Exact affine-arrangement projection-coefficient theorem imported as focal Theorem 2.5.

Kabluchko's coefficient identification uses the Whitney subset formula and Zaslavsky region formula in Stanley's chapter. These convert the local central-arrangement counts into the coefficients of the original affine arrangement.

• Theorem 1.2Exact affine-arrangement projection-coefficient theorem imported as focal Theorem 2.5.

For finite arrangements, Lofano–Paolini prove minimality because the number of critical cells equals the sum of Betti numbers. Orlik–Solomon supplies the cohomological Poincaré formula behind that equality; the focal use is finite and central.

• Corollary 5.14Counts chambers by the codimension of the metric-projection face, yielding the Poincaré coefficients.

The focal theorem indexes by face dimension and characteristic coefficients, whereas Corollary 5.14 indexes by face codimension and Poincaré coefficients. The standard conversion cited to Orlik–Terao supplies the exact change of variables.

• Corollary 5.14Counts chambers by the codimension of the metric-projection face, yielding the Poincaré coefficients.

The finite minimality count equates the number of chambers with the total Betti number. Zaslavsky supplies the chamber side of that equality, paired with Orlik–Solomon's Poincaré formula.

• Corollary 5.14Counts chambers by the codimension of the metric-projection face, yielding the Poincaré coefficients.

Orlik–Solomon Theorem 5.2 uses Brieskorn's Lemma 5 to identify the generated logarithmic forms with the cohomology of each restricted arrangement complement.

• Theorem 5.2 and formula (1.4)Identifies hyperplane-complement cohomology with the Orlik–Solomon algebra and its Möbius Poincaré polynomial.

The proof of KVZ Lemma 3.5 needs general-position subspaces to be dense in the Grassmannian. It cites the Haar-null exceptional-set result in Schneider–Weil Lemma 13.2.1.

• Theorem 3.3 and Lemma 3.5Counts generic chamber sections and compares arbitrary sections, giving Kabluchko's polar-cone count.

KVZ Theorem 3.3 applies Zaslavsky's region formula to the arrangement induced on a general-position subspace, after computing its characteristic polynomial in Lemma 3.1.

• Theorem 3.3 and Lemma 3.5Counts generic chamber sections and compares arbitrary sections, giving Kabluchko's polar-cone count.

The first proof explicitly invokes Zaslavsky 1982 Theorem 8B.1 to identify the incidence vectors' dependence matroid with $G(\Sigma)$; Theorems 3.3, 4.1, and duality then yield Theorem 4.4. The marked Theorem 8B.1 branch is not affected by the known erratum.

• Theorem 4.4Bijection between signed-graphic arrangement regions and acyclic signed orientations.

Lemma 4.3 compares two subset expansions through Lemma 3.11. That lemma is vector-space duality applied to Theorem 3.5, whose proof is explicitly deferred to Zaslavsky 1982 Theorem 8B.1.

• Lemma 4.3Equality of the signed chromatic polynomial and the signed-graphic hyperplane-arrangement characteristic polynomial.