Counting chambers in restricted Coxeter arrangements
aa r X i v : . [ m a t h . C O ] J un COUNTING CHAMBERS IN RESTRICTED COXETERARRANGEMENTS
TILMAN M ¨OLLER AND GERHARD R ¨OHRLE
Abstract.
Solomon showed that the Poincar´e polynomial of a Coxeter group W satisfiesa product decomposition depending on the exponents of W . This polynomial coincides withthe rank-generating function of the poset of regions of the underlying Coxeter arrangement.In this note we determine all instances when the analogous factorization property of the rank-generating function of the poset of regions holds for a restriction of a Coxeter arrangement.It turns out that this is always the case with the exception of some instances in type E . Introduction
Much of the motivation for the study of arrangements of hyperplanes comes from Coxeterarrangements. They consist of the reflecting hyperplanes associated with the reflections ofthe underlying Coxeter group. Solomon showed that the Poincar´e polynomial W ( t ) of aCoxeter group W satisfies a product decomposition depending on the exponents of W , see(1.2). This polynomial coincides with the rank-generating function of the poset of regionsof the underlying Coxeter arrangement, see § E , see Theorem 1.3.The analogous factorization property for a localization of a Coxeter arrangement is an im-mediate consequence of Solomon’s theorem and a theorem of Steinberg [Ste60, Thm. 1.5],see Remark 1.5(iv).1.1. The Poincar´e polynomial of a Coxeter group.
Let (
W, S ) be a Coxeter groupwith a distinguished set of generators, S , see [Bou68]. Let ℓ be the length function of W with respect to S . The Poincar´e polynomial W ( t ) of the Coxeter group W is the polynomialin Z [ t ] defined by(1.1) W ( t ) := X w ∈ W t ℓ ( w ) . The following factorization of W ( t ) is due to Solomon [Sol66]:(1.2) W ( t ) = n Y i =1 (1 + t + . . . + t e i ) , Mathematics Subject Classification.
Key words and phrases.
Coxeter arrangement, restriction of a Coxeter arrangement, poset of regions ofa real arrangement, factorization of rank-generating function. here { e , . . . , e n } is the set of exponents of W . See also Macdonald [Mac72].1.2. The rank-generating function of the posets of regions.
Let A = ( A , V ) be ahyperplane arrangement in the real vector space V = R n . A region of A is a connectedcomponent of the complement V \ ∪ H ∈ A H of A . Let R := R ( A ) be the set of regions of A . For R, R ′ ∈ R , we let S ( R, R ′ ) denote the set of hyperplanes in A separating R and R ′ . Then with respect to a choice of a fixed base region B in R , we can partially order R as follows: R ≤ R ′ if S ( B, R ) ⊆ S ( B, R ′ ) . Endowed with this partial order, we call R the poset of regions of A (with respect to B ) and denote it by P ( A , B ). This is a ranked poset of finite rank, where rk( R ) := |S ( B, R ) | ,for R a region of A , [Ed84, Prop. 1.1]. The rank-generating function of P ( A , B ) is definedto be the following polynomial in Z [ t ] ζ ( P ( A , B ) , t ) := X R ∈ R t rk( R ) . Let W = ( W, S ) be a Coxeter group with associated reflection arrangement A = A ( W )which consists of the reflecting hyperplanes of the reflections in W in the real space V = R n ,where | S | = n . Note that the Poincar´e polynomial W ( t ) associated with W given in (1.1)coincides with the rank-generating function of the poset of regions of the underlying reflectionarrangement A ( W ) with respect to B being the dominant Weyl chamber of W in V ; see[BEZ90] or [JP95].Thanks to work of Bj¨orner, Edelman, and Ziegler [BEZ90, Thm. 4.4] (see also Paris [Pa95]),respectively Jambu and Paris [JP95, Prop. 3.4, Thm. 6.1], in case of a real arrangement A which is supersolvable (see see § § B so that ζ ( P ( A , B ) , t ) admits a multiplicative decompositionwhich is equivalent to (1.2) determined by the exponents of A , see Theorem 2.2.1.3. Restricted Coxeter arrangements.
