Reflection Groups and Polytopes over Finite Fields, III

Reflection Groups and P olytop es o v er Finite Fields, II I B. Monson ∗ Universit y of New Brunsw ic k F rederict on, New Br unswic k , Canada E3 B 5A3 and Egon Sc h ulte † Northea stern Universit y Boston, Massach ussetts, USA, 0 2115 No v em b er 2, 2018 With b est wishes for our friend and c ol le ague, J¨ or g Wil ls Abstract When the standard represen tation of a crystallo graphic Co xeter group Γ is reduced mo dulo an o d d prime p , one obtains a fin it e group G p acting on some orth ogonal space o ver Z p . If Γ has a string d i agram, then G p will often b e the automorphism group of a fi n it e abstr act regular p olytop e. In p arts I and I I w e established the basics of this construction and en umerated the p olytop es asso ciated to groups of rank at most 4, as w ell as all groups of sp herical or Euclidean t yp e. Here we extend the range of our earlier criteria f or the p olytopalit y of G p . Building on this w e inv estigate the class of ‘3–infinit y’ groups of general r ank, and then complete a sur v ey of those lo cally toroidal p olytop es which can b e describ ed b y our construction. Key W ord s : reflection groups, abstract regular p olytop es AMS Sub j ect Classifi cation (2000 ): Pr imary: 51M20. Secondary: 20F55. 1 In t r o d uction and No t ation The regular p olytop es con tinue to b e a ric h source of b eautiful mathematical ideas. Their com binatorial f eat ur es, fo r instance, hav e b een generalized in the theory of abstr act r e gular p olytop es . Here we conclude a series of three pa p ers concerning the prop erties o f (abstract) regular p olytop es, as constructed from orthog onal groups ov er finite fields. Our main goal is ∗ Suppo rted by NSE RC o f Cana da Grant # 48 18 † Suppo rted by NSA-g rants H98230- 05-1- 0027 and H98230 - 07-1- 0005 1 to complete a description of the lo c al ly tor oidal p olytop es pro vided b y our construction (see Section 4). T o that end, in Section 2 w e establish some new structural theorems concerning the ‘p olytopality ’ of orthogonal groups. As a test case, w e also apply our metho ds in Section 3 to a n in teresting f amily of p olytop es of general rank n . Let us b egin with a review o f the basic set up and k ey results from parts I and I I ([14] and [15], resp ective ly). In [14], we first surv ey ed some of the esse n tial pro p erties of an abstract regular p olytop e P , referring to [12] for details. Crucially , for eac h suc h P the automorphism group Γ( P ) is equipp ed with a natura l list of inv olutory generators and is further a ve ry sp ecial quotien t of a certain Co xeter group G . (W e say that Γ( P ) is a string C-gr oup .) Since P can b e uniquely reconstructed from Γ( P ), w e may therefore shift our fo cus. Throughout, then, G = h r 0 , . . . , r n − 1 i will b e a p o ssibly infinite, crystallographic Cox - eter gro up [ p 1 , p 2 , . . . , p n − 1 ] with a string Coxete r diagram ∆ c ( G ) (with br a nc hes lab eled p 1 , p 2 , . . . , p n − 1 , resp ectiv ely), o bta ined from the corresp o nding abstract Co xeter group Γ = h ρ 0 , . . . , ρ n − 1 i via the standard represen tation on real n -space V . (V ery often G will b e infinite.) F or a ny o dd prime p , w e ma y reduce G mo dulo p to obtain a subgroup G p of GL n ( Z p ) generated b y the mo dular imag es o f the r i ’s. W e shall abuse nota t io n by referring to the mo dular images of o b jec ts b y the same na me (suc h a s r i , b i , B = [ b i · b j ], V , etc.). In particular, { b i } will denote the standard basis for V = Z n p . In an y eve n t, G p is a subgroup of t he orthogonal g roup O ( Z n p ) of isometries for the (p ossibly singular) symm etric bilinear form x · y , t he latter b eing defined o n Z n p b y means of the G ram matrix B . Lik ewise, eac h r i remains a reflection, altho ug h w e may write r i ( x ) = x − 2 x · b i b i · b i b i only if b 2 i := b i · b i 6≡ 0 mo d p . Concerning this situation, w e now mak e a conv enien t definition: if p ≥ 5, or p = 3 