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Enumerating Chambers of Hyperplane Arrangements with Symmetry

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Enumerating Chambers of Hyperplane Arrangements with Symmetry ENUMERATING CHAMBERS OF HYPERPLANE ARRANGEMENTS WITH SYMMETRY TAYLOR BRYSIEWICZ, HOLGER EBLE, AND LUKAS KUHNE¨ Abstract. We introduce a new algorithm for enumerating chambers of hyperplane arrangements which exploits their underlying symmetry groups. Our algorithm counts the chambers of an arrangement as a byproduct of computing its characteristic polynomial. We showcase our julia implementation, based on OSCAR, on examples coming from hyper- plane arrangements with applications to physics and computer science. Keywords. Hyperplane arrangement, chambers, algorithm, symmetry, resonance arrangement, separability 1. Introduction The problem of enumerating chambers of hyperplane arrangements is a well-known challenge in computational discrete geometry [19, 28, 33, 42]. We develop a novel chamber-counting algorithm which takes advantage of the combinatorial symmetries of an arrangement. This count is derived from the computation of other important combinatorial invariants: Betti numbers and characteristic polynomials [1, 20, 29, 37, 43, 45]. While most arrangements admit few combinatorial symmetries [39], most arrangements of interest do [18, 40, 46]. We implemented our algorithm in julia [3] and published it as the pack- age CountingChambers.jl1. Our implementation relies heavily on the cor- nerstones of the new computer algebra system OSCAR [38] for group theory computations (GAP [16]) and the ability to work over number fields (Hecke and Nemo [15]). To the best of our knowledge, it is the first publicly available software for counting chambers which uses symmetry. We showcase our algorithm and its implementation on a number of well- known examples, such as the resonance and discriminantal arrangements. Additionally, we study sequences of hyperplane arrangements which come from the problem of linearly separating vertices of regular polytopes. In particular, we investigate one corresponding to the hypercube [0; 1]d whose arXiv:2105.14542v2 [math.CO] 25 Jun 2021 chambers are in bijection with linearly separable Boolean functions. In the presence of symmetry, our implementation outperforms the exist- ing software by several orders of magnitude (cf. Table 1). Moreover, its output is guaranteed to be accurate since we compute symbolically over the integers or exact number fields and avoid overflow errors thanks to the package SaferIntegers.jl [41]. 9 The ninth resonance arrangement (511 hyperplanes in R ) approaches the limit of what is possible with our implementation: the computation of its 2010 Mathematics Subject Classification. 52C35, 52B15. 1available at https://mathrepo.mis.mpg.de/CountingChambers 1 2 T. BRYSIEWICZ, H. EBLE, AND L. KUHNE¨ characteristic polynomial took 10 days on 42 processors. Our computation confirms that its chamber-count is 1955230985997140 as independently and concurrently computed by Chroman and Singhar with different methods [9]. We first give background on hyperplane arrangements in Section 2. The ideas outlined in Section 3, regarding deletion and restriction algorithms, form the basic structure of our algorithm. We explain the relevant results regarding symmetries of arrangements in Section 4. The algorithm and its implementation details reside in Section 5. In Section 6 we construct and discuss examples of arrangements exhibiting large symmetry groups. We conclude in Section 7 with timings and comparisons to other software. Acknowledgements We are very grateful to Tommy Hofmann, Christopher Jefferson, and Marek Kaluba for their support regarding the implementation and to Michael Cuntz for initial verifications of our computations. We would also like to thank Michael Joswig for his helpful comments throughout the project and Bernd Sturmfels for suggesting the discriminantal arrangement. 2. Hyperplane arrangements We begin by discussing background on the theory of hyperplane arrange- ments related to the problem of enumerating chambers: the main goal of this article and the associated software. Our notation will mostly follow the textbook by Orlik and Terao [37]. d For any field K, a hyperplane in K is an affine linear space of codi- mension one. Throughout this article, we denote by = H1;:::;Hn a A d f g (hyperplane) arrangement where Hi is a hyperplane in K . d Definition 1. Suppose is an arrangement in R . The connected com- A d S ponents of the complement R H2A H are called chambers of and are denoted ch( ). n A A Example 2. We use the arrangement y x = 1 ; x = 0 ; x + y = 1 ; y = 0 ff| −{z g} |f {z g} |f {z g} |f {z g}g H1 H2 H3 H4 2 in R as a running example. This arrangement is depicted in Figure 1. It has 10 chambers: 2 bounded and 