Let W be a Coxeter group with reflection ar-rangement A = A ( W ) in V = R n . We consider the following generalization of the Poincar´epolynomial W ( t ) of W . Let X be in the intersection lattice L ( A ) of A , i.e. X is the sub-space in V given by the intersection of some hyperplanes in A . Then we can consider therestricted arrangement A X which is the induced arrangement in X from A , see § A X is always free, so we can speak of the exponents of A X , see [OT92, § W is a Weylgroup, Douglass [Dou99, Cor. 6.1] gave a uniform proof of this fact by means of an elegant,conceptual Lie theoretic argument.It follows from the discussion above that in the special instances when either A X is super-solvable (which is for instance always the case for X of dimension at most 2) or inductivelyfactored, or else if X is just the ambient space V (so that A V = A ), then ζ ( P ( A X , B ) , t )is known to factor analogous to (1.2) involving the exponents of A X . adell and Neuwirth [FN62] showed that the braid arrangement is fiber type and Brieskorn[Br73] proved this for the reflection arrangement of the hyperoctahedral group. This prop-erty is equivalent to being supersolvable, see [Ter86]. Therefore, since any restriction of asupersolvable arrangement is again supersolvable, [Sta72], in case of the symmetric or hy-peroctahedral group W , we see that A ( W ) X is supersolvable for any X . Thus in each ofthese cases the rank generating function of the poset of regions of A ( W ) X factors as in (1.2),thanks to Theorem 2.2.Therefore, it is natural to study the rank-generating function of the poset of regions of anarbitrary restriction of a Coxeter arrangement. The following gives a complete classificationof all instances when ζ ( P ( A X , B ) , t ) factors analogous to (1.2). Theorem 1.3.
Let W be a finite, irreducible Coxeter group with reflection arrangement A = A ( W ) . Let A X be the restricted arrangement associated with X ∈ L ( A ) \ { V } . Thenthere is a suitable choice of a base region B so that the rank-generating function of the posetof regions of A X satisfies the multiplicative formula (1.4) ζ ( P ( A X , B ) , t ) = n Y i =1 (1 + t + . . . + t e i ) , where { e , . . . , e n } is the set of exponents of A X if and only if one of the following holds: (i) W is not of type E ; (ii) W is of type E and either the rank of X is at most , but A X = ( E , A A ) and A X = ( E , A A ) , or else A X ∼ = ( E , D ) . We prove Theorem 1.3 in Section 3. For classical W , either A ( W ) X is supersolvable and theresult follows from Theorem 2.2, or else W is of type D and A ( W ) X belongs to a particularfamily of arrangements D kp for 0 ≤ k ≤ p studied by Jambu and Terao, [JT84, Ex. 2.6]. Weprove Theorem 1.3 for the family D kp in Lemma 3.11.For W of exceptional type, there are 31 restrictions A ( W ) X of rank at least 3 (up toisomorphism) that need to be considered. These are handled by computational means, seeRemark 3.13. Remarks 1.5. (i). In the statement of the theorem and later on we use the conventionto label the W -orbit of X ∈ L ( A ) by the Dynkin type T of the stabilizer W X of X in W which is itself a Coxeter group, by Steinberg’s theorem [Ste60, Thm. 1.5]. So we denote therestriction A X just by the pair ( W, T ); see also [OT92, App. C, D].(ii). Among the restrictions A ( W ) X all supersolvable and all inductively factored instancesare known, see Theorems 3.1 and 3.2 below. Thus, by Theorem 2.2, in each of these cases ζ ( P ( A X , B ) , t ) factors as in (1.4).