Positive Chamber Complex: Toric & Arithmetic

• Positive chamber complex is a combinatorial structure endowed with a positivity feature that encodes orbit geometry and combinatorial arrangements.
• It employs hyperplane arrangements and graph-theoretic admissibility to partition moment polytopes and elucidate toric orbit closures and secondary fans.
• In arithmetic settings, the framework underpins zeta functions through non-negative gallery weights and transfer operators, linking spectral properties with weighted chamber dynamics.

In current usage around chamber-based combinatorics, a positive chamber complex refers to a chamber structure endowed with a positivity feature rather than to a single universally fixed axiomatic object. One important usage is the cell or chamber structure on the hypersimplex $\Delta_{n,2}$ arising from torus actions of positive complexity on $G_{n,2}$ and on $\mathbb{C}P^{N_2}$, where chambers refine the moment polytope so as to record non-toric orbit geometry. A second usage appears in chamber complexes of groups attached to $\Gamma\backslash \mathcal B$ for $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$, where positivity takes the form of non-negative gallery weights, non-negative chamber transfer operators, and non-negative closed-gallery counts. The underlying chamber-combinatorial language is supplied by the theory of thin chamber complexes and zigzags, in which chambers are facets of a simplicial complex and special gallery-like flag orbits encode global structure (Sergeev, 5 Jun 2026, Hong et al., 29 Dec 2025, Deza et al., 2015).

1. Chamber complexes as the combinatorial substrate

A thin chamber complex is a finite abstract simplicial complex $\Delta$ that is pure of rank $n$ and has the property that every ridge, equivalently every face of rank $n-2$, is contained in precisely two distinct facets. Following Tits, $\Delta$ is a chamber complex when the chamber graph $\Gamma_{n-1}(\Delta)$, whose vertices are the facets and whose edges join adjacent facets, is connected. In this setting the facets are the chambers, and the graphs $G_{n,2}$0 of $G_{n,2}$1-face adjacency are connected for all $G_{n,2}$2 once the chamber graph is connected (Deza et al., 2015).

The flag structure of a thin chamber complex is central. A flag is a maximal chain

$G_{n,2}$3

with $G_{n,2}$4 a $G_{n,2}$5-face. Thinness implies that for each $G_{n,2}$6 there is a unique flag operation $G_{n,2}$7 replacing $G_{n,2}$8 by the unique adjacent $G_{n,2}$9-face incident to all other members of the flag. The zigzag operator is

$\mathbb{C}P^{N_2}$0

and a zigzag is an orbit of $\mathbb{C}P^{N_2}$1 on flags. Its $\mathbb{C}P^{N_2}$2-shadow is the induced cyclic sequence of $\mathbb{C}P^{N_2}$3-faces; any shadow determines the zigzag uniquely. This produces a rank-sensitive notion of gallery that is more rigid than ordinary chamber adjacency (Deza et al., 2015).

In Coxeter complexes, zigzags are controlled by Coxeter elements. If $\mathbb{C}P^{N_2}$4 is a finite Coxeter system of rank $\mathbb{C}P^{N_2}$5 with Coxeter number $\mathbb{C}P^{N_2}$6, then every zigzag in the Coxeter complex $\mathbb{C}P^{N_2}$7 is simple and has length $\mathbb{C}P^{N_2}$8. Generalized zigzags in the simplex $\mathbb{C}P^{N_2}$9, the cross-polytope $\Gamma\backslash \mathcal B$0, the $\Gamma\backslash \mathcal B$1-cell, the icosahedron, and the $\Gamma\backslash \mathcal B$2-cell have lengths equal to the Coxeter numbers of $\Gamma\backslash \mathcal B$3, $\Gamma\backslash \mathcal B$4, $\Gamma\backslash \mathcal B$5, and $\Gamma\backslash \mathcal B$6, $\Gamma\backslash \mathcal B$7, respectively. This establishes the basic chamber-theoretic framework in which later notions of positivity are formulated (Deza et al., 2015).