but no branch o f ∆ c ( G ) is mark ed 6, then w e sa y that p is gen e ric for G . Indeed, in suc h cases, no no de lab el b 2 i of the diagram ∆( G ) (for a basic system) is zero mo d p , and the corresp onding r o o t b i is anisotro pic. Also, a c hang e in the underlying basic system for G has the effect of merely conjug a ting G p in GL n ( Z p ). On the other hand, in the non-g eneric case, in whic h p = 3 and ∆ c ( G ) has some branc h marked 6, the group G p ma y dep end essen tia lly on the actual diagra m ∆( G ) tak en f o r the reduction mo d p . (Note that p g eneric do es not necessarily mean t ha t p ∤ | G | , o r that certain subspaces of V are non-singular, etc.) No w we confront tw o questions: what exactly is the finite reflection group G p and when is it a string C -gro up (i.e. the automorphism group of a finite, abstract r egula r n -p o lytop e P = P ( G p ))? T o help answ er the first question, w e recall from [14, Thm. 3.1] that an irreducible group G p of the a b o v e sort, generated b y n ≥ 3 reflections, must necessarily b e one of the following: • an orthogonal group O ( n, p, ǫ ) = O ( V ) o r O j ( n, p, ǫ ) = O j ( V ), excluding the cases O 1 (3 , 3 , 0), O 2 (3 , 5 , 0), O 2 (5 , 3 , 0) (supp osing for these three that disc ( V ) ∼ 1), and also excluding the case O j (4 , 3 , − 1); or • the reduction mod p of one of the finite linear Co xeter groups of type A n ( p ∤ n + 1), B n , D n , E 6 ( p 6 = 3), E 7 , E 8 , F 4 , H 3 or H 4 . 2 W e shall sa y in these t wo cases that G p is of ortho g o nal or spheric a l typ e , res p ectiv ely , although there is some o v erlap for small primes. Concerning our groups G p , it is only a sligh t abuse of notation to let [ p 1 , . . . , p n − 1 ] p denote the mo dular repres en tation of a group [ p 1 , . . . , p n − 1 ], so long a s p is generic for the group. Let us turn to our second question. The generato r s r i of G p certainly satisfy t he Co xeter- t ype relations inherited from G . Th us G p is a string C-gr oup if and only if it satisfies the follo wing in tersec tion prop ert y on standard subgroups: h r i | i ∈ I i ∩ h r i | i ∈ J i = h r i | i ∈ I ∩ J i , (1) for all I , J ⊆ { 0 , . . . , n − 1 } (see [12, § 2E]). Our main problem is therefore to determine when G p satisfies (1). Before reviewing a few preliminary results in this direction, we establish some not a tion. F or an y J ⊆ { 0 , . . . , n − 1 } , w e let G p J := h r j | j 6∈ J i ; in particular, for k , l ∈ { 0 , . . . n − 1 } w e let G p k := h r j | j 6 = k i and G p k ,l := h r j | j 6 = k , l i . W e also let V J b e the subsp ace of V = Z n p spanned b y { b j | j 6∈ J } , and similarly for V k , V k ,l . Note that V J is G p J -in v arian t. In particular, G p j acts on V j , for j = 0 or n − 1. The upshot of Lemma 3.1 in [15] is that this a ctio n is faithful when p is generic for G . Referring t o [14, Eq. 12 ], w e record here a useful rule f or inductiv ely computing the determinan t o f the Gram matrix B = [ b i · b j ]. Letting B J b e the submatrix obtained b y deleting all rows and columns indexed by J , w e ha v e, for example, det( B ) = b 2 0 det( B 0 ) − ( b 0 · b 1 ) 2 det( B 0 , 1 ) . (2) W e will frequen t ly refer to the follow ing general prop erties of string C -g roups, here as they apply to t he gro ups G p : Prop osition 1.1 (a) G p is a string C -gr oup if an d on ly if G p 0 , G p n − 1 ar e string C -gr oups and G p 0 ∩ G p n − 1 = G p 0 ,n − 1 . (b) If G p is a s tring C -g r oup, then so to o is any sub gr oup G p J , for J ⊆ { 0 , . . . , n − 1 } . Pro of . See [12, 2E16 and 2E12].  In the next section we extract from [14] and [15] v a rious more sp ecialized criteria for G p to b e a string C-group. These concern the features of V as a n o rthogonal space, a s we ll as the action of standard subgroups of G p on V . Using them, w e w ere able in [14] to classify all gro ups G p , and their p olytop es, whenev er n ≤ 3, a s w ell as when G is of spherical or Euclidean t yp e, fo r a ll ranks n . Then in [1 5] we extended the classification to all cases in rank 4 . After generalizing these criteria, it will b e clear that w e hav e enough mac hinery to systematically extend our efforts to p olytop es of still higher rank. How ev er, already in rank 4 there is a b ewildering v ariet y of p ossibilities, so that b elow w e shall inv estigate only a few families of sp ecial in terest. 