(iii). Hoge checked that the exceptional case ( E , A A ) from Theorem 1.3 is isomorphic tothe real simplicial arrangement “ A (17)” from Gr¨unbaum’s list [Gr71]. It was observed byTerao that the latter does not satisfy the product rule (1.4), [BEZ90, p. 277]. It is ratherremarkable that this arrangement makes an appearance as a restricted Coxeter arrangement.In contrast, according to Theorem 1.3, the rank-generating function of the poset of regions f ( E , A A ) does factor according to (1.4). In particular, these two arrangements are notisomorphic, as claimed erroneously in [OT92, App. D].(iv). For X in L ( A ( W )) consider the localization A ( W ) X of A ( W ) at X , which consists ofall members of A ( W ) containing X , see § W X in W of X is itselfa Coxeter group, by Steinberg’s theorem [Ste60, Thm. 1.5], and since A ( W ) X = A ( W X ), by[OT92, Cor. 6.28(2)], it follows from Solomon’s factorization (1.2) that the rank generatingfunction of the poset of regions of A ( W ) X (with respect to the base chamber being theunique chamber of A ( W ) X containing the dominant Weyl chamber of W ) factors analogousto (1.2) involving the exponents of W X .(v). In Lie theoretic terms, for W a Weyl group, W ( t ) is the Poincar´e polynomial of theflag variety of a semisimple linear algebraic group with Weyl group W . The formula (1.2)then gives a well-known factorization of the Poincar´e polynomial of the flag variety.If W is of type A or B , then each restriction A ( W ) X is the Coxeter arrangement of the sameDynkin type of smaller rank, cf. [OT92, Props. 6.73, 6.77]. Thus, by the previous paragraph,in these instances, ζ ( P ( A X , B ) , t ) is just the Poincar´e polynomial of the flag variety of asemisimple linear algebraic group of the same Dynkin type as W but of smaller rank.In view of these examples, it is natural to wonder whether in general there is a suitableprojective variety associated with a fixed semisimple group G with Weyl group W whosePoincar´e polynomial is related to the rank-generating function of the poset of regions forany restriction of A ( W ) in the same manner as in these special instances above, relating toand generalizing the flag variety of G .For general information about arrangements and Coxeter groups, we refer the reader to[Bou68] and [OT92]. 2. Recollections and Preliminaries
Hyperplane arrangements.
Let V = R n be an n -dimensional real vector space. A (real) hyperplane arrangement A = ( A , V ) in V is a finite collection of hyperplanes in V each containing the origin of V . We denote the empty arrangement in V by Φ n .The lattice L ( A ) of A is the set of subspaces of V of the form H ∩ . . . ∩ H i where { H , . . . , H i } is a subset of A . For X ∈ L ( A ), we have two associated arrangements, firstly A X := { H ∈ A | X ⊆ H } ⊆ A , the localization of A at X , and secondly, the restriction of A to X , A X = ( A X , X ), where A X := { X ∩ H | H ∈ A \ A X } . Note that V belongs to L ( A ) asthe intersection of the empty collection of hyperplanes and A V = A . The lattice L ( A ) isa partially ordered set by reverse inclusion: X ≤ Y provided Y ⊆ X for X, Y ∈ L ( A ).Throughout, we only consider arrangements A such that 0 ∈ H for each H in A . Theseare called central . In that case the center T ( A ) := ∩ H ∈ A H of A is the unique maximalelement in L ( A ) with respect to the partial order. A rank function on L ( A ) is given by r ( X ) := codim V ( X ). The rank of A is defined as r ( A ) := r ( T ( A )). .2. Free arrangements.
Free arrangements play a fundamental role in the theory of hy-perplane arrangements, see [OT92, §
4] for the definition and properties of this notion. Cru-cial for our purpose is the fact that associated with a free arrangement is a set of importantinvariants, its (multi)set of exponents , denoted by exp A .2.3. Supersolvable arrangements.