2. Positive complexity and chamber decomposition of the hypersimplex

For the standard torus action on the Grassmannian $\Gamma\backslash \mathcal B$8, the relevant moment polytope is the hypersimplex

$\Gamma\backslash \mathcal B$9

equivalently the convex hull of the points $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$0 for $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$1. The Plücker embedding

$\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$2

is $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$3-equivariant, and the induced moment map is

$\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$4

By the Atiyah–Guillemin–Sternberg convexity theorem, $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$5. The effective action of

$\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$6

on $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$7 has complexity

$\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$8

which is positive for $\Gamma=\operatorname{PGL}_3(\mathbb F_q[t])$9. In this regime, the moment polytope alone no longer determines the orbit space, because different orbits can have the same moment image and interiors of different orbit closures can intersect (Sergeev, 5 Jun 2026).

The resulting chamber decomposition is built from torus-invariant strata

$\Delta$0

where $\Delta$1 is admissible when $\Delta$2. The associated admissible polytope is

$\Delta$3

and the moment image of $\Delta$4 is the relative interior $\Delta$5. Chambers, in the sense of Goresky–MacPherson used here, are maximal intersections

$\Delta$6

such that $\Delta$7 and $\Delta$8 for $\Delta$9. Their interiors are disjoint and cover $n$0 (Sergeev, 5 Jun 2026).

A hyperplane arrangement $n$1 gives an explicit combinatorial model of this chamber complex: $n$2 The first two families define boundary facets of $n$3, while the third family cuts admissible $n$4-dimensional polytopes in the interior. For each chamber $n$5 lying in the interior of $n$6, there is a face $n$7 of the arrangement such that

$n$8

and conversely any face of the arrangement intersecting $n$9 yields such a chamber. In this sense, the positive chamber complex is literally a chamber decomposition of the moment polytope by an explicit arrangement (Sergeev, 5 Jun 2026).

3. Admissible graphs, toric orbit closures, and secondary fans

The chamber decomposition on $n-2$0 admits a graph-theoretic description. For $n-2$1, the graph $n-2$2 on vertex set $n-2$3 with edge set $n-2$4 is called an admissible graph when $n-2$5 is admissible. The key classification theorem states that $n-2$6 is admissible for the $n-2$7-action on $n-2$8 if and only if there exists a partition

$n-2$9

such that $\Delta$0 exactly when $\Delta$1 and $\Delta$2 lie in different parts $\Delta$3 with $\Delta$4. Equivalently,

$\Delta$5

is a complete multipartite graph together with isolated vertices. This encodes the Plücker-coordinate condition that, after a $\Delta$6 change of basis, the nonzero rows of a matrix representing $\Delta$7 fall into proportionality classes (Sergeev, 5 Jun 2026).

The dimension of an admissible polytope is determined directly from the graph. If

$\Delta$8

then

$\Delta$9

In particular, the $\Gamma_{n-1}(\Delta)$0-dimensional admissible polytopes meeting the interior of $\Gamma_{n-1}(\Delta)$1 correspond exactly to complete bipartite graphs $\Gamma_{n-1}(\Delta)$2 with $\Gamma_{n-1}(\Delta)$3 and $\Gamma_{n-1}(\Delta)$4. Their supporting hyperplanes are

$\Gamma_{n-1}(\Delta)$5

their polytopes are products $\Gamma_{n-1}(\Delta)$6, and the corresponding orbit closures are diffeomorphic to $\Gamma_{n-1}(\Delta)$7 (Sergeev, 5 Jun 2026).