2 More on t he In tersecti on Prop er ty Let us review v arious situatio ns in which G p is guarantee d t o b e a string C - group. First of all, this will b e the case if one of the subgroups G p 0 or G p n − 1 is spherical and the other is a 3 string C-group: Theorem 2.1 [1 4, Th. 4.2] L et G = h r 0 , . . . , r n − 1 i b e a crystal lo gr aphic line ar Co xeter g r oup with string diagr am, and supp ose the prim e p ≥ 3 . If G n − 1 is of sph eric al typ e and G p 0 is a string C-gr oup, or (dual ly) i f G 0 is of spheric a l typ e an d G p n − 1 is a string C-gr oup, then G p is a string C-gr oup. W e note t ha t the pro of supplied in [14] is inadequate for the groups G = [6 , k ] with p = 3, though o nly for some of the p ossible basic systems (whic h need not b e equiv alent in these non-generic cases). A familiar example, taking k = 3, is the Euclidean group with diagram 1 • 3 • 3 • . Nev ertheless, t he inte rsection condition can b e ve rified for all these groups, using GAP [4] or by hand. In fact, suc h p eculiar exceptions app ear o nly p eripherally in this pap er. The next tw o theorems utilize the o ccurence o f gr oups of ort hogonal type. Theorem 2.2 [1 4, Th. 4.1] L et G = h r 0 , . . . , r n − 1 i b e a crystal lo gr aphic line ar Co xeter g r oup with string diagr am, a nd supp ose the prim e p ≥ 3 . Supp ose that G p 0 and G p n − 1 ar e string C-gr oups, and that the s ubsp ac e V 0 ,n − 1 is non-s i n gular. Then if G p 0 ,n − 1 is the ful l ortho gonal gr oup O ( n − 2 , p, ǫ ) on V 0 ,n − 1 , G p must b e a string C-gr oup. W e no w take a closer lo ok at w a ys in whic h the geometry o f the v arious subspaces V , V 0 , V n − 1 or V 0 ,n − 1 affects the inte raction of the corresp onding subgroups of G p . The ful ly non-singular case, pro v ed in [1 5, Th. 3.2 ] and generalized in part (a) b elow , sometimes allo ws us to r eje ct large classes of gr o ups G p as C- g roups b ecause of the size of their subgroups G p 0 ∩ G p n − 1 . When w e lea ve the fully non- singular case, w e m ust adjust our a pproac h in v arious w ays , dep ending on whic h of the v ar io us subs paces is singular. The case in whic h just the midd le se c tion is singular, pro v ed in [15, Th. 3.3 ] and rep eated in (b) b elo w, can sometimes b e used to affirm the p olytopalit y of G p (see [15, Cor. 3 .2]). In an y am bien t space V , each non-singular subsp ace W induces an orthogonal direct sum V = W ⊥ W ⊥ ([2, Ch. 6, L emma 2.1]). No w consider O ( W ), the orthogona l group for W (equipp ed with the bilinear form inherited from V ). It is easy to c hec k that t he ma pping λ : O ( W ) − → Stab O ( V ) W ⊥ g − → g ⊥ 1 W ⊥ (3) establishes a n isomorphism b etw een O ( W ) a nd a subgroup of the p ointwise stabilizer of W ⊥ in O ( V ). W e may therefore iden tify O ( W ) with this subgroup; this is done without muc h commen t for sev eral subspac es W in Theorem 2.3 b elo w. If V happ ens to b e non-singular, then the spinor norm on O ( W ) is also inv ariant under this iden tification [1, Th. 5.13], and w e clearly hav e O ( W ) ≃ Stab O ( V ) W ⊥ . Let us now turn t o the subgroup O 1 ( W ) := h r a | a ∈ W , a 2 = 1 i . By [14, Prop. 3.1], O 1 ( W ) almost alwa ys coincides with the k ernel of the spinor norm on O ( W ) and so t hen 4 has index 2 in O ( W ). Ho w ever, for dim( W ) ≥ 2 there are t w o exceptions to this: if O ( W ) is isomor phic to either [ B 3 ] 3 ≃ O (3 , 3 , 0) (with disc ∼ 1) or [ F 4 ] 3 ≃ O (4 , 3 , +1) , (4) then O 1 ( W ) has index 3 in the spinor kerne l [14, p. 30 1]. In similar fashion w e can w ork with a singular subspace W of a non- singular space V . Here we let b O ( W ) denote t he subgroup of O ( W ) consisting of those isometries whic h act trivially on r a d W (see [15, Section 3]). It is not hard to sho w that b O ( W ) contains a nd is generated by all reflections with non- isotropic ro ots in W . F urthermore, we may define a spinor n orm θ on O ( W ); and b O 1 ( W ) will denote the subgroup of b O ( W ) g enerated b y reflections in O ( W ) with square spinor norm. In the pro of of [1 5, Th. 3.3], w e emplo y ed a v ariant of the mapping in ( 3) to sho w that b O ( W ) can also b e iden tified with a suitable subgroup of the p oin t wise stabilizer of W ⊥ in O ( V ), so long as W is a subspace V 0 ,n − 1 (of codimension 2 in V ); this is the only case that w e require. Again w e find that b O 1 ( W ) usually has index 2 in b O ( W ) ; for dim ( W ) ≥ 2, exceptions o ccur when O ( W / rad( W )) is either O (2 , 3 , +1 ) or one of the g r o ups in (4). Let us a ssem ble our old results, along with some new criteria, in to o ne pac k a ge: Theorem 2.3 L et G = h r 0 , . . . , r n − 1 i b e a crystal l o gr aphic li n e ar Coxeter gr oup with string diagr am. Supp o s e that n ≥ 3 , that the p rime p is generic for G a nd that ther e is a squar e among the lab els of the no des 1 , . . . , n − 2 of the diag r am ∆( G ) (this c an b e achieve d by r e adjusting the no de lab els). F or various subsp ac es W o f V we identify O ( W ) , b O ( W ) , etc. with suitable sub gr o ups of the p ointwise stabilizer of W ⊥ in O ( V ) . (a) L et the subsp ac es V 0 , V n − 1 and V 0 ,n − 1 b e non-sing ular, and let G p 0 , G p n − 1 b e of ortho gonal typ e. (i) Then G p 0 ∩ G p n − 1 acts trivial ly on V ⊥ 0 ,n − 1 and O 1 ( V 0 ,n − 1 ) ≤ G p 0 ∩ G p n − 1 ≤ O ( V 0 ,n − 1 ) . (ii) If G p 0 = O ( V 0 ) and G p n − 1 = O ( V n − 1 ) , then G p 0 ∩ G p n − 1 = O ( V 0 ,n − 1 ) . (iii) If either G p 0 = O 1 ( V 0 ) or G p n − 1 = O 1 ( V n − 1 ) , then G p 0 ∩ G p n − 1 = O 1 ( V 0 ,n − 1 ) . (b) L et V , V 0 , V n − 1 b e non-singular, let V 0 ,n − 1 b e sin gular (so that n ≥ 4 ), and let G p 0 , G p n − 1 b e of ortho gonal typ e. (i) Then G p 0 ∩ G p n − 1 acts trivial ly on V ⊥ 0 ,n − 1 , and b O 1 ( V 0 ,n − 1 ) ≤ G p 0 ∩ G p n − 1 ≤ b O ( V 0 ,n − 1 ) . (ii) If G p 0 = O ( V 0 ) and G p n − 1 = O ( V n − 1 ) , then b O ( V 0 ,n − 1 ) = G p 0 ∩ G p n − 1 . (iii) If either G p 0 = O 1 ( V 0 ) or G p n − 1 = O 1 ( V n − 1 ) , then b O 1 ( V 0 ,n − 1 ) = G p 0 ∩ G p n − 1 . (c) Supp ose V , V 0 ,n − 1 ar e non-singular while at le ast one of V 0 , V n − 1 is si n gular. Also supp os e that G p 0 ,n − 1 is of ortho gonal typ e, with G p = O 1 ( V ) when G p 0 ,n − 1 = O 1 ( V 0 ,n − 1 ) . Then G p 0 ∩ G p n − 1 = G p 0 ,n − 1 . Pro of . When V is non-singular, parts (a)–(i),(ii) app ear as Theorem 3.2 in [1 5 ]. F or V singular, r ad( V ) = h c i is 1-dimensional, and w e may c ho o se a basis w , w ′ for V ⊥ 0 ,n − 1 so that c = w + w ′ , V n − 1 = V 0 ,n − 1 ⊥ h w i , V 0 = V 0 ,n − 1 ⊥ h w ′ i and V = V n − 1 ⊥ h v i = V 0 ⊥ h v ′ i , with 5 v = v ′ = c . Then g ∈ G p 0 ∩ G p n − 1 implies that g ( w ) = α w , g ( w ′ ) = α ′ w ′ , where α, α ′ ∈ {± 1 } . Since g ( c ) = c , w e hav e α = α ′ = 1, so that g ∈ O ( V 0 ,n − 1 ). The rest of the pro of of (i) and (ii) pro ceeds as in [15, Th. 3.2]. F or (a)–(iii) w e may supp ose G p 0 = O 1 ( V 0 ). When n = 3 t here is nothing t o pro v e, since no de 1 has a square lab el and so O ( V 0 , 2 ) = O 1 ( V 0 , 2 ). No w supp ose n − 2 ≥ 2, so that there exists a r eflection r ∈ O ( V 0 ,n − 1 ) with non-squar e spinor norm. Since r 6∈ O 1 ( V 0 ) = G p 0 , we m ust b y (i) ha ve G p 0 ∩ G p n − 1 = O 1 ( V 0 ,n − 1 ), so long as O 1 ( V 0 ,n − 1 ) has index 2 in O ( V 0 ,n − 1 ). As we observ ed earlier, this almost alw ays holds. In fa ct, neither of the groups indicated in (4) can o ccur in our setup (as O ( V 0 , 4 ) , O ( V 0 , 5 ), respectiv ely). Indeed, since p = 3 in either case and since G p 0 = G 3 0 = O 1 ( V 0 ), no des 1 , 2 , . . . , n − 1 m ust all b e lab elled by squares mo d 3. Also, p is generic for G . These t w o restrictions