We say that X ∈ L ( A ) is modular provided X + Y ∈ L ( A ) for every Y ∈ L ( A ), [OT92, Cor. 2.26]. Definition 2.1 ([Sta72]) . Let A be a central arrangement of rank r . We say that A is supersolvable provided there is a maximal chain V = X < X < . . . < X r − < X r = T ( A )of modular elements X i in L ( A ), cf. [OT92, Def. 2.32].Note that arrangements of rank at most 2 are always supersolvable, e.g. see [OT92, Prop. 4.29(iv)]and supersolvable arrangements are always free, e.g. see [OT92, Thm. 4.58]. Also, restrictionsof a supersolvable arrangement are again supersolvable, [Sta72, Prop. 3.2].2.4. Nice and inductively factored arrangements.
The notion of a nice or factored arrangement is due to Terao [Ter92]. It generalizes the concept of a supersolvable arrange-ment, e.g. see [OT92, Prop. 2.67, Thm. 3.81]. Terao’s main motivation was to give a generalcombinatorial framework to deduce tensor factorizations of the underlying Orlik-Solomonalgebra, see also [OT92, § § inductively factored arrangement, see[JP95], [HR16, Def. 3.8] for further details on this concept.The connection with the previous notions is as follows. Supersolvable arrangements arealways inductively factored ([HR16, Prop. 3.11]) and inductively factored arrangements arealways free ([JP95, Prop. 2.2]) so that we can talk about the exponents of such arrangements.The following theorem due to Jambu and Paris, [JP95, Prop. 3.4, Thm. 6.1], was first shownby Bj¨orner, Edelman and Ziegler for A supersolvable in [BEZ90, Thm. 4.4] (see also Paris[Pa95]). Theorem 2.2. If A is inductively factored, then there is a suitable choice of a base region B so that ζ ( P ( A , B ) , t ) satisfies the multiplicative formula (2.3) ζ ( P ( A , B ) , t ) = n Y i =1 (1 + t + . . . + t e i ) , where { e , . . . , e n } = exp A is the set of exponents of A . .5. Restricted root systems.
Given a root system for W , associated with a member X from L ( A ( W )) we have a restricted root system which consists of the restrictions of theroots of W to X , see [BG07, § A ( W ) X , [BG07, Cor. 7].More specifically, let Φ be a root system for W and let ∆ ⊂ Φ be a set of simple roots. Inview of Remark 1.5(i), choosing X ∈ L ( A ( W )) amounts to specifying the Dynkin type T of the parabolic subgroup W X , so that the pair ( W, T ) characterizes A ( W ) X . Let B T be theset of all subsets of ∆ that generate a root system of Dynkin type T . Fixing an element∆ J ∈ B T , the bases for Φ containing ∆ J are in bijective correspondence with the bases forthe restricted root system, [BG07, Thm. 10].Furthermore, the set B T characterizes a set of representatives for the action of the restrictedWeyl group on the set of chambers of the arrangement A ( W ) X , [BG07, Lem. 11]. Thusthere is a suitable choice of a base region B such that ζ ( P ( A ( W ) X , B ) , t ) factors accordingto (1.4), if and only if there is such a choice among regions that arise from elements in B T .3. Proof of Theorem 1.3