The Plücker ambient space $\Gamma_{n-1}(\Delta)$8 carries a different $\Gamma_{n-1}(\Delta)$9-chamber decomposition. Here any subset $G_{n,2}$00 is allowed, with strata

$G_{n,2}$01

and the same convex hulls $G_{n,2}$02 occur as moment images. For $G_{n,2}$03 this chamber decomposition coincides with the one on $G_{n,2}$04, but for $G_{n,2}$05 it differs. The paper identifies the chamber cones with the GKZ secondary fan $G_{n,2}$06 for

$G_{n,2}$07

and proves that for the $G_{n,2}$08-action on $G_{n,2}$09, the cones spanned by the chambers form the secondary fan of the cone spanned by the vertices of $G_{n,2}$10. A refined arrangement $G_{n,2}$11 describes the maximal-dimensional chambers; for $G_{n,2}$12 it includes, for example, hyperplanes $G_{n,2}$13, part of the braid arrangement. Thus the positive chamber complex in the projective setting is the intersection of a secondary fan with the hypersimplex (Sergeev, 5 Jun 2026).

4. Chamber complexes of groups in the $G_{n,2}$14 building

A different chamber-theoretic positivity arises for the Bruhat–Tits building $G_{n,2}$15 of

$G_{n,2}$16

a $G_{n,2}$17-dimensional affine building of type $G_{n,2}$18. Its vertices are homothety classes $G_{n,2}$19 of rank-$G_{n,2}$20 $G_{n,2}$21-lattices $G_{n,2}$22, its edges correspond to lattice inclusions of index $G_{n,2}$23, and its chambers are $G_{n,2}$24-simplices $G_{n,2}$25 satisfying

$G_{n,2}$26

for suitable representatives, where $G_{n,2}$27. The building is thick of thickness $G_{n,2}$28: each panel is contained in exactly $G_{n,2}$29 chambers. A color function

$G_{n,2}$30

assigns three distinct colors to the vertices of every chamber, and a pointed chamber is an ordered triple of its vertices with a type $G_{n,2}$31 determined by the color differences. Type $G_{n,2}$32 pointed chambers play the primary role (Hong et al., 29 Dec 2025).

A type $G_{n,2}$33 gallery is a sequence of type $G_{n,2}$34 pointed chambers

$G_{n,2}$35

with oriented edge adjacency

$G_{n,2}$36

It is tailless when

$G_{n,2}$37

so it never backtracks across the panel just crossed, and closed when

$G_{n,2}$38

Closed galleries are considered modulo cyclic shift, and primitive classes are those that are not powers of shorter classes. This is the directed chamber system that the source explicitly identifies as what one might call a chamber complex at the combinatorial level (Hong et al., 29 Dec 2025).

The arithmetic quotient

$G_{n,2}$39

is a standard non-uniform complex. It has finite volume in the sense of $G_{n,2}$40-invariant measures but is not compact; combinatorially it has a finite core and cuspidal directions where chambers go off to infinity. Because $G_{n,2}$41 is non-cocompact, adjacency in the quotient does not lift uniquely. The paper therefore introduces weights

$G_{n,2}$42

and hence well-defined weights $G_{n,2}$43 between pointed chambers of $G_{n,2}$44. An admissible gallery in $G_{n,2}$45 is a projection of a tailless gallery in $G_{n,2}$46, and its weight is the product of the step weights. Positivity here is literal: the weights are non-negative integers encoding multiplicities of lifts (Hong et al., 29 Dec 2025).

5. Chamber zeta functions, transfer operators, and weighted counting

The chamber zeta function in the arithmetic $G_{n,2}$47 setting is defined by an Euler product over primitive type $G_{n,2}$48 admissible closed gallery classes, in direct analogy with the Ihara–Bass zeta function for graphs, but with chambers replacing edges and tailless chamber galleries replacing non-backtracking cycles. To connect this Euler product with spectral data, the paper introduces the chamber transfer operator

$G_{n,2}$49

where $G_{n,2}$50 is the set of type $G_{n,2}$51 pointed chambers in $G_{n,2}$52. Its matrix entries are $G_{n,2}$53. The operator is a non-negative, locally finite infinite matrix encoding weighted non-backtracking chamber adjacency, and its powers $G_{n,2}$54 represent weighted walks of length $G_{n,2}$55 (Hong et al., 29 Dec 2025).