imply that each of the standard rotations r j − 1 r j in G , exce pt p ossibly for j = 1, must hav e p erio d 3 or ∞ (or 2, if ∆( G ) is disconnected). In any case, G 3 0 ≃ S a 1 × . . . × S a k is a direct pro duct of k ≥ 1 symmetric groups, where ( a 1 − 1) + . . . + ( a k − 1) = n − 1 (= 4 o r 5 in the t w o cases). A direct ch ec k of the p o ssible orders sho ws that G 3 0 could not t hen b e of orthogo na l t yp e in dimension 4 or 5 resp ectiv ely . In (b) w e ha ve n ≥ 4; indeed, for n = 3 w e note that V 0 , 2 m ust b e non-singular, since p is generic for G . P ar t s (i),(ii) app ear as Theorem 3.3 in [15]. W e settle part (iii) in muc h the same w a y as for (a)–(iii) ab ov e. Supp ose that G p 0 = O 1 ( V 0 ) and let X := V 0 ,n − 1 / rad( V 0 ,n − 1 ), a non-singular space of dimension n − 3. (Note that X ≃ V 0 , 1 ,n − 1 ≃ V 0 ,n − 2 ,n − 1 .) If n = 4, the in v ar ia n t quadratic form induced on X m ust b e equiv alent to x 2 1 ; then b O ( V 0 , 3 ) = b O 1 ( V 0 , 3 ) and (b)–(iii) follo ws trivially . Now supp ose that n ≥ 5 . By our earlier remarks, b O 1 ( V 0 ,n − 1 ) usually has index 2 in b O ( V 0 ,n − 1 ), in whic h case (b)–(iii) f ollo ws easily . The three exceptional cases ha v e p = 3 with n = 5 , 6 , 7. But as in pa r t (a)–(iii) ab ov e, these groups cannot o ccur as O ( X ) when G p 0 = O 1 ( V 0 ) or G p n − 1 = O 1 ( V n − 1 ) . In pa r t (c) there are t w o v ery similar cases, dep ending on whether one or b oth of V 0 , V n − 1 are singular. T o b egin with, eac h g ∈ G p 0 ∩ G p n − 1 certainly fixes rad( V 0 ) and rad( V n − 1 ) p oin t wise. It is then easy to show in the t w o cases that g fixes V ⊥ 0 ,n − 1 p oin t wise, so that g ∈ O ( V 0 ,n − 1 ). Next one show s that O 1 ( V ) ∩ O ( V 0 ,n − 1 ) = O 1 ( V 0 ,n − 1 ), using the assumption on square lab els and [1, Th. 5.13]. (Here, to o, we m ust consider, and aga in exclude, the p ossibilit y that O ( V 0 ,n − 1 ) is one of the groups in (4).) Since either G p = O 1 ( V ) or G p 0 ,n − 1 = O ( V 0 ,n − 1 ), w e now ha v e g ∈ G p 0 ,n − 1 .  Theorem 2.3 has sev eral immediate and useful consequences. F or example, in [15, Cor. 3.2], we used a preliminary ve rsion of part (b)(iii) to prov e that [ k , ∞ , m ] p is a C -group for an y o dd prime p and in tegers k , m ≥ 2. On the other hand, part (a )(iii) led just as easily to a pro of that [ ∞ , 3 , ∞ ] p is a C -group only when p = 3 , 5 , 7. Next w e g eneralize [15, Th. 3.4], whic h concerns 4- p olytop es for whic h the facet group G 3 (sa y) is Euclidean and so situated tha t the ‘p oint group’ acts on the middle section of the p olytop e. Our first step is a closer lo ok at the geometric action of groups of affine Euclidean isometries. In the bac kground w e typically hav e an abstract Co xeter gro up of Euclidean (or ‘affine’) type, f aithfully represen ted in the standard w a y as a linear reflection group E = h r 0 , . . . , r m i on real ( m + 1)-space W . R ecall that E preserv es a p ositiv e semidefinite form x · y , so that rad( W ) = h c i is 1- dimensional. Since r j ( c ) = c , for 0 ≤ j ≤ m , E is in 6 fact a subgro up o f b O ( W ) . T o actually exploit the structure of E as a group of isometries on Euclide an m -space, w e pass to the contragredien t represen t a tion of E in the dual space ˇ W (a s in [7, 5 .1 3]). Since c is fixed b y E , w e see t ha t E lea v es inv ariant any translate of the m -space U = { µ ∈ ˇ W : µ ( c ) = 0 } . Next, for eac h w ∈ W define µ w ∈ ˇ W b y µ w ( x ) := w · x . The mapping w 7→ µ w factors to a linear isomorphism b etw een W / rad( W ) and U , and so w e t ransfer to U the p ositive definite form induced b y W on W / rad( W ). No w c ho ose a n y α ∈ ˇ W suc h that α ( c ) = 1 , and let A m := U + α . Putting all this together w e may now think of A m as Euclide an m -sp ac e , with U as its sp ac e of tr a nslations . Indeed, each fixed τ ∈ U defines an isometric translation on A m : µ 7→ µ + τ , ∀ µ ∈ A m . It is easy to c hec k that this mapping on A m is induced by a unique isometry t ∈ b O ( W ) , namely the tr ansve ction t ( x ) = x − τ ( x ) c, = x − ( x · a ) c, where τ = µ a for suitable a ∈ W . (R emem b er here that we emplo y the con tragredien t represen tation of b O ( W ) on ˇ W , not just tha t of E .) In summary , w e can therefore safely think o f translations as tr a nsv ections. In the fo llowing t able we list those Euclidean Co xeter groups whic h are relev ant to our analysis (see [14, § 6B]). Concerning the group E = [4 , 3 m − 2 , 4] (for the f a milar cubical tessellation of A m ), w e recall our con ven tion tha t 3 m − 2 indicates a string of m − 2 ≥ 0 consecutiv e 3’s. The group E m = dim( A m ) One p ossible diagram The corresp onding v ector ∆( E ) c ∈ rad( W ) [4 , 3 m − 2 , 4] m ≥ 2 2 • 1 • 1 • · · · 1 • 1 • 2 • c = b 0 + 2( b 1 + . . . + b m − 1 ) + b m [3 , 3 , 4 , 3] 4 1 • 1 • 1 • 2 • 2 • c = b 0 + 2 b 1 + 3 b 2 + 2 b 3 + b 4 [3 , 6] 2 1 • 1 • 3 • c = b 0 + 2 b 1 + b 2 [ ∞ ] 1 1 • = = = 1 • c = b 0 + b 1 T a ble 1. Euclidean Co xeter Groups An in v estigation of the action of these discrete reflec tion g roups o n the Euclidean m -space A m sho ws, in eac h case, that E splits as the semidirect pro duct of the (normal) subgroup T of t r a nslations with a certain ( finite) p oint gr o up group H : E ≃ T ⋊ H . (5) 7 (See [7, Prop. 4 .2].) W e can and do displa y each group in the table so that H = E 0 = h r 1 , . . . , r m i . Returning now to our generalization of [1 5, Th. 3.4 ], w e supp ose that G n − 1 is of Euclidean t ype. Of course, a dual result holds when G 0 is Euclidean. Theorem 2.4 L et G = h r 0 , . . . , r n − 1 i b e a crystal l o gr aphic li n e ar Coxeter gr oup with string diagr am. Supp ose that G n − 1 is Euclide an, with G n − 1 = T ⋊ G 0 ,n − 1 , wh e r e T is the tr ans l a tion sub gr oup of G n − 1 . Supp os e also that the prime p is gene ric for G , and that G p 0 is a C -g r oup. Then G p is a C -g r oup. Pro of. The subgroup G p n − 1 of G p lea v es in v arian t the subs pace V n − 1 of V . Since p is g eneric for G , w e may conclude from [15, Lemma 3.1] that this action is faithf ul. Th us G p n − 1 is a string C -group of Euclidean ty p e, as described in [14, § 6B]. By Prop osition 1.1(a), our task is therefore to sho w that G p 0 ∩ G p n − 1 = G p 0 ,n − 1 ; so consider a ny g ∈ G p 0 ∩ G p n − 1 . Now since G n − 1 = T ⋊ G 0 ,n − 1 pro jects o nto G p n − 1 , w e can m ultiply g b y a suitable eleme n t of G p 0 ,n − 1 , and thereb y assume that g ∈ T p . W e wan t to show that g = e . W e observ ed earlier that g acts as a tra nsv ection on V n − 1 , with g ( x ) − x ∈ h c i = rad( V n − 1 ) for all x ∈ V n − 1 . On the other hand, since g ∈ G p 0 ∩ G p n − 1 w e ha v e g ( x ) − x ∈ V 0 ,n − 1 . Finally , w e observ e that h c i ∩ V 0 ,n − 1 = { 0 } by direct insp ection o f the v arious cases exhibited in T able 1, taking m = n − 1. (In most cases this trivial in tersection is implied directly b y the f act that G 0 ,n − 1 is o f spherical type.) Th us g ( x ) = x for all x ∈ V n − 1 . Inv ariably f or us the final no de n − 1 in the diagram ∆( G ) will b e connected to no de n − 2, so that disc( V ) ∼ − disc( V n − 2 ,n − 1 ) by a dual vers ion of (2). The latter discriminan t is non-zero for all groups G n − 1 encoun tered here, again b ecause p is generic for G . Since V n − 1 is therefore a singular subspace of the non-singular space V , w e conclude from [1, Th. 3.17] t ha t g = e . This completes the pro of in all imp ortant cases. (It is p ossible that no des n − 1, n − 2 b e non-adja cen t; but then it is easy to c hec k directly that G p ≃ G p n − 1 × C 2 is a C -group.)  3 The 3 -infi n it y g roups The large num b er of crystallographic Cox eter groups G = [ p 1 , . . . , p n − 1 ] of higher ranks mak es it difficult to fully en umerate the regular p olytop es obtained by our metho d. How ev er, it is clear tha t man y in teresting examples o ccur. As a test o f our metho ds, w e surv ey in this section groups G = [ . . . , 3 k , ∞ l , 3 m , . . . ] , of g eneral rank n a nd having a ll p erio ds p j ∈ { 3 , ∞} . When p j = ∞ , it is con ve nien t to emplo y the basic sys tem defined by the sub diagra m . . . 