8 unbounded. H1 H3 H2 H4 Figure 1. The arrangement introduced in Example 2. Given a subset I [n] := 1; : : : ; n , we write the set Hi i2I as HI and ⊆ T f g f g its intersection as LI = i2I Hi. The collection of these intersections form ENUMERATING CHAMBERS OF ARRANGEMENTS WITH SYMMETRY 3 the set L( )= LI I [n] ;LI = , a combinatorial shadow of known as its intersectionA f posetj ⊆. This poset6 ;g is ordered by reverse inclusionA and graded by the rank function, r : L( ) Z≥0, where r(LI ) = codim (LI ). A ! As a notational convention, we set r(I) = r(LI ) for I [n] whenever LI = . ⊆ 6 ; 2.1. The characteristic polynomial. Our algorithm counts chambers of an arrangement by computing a more refined count, namely the Betti num- bers. Knowing the Betti numbers of an arrangement is equivalent to knowing its characteristic polynomial. d Definition 3. The characteristic polynomial of an arrangement in K is the polynomial A d X jIj d−r(I) X i d−i (1) χA(t)= ( 1) t = ( 1) bi( )t : − − A I⊆[n]:LI 6=; i=0 d The integers bi( ) i=0, defined via (1), are non-negative and are called the Betti numbersf ofA g. We denote the vector of Betti numbers by b( ). A A The characteristic polynomial and Betti numbers of an arrangement depend only on the intersection poset L( ) and have various interpretationsA A depending on the field K as detailed below. d Real: For an arrangement in R , Zaslavsky [45] proved that A d d X ch( ) = ( 1) χA( 1) = bi( ): j A j − − A i=0 Thus, the Betti numbers are a refined count of the chambers of . They have the following geometric interpretation. Given a genericA d flag • : F0 F1 Fd = R of affine linear subspaces Fi F ⊂ ⊂ · · · ⊂ (where dim(Fi) = i) the number of chambers of which meet Fi A but do not meet Fi−1 is equal to bi( ) [44, Proposition 2.3.2]. Ad Complex: If is an arrangement in C where all hyperplanes contain A the origin, then bi( ) is the i-th topological Betti number of the d S A complement C H with rational coefficients [36]. n H2A Finite: When is an arrangement over a finite field Fq, Crapo and A d S Rota proved that χA(q) = Fq H2A H [11]. Moreover, if is a hy- jd n j A perplane arrangement in Q one may consider its reduction modulo q: Fq = H1 Fq;:::;Hn Fq . When q is sufficiently large, we A ⊗ f ⊗ ⊗ g have that L( ) = L( Fq) and thus computing χA(t) for rational arrangementsA also yieldsA ⊗ the number of points in the complement after reducing modulo large primes. Example 4. Let be the arrangement introduced in Example 2. Its char- A 2 acteristic polynomial is χA(t) = t 4t + 5. Figure 2 shows a generic flag • intersecting this arrangement verifying− that b( ) = (1; 4; 5). F A 3. A deletion-restriction algorithm d To compute the Betti numbers of an arrangement in K , we take ad- A vantage of the behavior of χA(t) under the operations of deletion and re- striction. These operations reduce computations about to computations A 4 T. BRYSIEWICZ, H. EBLE, AND L. KUHNE¨ F2 F1 F0 2 Figure 2. The intersections of a generic flag (purple) in R with the chambers of . The point F0 intersects one cham- A ber, F1 intersects four others, and F2 intersects the remain- ing 5, and so b( ) = (1; 4; 5). A about two smaller arrangements. Thus at its core, our main algorithm is a divide-and-conquer algorithm. Given a hyperplane H , the deletion of H in is the arrangement 2 A A d−1 H . The restriction of H in is the arrangement in H ∼= K defined byAnf Hg= K H K H A. The following lemma provides the basic foundationA f of\ our algorithm.j 2 Anf gg Lemma 5 [37, Corollary 2.57]. Given a hyperplane H , we have that 2 A H χA(t) = χAnfHg(t) χAH (t): In particular, b( ) = b( H ) + 0 b( ) where 0 b means prepending− the vector b with aA zero. Anf g j A j 3.1. A simple deletion-restriction algorithm. Lemma 5 along with the d d+1 fact that the empty arrangement in K has Betti numbers (1; 0;:::; 0) N suggests the following well-known recursive algorithm for computing2b( ). A Algorithm 1: Betti numbers via simple deletion and restriction d Input: A hyperplane arrangement in K Output: The Betti numbers b( ) A BettiNumbers ( ) A A 1 if = then ; 6 A 2 choose H 2 A 3 return BettiNumbers( H ) + 0 BettiNumbers( H ) Anf g j A 4 else 5 return (1; 0;:::; 0) Structurally, Algorithm 1 is a depth-first binary tree algorithm on ar- rangements, rooted at the initial input: one child represents a deletion and the other a restriction, as shown in Figure 3. The implementation of Algorithm 1 is already nontrivial as it is often the case that some hyperplanes become the same after a restriction. Thus, its proper implementation requires care in representing an arrangement on a computer. 3.2. Computationally representing deletions and restrictions. An arrangement coming from via deletions and restrictions may be repre- sented by an encodingB of theA restricted hyperplanes.
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