A key trace formula identifies powers of $G_{n,2}$56 with weighted closed galleries: $G_{n,2}$57 where $G_{n,2}$58 is the primitive class underlying $G_{n,2}$59. A crucial finiteness lemma states that for any positive integer $G_{n,2}$60, there are only finitely many type $G_{n,2}$61 admissible closed gallery classes of length $G_{n,2}$62 in $G_{n,2}$63. This relies on monotonicity along cusps: certain directed steps strictly increase height in the cuspidal direction, so a closed gallery of fixed length cannot drift arbitrarily far. The resulting determinant identity is the higher-rank analogue of Ihara–Bass,

$G_{n,2}$64

and yields rationality of the chamber zeta function, with poles among $G_{n,2}$65 where $G_{n,2}$66 is an eigenvalue of $G_{n,2}$67 (Hong et al., 29 Dec 2025).

For the standard non-uniform quotient, the zeta function is computed explicitly: $G_{n,2}$68 If

$G_{n,2}$69

then the exact counting formulas are

$G_{n,2}$70

Only lengths divisible by $G_{n,2}$71 occur, with additional contributions at multiples of $G_{n,2}$72. The paper identifies this as a strong positivity phenomenon: the weights are non-negative, the transfer matrix has non-negative entries, and the counting functions $G_{n,2}$73 are non-negative integers. Spectrally, the poles of $G_{n,2}$74, located at the zeros of $G_{n,2}$75 or $G_{n,2}$76, govern the asymptotic growth of closed tailless galleries (Hong et al., 29 Dec 2025).

6. Terminological range, related structures, and conceptual synthesis

The three cited frameworks use chamber language in distinct senses. In the positive-complexity torus-action setting, chambers are regions inside the moment polytope $G_{n,2}$77, obtained as maximal intersections of relative interiors of admissible polytopes. In the arithmetic $G_{n,2}$78 setting, chambers are $G_{n,2}$79-simplices in the quotient of a Bruhat–Tits building, and the central objects are tailless galleries, chamber transfer operators, and zeta functions. In the thin-simplicial setting, chambers are facets, and zigzags are orbits of a flag operator $G_{n,2}$80 on the flag complex (Sergeev, 5 Jun 2026, Hong et al., 29 Dec 2025, Deza et al., 2015).

This distinction resolves a common ambiguity. In the torus-action literature, positivity refers to positive complexity: for an effective Hamiltonian $G_{n,2}$81-action on a compact symplectic manifold of real dimension $G_{n,2}$82, the complexity is $G_{n,2}$83, and the chamber decomposition is introduced because the action is not toric. In the arithmetic gallery setting, positivity refers instead to non-negative weights, non-negative transfer operators, positive spectral radii on finite truncations, and non-negative closed-gallery counts. The thin-chamber-complex theory, by contrast, does not use the phrase positive chamber complex, but it provides the foundational notions of chambers, adjacency, galleries, and specially structured cyclic motions through the complex (Sergeev, 5 Jun 2026, Hong et al., 29 Dec 2025, Deza et al., 2015).

Taken together, these works suggest that positive chamber complex is best understood as an umbrella expression for chamber-based combinatorial geometries in which positivity enters through either positive complexity or positively weighted chamber dynamics. In the hypersimplex model, the chamber complex records orbit combinatorics, toric orbit closures, and the intersection of the secondary fan with $G_{n,2}$84. In the arithmetic $G_{n,2}$85 model, the chamber complex of groups supports an Ihara–Bass type zeta theory for weighted tailless galleries. In the background, thin chamber complexes and Coxeter complexes show how chamber systems admit rigid cyclic structures such as zigzags whose lengths reflect Coxeter numbers. A plausible implication is that these are complementary manifestations of the same general principle: chamber combinatorics becomes especially tractable when it is constrained by a positivity condition strong enough to force explicit hyperplane descriptions, determinant formulas, or uniform cyclic gallery behavior (Sergeev, 5 Jun 2026, Hong et al., 29 Dec 2025, Deza et al., 2015).