1 • = = = 1 • . . . on no des j − 1 , j . (Th us, 1 = b 2 j − 1 = b 2 j = − b j − 1 · b j .) T ypically then, ∆( G ) consists of alternat ing strings o f single and doubled branche s, as in . . . 1 • = = = 1 • − − − 1 • − − − 1 • = = = 1 • = = = 1 • = = = 1 • . . . . 8 F or the prime p = 3, each rotation r j − 1 r j in G 3 has p erio d 3, and w e clearly obtain G 3 ≃ A n ≃ S n +1 , regardless of the a llo cation of branches [14, 6.1]. Lik ewise, if no p j = ∞ , t hen G p ≃ A n for an y prime p ≥ 3. Th us, w e may henceforth a ssume when it suits us that p ≥ 5 and that ∆( G ) has at least o ne doubled branc h. If in this case V is non-singular, then G p ≃ O 1 ( V ) is of o rthogonal t yp e (see [14, Th. 3.1]). Our approa ch now m ust b e inductiv e on the size o f certain classes of sub diagrams in ∆( G ); but first w e mus t determine the orthogo nal structures on V , V 0 , V n − 1 and V 0 ,n − 1 . F or n ≥ 1 w e let d n := disc( V ) b e the discriminan t of the underlying basic sys tem for G = [ ∞ n − 1 ], as enco ded in t he dia gram ∆( G ) = 1 • = = = 1 • = = = 1 • = . . . = 1 • = = = 1 • = = = 1 • (on n no des). F rom (2) w e ha ve d n = 1 d n − 1 − 1 2 d n − 2 = d n − 1 − d n − 2 , for n ≥ 2 and taking d 0 := 1 . Th us d n =    1 if n ≡ 0 , 1 mo d 6 , 0 if n ≡ 2 , 5 mo d 6 , − 1 if n ≡ 3 , 4 mo d 6 . (6) Again using (2), w e find that t he basic system underpinning [3 n − 1 ] has disc( V ) = ( n + 1) / 2 n . A routine induction then giv es the discriminan t e k ,l,m corresp onding to the basic system for the group G = [3 k , ∞ l , 3 m ], with k + l + m = n − 1 and k , l, m ≥ 0. Th us e k ,l,m = 1 2 k + m +2 [ d l +1 (4 + 2 k + 2 m ) − d l − 1 (2 k + 2 m + 3 k m ) ] . (7) In certain singular cases we ha v e this Lemma 3.1 L et G = [3 k , ∞ l ] , with k + l = n − 1 and l ≥ 1 . Supp ose that the c orr esp onding sp ac e V is singular for the p rim e p . Then G p = b O 1 ( V ) . Pro of . Clearly G p ≤ b O 1 ( V ). Now supp ose that c = P n − 1 j =0 x j b j ∈ ra d( V ). Note that eac h scalar b j − 1 · b j ∈ {− 1 / 2 , − 1 } and is t herefore in v ertible in Z p . Th us, fro m 0 = b 0 · c = x 0 + ( b 0 · b 1 ) x 1 w e o bta in x 1 = α 1 x 0 , where α 1 ∈ { 1 , 2 } . Since ∆( G ) is a tree, w e can con tin ue to solv e for x 2 , . . . , x n − 1 as m ultiples of x 0 to obtain x j = α j x 0 for v arious α j (with α 0 := 1 ) , where α 0 , . . . , α n − 1 are determined o nly b y the basic system for G . In the end, as V is singular, the equation 0 = b n − 1 · c m ust b e redundan t, so w e ha v e rad( V ) = h c i (with x 0 6 = 0). Anyw a y , we ma y no w take c = 1 b 0 + α 1 b 1 + . . . + α n − 1 b n − 1 . Then V = h c i ⊥ V 0 , where V 0 is no n-singular a nd G p 0 ≃ O 1 ( V 0 ) (b ecause l ≥ 1 ). W e t h us ha v e b O 1 ( V ) = T ⋊ G p 0 , (8) where T ≃ Z n − 1 p is the ab elian group generated b y transv ections t 1 , . . . , t n − 1 satisfying t j ( b i ) = b i + δ i,j c and t j ( c ) = c , for 1 ≤ i, j ≤ n − 1. No w r 0 ∈ b O 1 ( V ) induces an isometry on V / rad( V ) ≃ V 0 . Let h ∈ G p 0 b e the isometry corresp onding to r 0 under the natural isomorphism b et w een O 1 ( V / rad( V )) and O 1 ( V 0 ) ≃ G p 0 . A short calculation sho ws 9 that t 1 = ( h − 1 r 0 ) q ∈ G p , where q = 1 or ( p + 1) / 2, according as r 0 r 1 has p erio d 3 or ∞ (in c ha racteristic 0 ). Finally w e show inductiv ely that t j ∈ G p for 1 ≤ j ≤ n − 1; this implies that G p = b O 1 ( V ). Fixing j < n − 1 w e ma y suppose t i ∈ G p for all 1 ≤ i ≤ j . F rom [14, Eq. (9)] w e note that r j ( b j − 1 ) = b j − 1 + αb j , r j ( b j ) = − b j and r j ( b j +1 ) = b j +1 + β b j , where in our case the Cartan inte gers α , β ∈ { 1 , 2 } ; otherwise, for | k − j | > 1, r j ( b k ) = b k . It is then a routine matter to c heck t hat t − α j − 1 t j r j t j r j = t β j +1 . It follo ws by induction that t β j +1 ∈ G p and hence t j +1 ∈ G p .  