It is well known that if W is of type A or B , then the Coxeter arrangement A ( W ) issupersolvable and so is every restriction thereof. So Theorem 1.3 follows in this case fromTheorem 2.2. Therefore, for W of classical type, we only need to consider restrictions for W of type D . The restrictions D kp for 0 ≤ k ≤ p of Coxeter arrangements of type D are givenby the defining polynomial Q ( D kp ) := x p − k +1 · · · x p Y ≤ i Theorem 3.2 ([MR17, Thms. 1.5, 1.6]) . Let W be a finite, irreducible Coxeter group withreflection arrangement A = A ( W ) and let X ∈ L ( A ) \ { V } with dim X ≥ . Then therestricted arrangement A X is inductively factored if and only if one of the following holds: (i) A X is supersolvable, or ii) W is of type D n for n ≥ and A X ∼ = D p − p , where p = dim X ; (iii) A X is one of ( E , A A ) , ( E , A ) , or ( E , ( A A ) ′′ ) . It follows from Theorem 2.2 that in all instances covered in Theorem 3.2, ζ ( P ( A , B ) , t )satisfies the factorization property of (2.3) with respect to a suitable choice of base region B . In particular, Theorem 1.3 holds in all these instances.It is not apparent that the rank-generating function of the poset of regions of D kp factorsaccording to (1.4) for 1 ≤ k ≤ p − 3. For, these arrangements are neither reflection arrange-ments nor are they inductively factored, by the results above. To show that the factorizationproperty from (1.4) also holds in these instances, we first parameterize the regions R ( D kp )suitably and then prove a recursive formula for ζ ( P ( D kp , B ) , t ). Remark 3.3. Since the inequalities given by the hyperplanes do not change within a region,the set of regions is uniquely determined by specifying one interior point for each region. Let M kp := { ( x , . . . , x p ) ∈ {± , . . . , ± p } p | x , . . . , x p − k = − , | x i | 6 = | x j | ∀ i = j } . It is easy to verify that each region in R := R ( D kp ) contains exactly one element of M kp . Sothis gives a parametrization for the regions in R . Without further comment, we frequentlyidentify points in M kp with their respective regions in R . For x ∈ M kp , write R x ∈ R forthe unique region containing x . Once a base region B in R is chosen so that R becomes aranked poset, we may write ζ ( P ( D kp , B ) , t ) = X x ∈ M kp t rk( R x ) . Using this notation it is easy to see which regions are adjacent and which hyperplanes arewalls of a given region. Let x = ( x , . . . , x p ) ∈ M kp . If x j = x i ± 1, then ker( x i − x j ) is awall of R x and the corresponding adjacent region is obtained from x by exchanging x i and x j in x . If x j = − ( x i ± x i + x j ) is a wall of R x and the adjacent region againoriginates from x by exchanging x i and x j but maintaining their respective signs . Finally, if x i = ± p − k < i ≤ p , then ker( x i ) is a wall of R x and the adjacent region is obtainedby exchanging x i with − x i .For our subsequent results, we choose B p := R y ∈ R for y = ( p, p − , . . . , 1) as our basechamber independent of k . Lemma 3.4. Let p ≥ , k ∈ { , . . . , p } and B p ∈ R as above. For an arbitrary i ∈ { , . . . , p } ,we have (3.5) X x ∈ M kp x i = p t rk( R x ) = ( t i − · ζ ( P ( D kp − , B p − ) , t ) if i ≤ p − k,t i − · ζ ( P ( D k − p − , B p − ) , t ) if i > p − k, and (3.6) X x ∈ M kp x i = − p t rk( R x ) = ( t p − i − · ζ ( P ( D kp − , B p − ) , t ) if i ≤ p − k,t p − i · ζ ( P ( D k − p − , B p − ) , t ) if i > p − k. roof. Set N − := { x ∈ M kp | x i = − p } . Thanks to Remark 3.3, no hyperplane involving thecoordinate x i lies between any two regions of N − . Setting z = ( z , . . . , z i , . . . , z p ) := ( p − , p − , . . . , p − i + 1 , − p, p − i − , . . . , , ∈ N − , there are only hyperplanes involving x i between B p and R z . More precisely, we have S ( B p , R z ) = ( { ker( x i − x j ) | j ≤ i } ∪ { ker( x i ± x j ) | i < j ≤ p ) } for i ≤ p − k, { ker( x i − x j ) | j ≤ i } ∪ { ker( x i ± x j ) | i < j ≤ p ) } ∪ { ker( x i ) } for i > p − k. So if we choose an arbitrary x ∈ N − , we have S ( B p , R x ) = S ( B p , R z ) ˙ ∪ S ( R z , R x ) . Consequently, we obtain(3.7) rk( R x ) = |S ( B p , R z ) | + |S ( R z , R x ) | = ( t p − i − + |S ( R z , R x ) | for i ≤ p − k,t p − i + |S ( R z , R x ) | for i > p − k. Now set(3.8) A := ( D kp − if i ≤ p − k, D k − p − if i > p − k, and identify the set of regions R ( A ) of A with the corresponding set of ( p − i -th coordinate defines a map h : N − −→ R ( A )which is bijective, h ( R z ) = B p − and if e rk denotes the rank function on P ( A , B p − ), thenwe get |S ( R z , R x ) | = e rk( h ( R x )). Therefore, by (3.7), (3.8) and the bijectivity of h , we get X x ∈ M kp x i = − p t rk( R x ) = t |S ( B p ,R z ) | X x ∈ N − t |S ( R z ,R x ) | = t |S ( B p ,R z ) | X x ∈ N − t e rk( h ( R x )) = t |S ( B p ,R z ) | X x ∈ R ( A ) t e rk( R x ) = t |S ( B p ,R z ) | ζ ( P ( A , B p − ) , t )= ( t p − i − · ζ ( P ( D kp − , B p − ) , t ) if i ≤ p − k,t p − i · ζ ( P ( D k − p − , B p − ) , t ) if i > p − k. So (3.5) follows.Next let N + := { x ∈ M kp | x i = p } and set z = ( z , . . . , z i , . . . , z p ) := ( p − , p − , . . . , p − i + 1 , p, p − i − , . . . , , ∈ N + . Then S ( B p , R z ) = { ker( x i − x j ) | ≤ j < i } has cardinality i − 1. The proof of this case issimilar to the one above, and is left to the reader. So (3.6) follows. (cid:3) he next technical lemma is needed in the proof of Lemma 3.11. For ease of notation, weset F ( e , . . . , e m ) := m Y i =1 (1 + t + · · · + t e i ) ∈ Z [ t ]for any m ≥ e , . . . , e m ≥ 1. In particular, F ( e ) = 1 + t + · · · + t e . Also notethat for j > 0, we have(3.9) F ( j − t j ) = F (2 j − . Lemma 3.10. Let p ≥ and ≤ k ≤ p . Define ∆ kp := p − k X i =1 ( t i − + t p − i − ) F ( p + k − 2) + p X i = p − k +1 ( t i − + t p − i ) F ( p + k − . Then ∆ kp = F ( p + k − , p − . Proof. We argue by induction on k . First let k = 0. Then, using (3.9), we have∆ p = p X i =1 ( t i − + t p − i − ) F ( p − · · · + t p − ) F ( p − 2) + ( t p − + · · · + t p − ) F ( p − F ( p − , p − 2) + t p − F ( p − , p − F ( p − , p − (cid:0) t p − (cid:1) = F ( p − , p − . Now let k > k ′ < k . Then using the inductivehypothesis, we get∆ kp = ∆ k − p + t p + k − p − k X i =1 (cid:0) t i − + t p − i − (cid:1) + t p + k − p X i = p − k +1 (cid:0) t i − + t p − i (cid:1) − F ( p + k − (cid:0) t p − k + t p + k − (cid:1) + F ( p + k − (cid:0) t p − k + t p + k − (cid:1) = F (2 p − , p + k − 2) + ( t p + k − + · · · + t p − + t p +2 k − + · · · + t p + k − )+ ( t p − + · · · + t p + k − + t p + k − + · · · + t p +2 k − ) − ( t p − + t p + k − )= F (2 p − , p + k − 2) + t p + k − (1 + · · · + t p − )= F (2 p − F ( p + k − 2) + t p + k − )= F (2 p − , p + k − , as claimed. (cid:3) Finally, armed with Lemmas 3.4 and 3.10, we are able to prove the desired result for thearrangements D kp . Lemma 3.11. The rank-generating function of the poset of regions of D kp factors accordingto (1.4) for all ≤ k ≤ p − and p ≥ . roof. We argue by induction on n = p + k . For n = 3, the result holds vacuously. So let1 ≤ k ≤ p − p ≥ p ′ , k ′ , with 1 ≤ k ′ ≤ p ′ − p ′ ≥ n > p ′ + k ′ , the arrangement D k ′ p ′ satisfies (1.4). Note that(3.12) exp( D kp ) = exp( D p − p − ) ∪ { p + k − } , see [JT84, Ex. 2.6]. Then the inductive hypothesis together with Lemmas 3.4 and 3.10 and(3.12) imply ζ ( P ( D kp , B p ) , t ) = X x ∈ M kp t rk( R x ) = p X i =1 X x ∈ M kp x i = ± p t rk( R x ) = p − k X i =1 ( t i − + t p − i − ) ζ ( P ( D kp − , B p − ) , t ) + p X i = p − k +1 ( t i − + t p − i ) ζ ( P ( D k − p − , B p − ) , t )= p − k X i =1 ( t i − + t p − i − ) F (cid:0) exp( D kp − ) (cid:1) + p X i = p − k +1 ( t i − + t p − i ) F (cid:0) exp( D k − p − ) (cid:1) = F (cid:0) exp( D p − p − ) (cid:1) p − k X i =1 ( t i − + t p − i − ) F ( p + k − 2) + p X i = p − k +1 ( t i − + t p − i ) F ( p + k − ! = F (cid:0) exp( D p − p − ) (cid:1) ∆ kp = F (cid:0) exp( D p − p − ) (cid:1) F (2 p − , p + k − F (cid:0) exp( D kp ) (cid:1) . This completes the proof of the lemma. (cid:3) Remark 3.13. In view of Theorems 2.2, 3.1 and 3.2, Lemma 3.11 settles all the remain-ing classical instances of Theorem 1.3. It follows from Theorems 3.1 and 3.2 that thereare 31 instances for W of exceptional type to be checked (here we take the isomorphismsof rank 3 restrictions A ( W ) X into account, cf. [OT92, App. D]). We have verified that ζ ( P ( A ( W ) X , B ) , t ) satisfies the factorization property (1.4) precisely in all the instanceswhen W is of exceptional type, as specified in Theorem 1.3. In the listed exceptions, ζ ( P ( A ( W ) X , B ) , t ) does not factor according to this rule with respect to any choice ofbase region. This was checked using the computer algebra package SAGE , [S + SAGE -package HyperplaneArrangements which provides methods to compute ζ ( P ( A , B ) , t )) for given A and B . More specifically, the algorithm is initiated with a listcontaining the vector space V as a polytope and for each hyperplane in A splits each polytopein the current list into two polytopes, defined by a positive resp. negative inequality, whilediscarding all empty solutions. This results in a list of chambers implemented as polytopes.After specifying a base region B the algorithm checks for each region R and each hyperplane H whether H separates B from R .In addition, we used the results from [BG07, § E , A ), as the latter is simply too big for SAGE to compute all ts chambers at once. For this case we instead used the bijective correspondences recalledin 2.5 to compute the chambers directly from the elements of the Weyl group W ( E ). Byordering the group elements by length using a depth-first search algorithm implemented inthe SAGE -package ReflectionGroup , we were able to compute the chambers of the restrictedarrangement ordered by rank, so we could conclude that the rank-generating polynomial ofthe poset of regions for the restriction A X = ( E , A ) does not factor according to (1.4)after computing only a small portion of the entire polynomial ζ ( P ( A X , B ) , t )). Acknowledgments : We are grateful to T. Hoge for checking that the simplicial arrange-ment “ A (17)” from Gr¨unbaum’s list coincides with the restriction ( E , A A ). We wouldalso like to thank C. Stump for helpful discussions concerning computations in SAGE .The research of this work was supported by DFG-grant RO 1072/16-1. References [AHR14] N. Amend, T. Hoge and G. R¨ohrle, Supersolvable restrictions of reflection arrangements , J. Com-bin. Theory Ser. A, (2014), 336–352.[BEZ90] A. Bj¨orner, P. Edelman, and G. 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