F rom [14, § 5] and [15, § 5], w e a lr eady kno w that [3 k , ∞ l ] p is a string C - g roup for all p ≥ 3 and ra nks n ≤ 4 (so that k + l ≤ 3). F or example, [ ∞ , ∞ , ∞ ] p is the automorphism group o f a self-dual regula r 4-p olytop e P of t yp e { p, p, p } ; when p = 5 w e find tha t P is isomorphic to the classical star-p o lytop e { 5 , 5 2 , 5 } (see [3, Ch. XIV] and [8 ]) . Similarly , [3 , ∞ , ∞ ] 5 giv es bac k the regula r star- p olytop e { 3 , 5 , 5 2 } . Let us tak e sto ck of our prog ress so f a r. Keep in mind that whenev er G p is a string C -group, so to o is the dually generated group, with k and m in terc hanged. Theorem 3.1 L et G = [3 k , ∞ l ] , with k + l = n − 1 . Then for al l primes p ≥ 3 , the gr oup G p = [3 k , ∞ l ] p is a s tring C -g r oup. Pro of . F or l = 0 we hav e already observ ed that G p = [3 n − 1 ] p ≃ S n +1 , the g roup of the n -simplex. Let us no w disp ose of the case l = 1. Since the facet group [3 k ] is sphe rical, an induction on k , together with Theorem 2.1, sho ws that G p = [3 k , ∞ ] p is a string C -group. No w we may supp ose l ≥ 2. F rom (7), with m = 0, w e ha v e e k ,l, 0 = 1 2 k +1 [( k + 2) d l +1 − k d l − 1 ] . (9) W e may also supp ose that p ≥ 5. Then it is easy to c hec k that if an y o ne o f the spaces V , V 0 , V n − 1 , V 0 ,n − 1 is singular for a given prime p , all t he others m ust b e non-singular. W e also kno w, as a basis for induction, that [3 k , ∞ l ] p is a string C - group whenev er k + l ≤ 3. Thus w e ma y supp ose that G p 0 and G p n − 1 are string C -g r oups. If V 0 , V n − 1 , V 0 ,n − 1 are non-singular, then the corresp onding subgroups o f G p are all of t ype O 1 , since l ≥ 2 . By Theorem 2.3(a)–(iii) w e ha v e G p 0 ∩ G p n − 1 = O 1 ( V 0 ,n − 1 ) = G p 0 ,n − 1 . Th us G p is a string C -group b y Prop o sition 1.1(a). If V 0 or V n − 1 is singular, w e similarly apply Theorem 2 .3(c). Finally , if just V 0 ,n − 1 is singular, w e emplo y Theorem 2.3(b)– (iii) and a pply Lemma 3.1 to the subspace V 0 ,n − 1 .  As an example, consider the g roup G = [3 , 3 , 3 , ∞ ] of rank 5. Here we obtain regula r 5-p olytop es of type { 3 , 3 , 3 , p } with g roup O 1 (5 , p, 0). A particularly interesting case o ccurs when p = 5. Then the p olytop e can b e view ed as a regular tessellation of t yp e { 3 , 3 , 3 , 5 } on a h yp erb olic 4-manifo ld whose 7 8 000 tiles (facets) are 4-simplices and whose 65 0 vertex - figures are 600-cells (see [12, 6J]). The dual tessellation has 12 0-cells as tiles and 4 -simplices as v ertex-figures. (As a n aside, whe n p = 5, the group G = [4 , 3 , 3 , ∞ ] similarly gives a regular tessellation of type { 4 , 3 , 3 , 5 } with group O (5 , p, 0) on a h yperb olic 4-ma nif o ld, whose facets a re 4-cub es and whose v ertex-figures are 600-cells. The dual tessellation again has 120-cells as tiles and 4- crossp olytop es as v ertex-figures.) 10 No w let us generalize a little and consider G = [3 k , ∞ l , 3 m ], with k + l + m = n − 1. All cases with k l m = 0 are co v ered b y Theorem 3.1, so w e assume k , l , m ≥ 1. Then the o nly new string C -gro up of small rank is [3 , ∞ , 3]. Again, we tackle G p inductiv ely; but since the details a re more complicated, w e shall settle for sligh tly less comprehensiv e results. Theorem 3.2 L et G = [3 k , ∞ l , 3 m ] , with k + l + m = n − 1 and k , l , m ≥ 1 ; and supp ose p is an o dd prime. Then (a) G p = [3 , ∞ l , 3] p , with n = l + 3 , l ≥ 1 is a string C -g r oup, exc ept p ossibly when p = 7 and l ≥ 4 , with l ≡ 1 mo d 3 . (b) G p = [3 k , ∞ l , 3 m ] p , with k > 1 or m > 1 , and l ≥ 1 , is a string C -gr oup for al l but finitely many primes p . Pro of . The details in part (a) are ve ry similar to those for Theorem 3.1, whic h also serv es as a basis for our induction. The analysis there fails only when the subspaces V , V 0 , V n − 1 , V 0 ,n − 1 are singular in more than one of dimensions n, n − 1 and n − 2; t his can only o ccur when l ≡ 1 mo d 3 and p = 7 (in this case exactly V and V 0 ,n − 1 are singular). In fact, when p = 7, l = 1, we can use GAP to v erify t hat G 7 is a string C -group any w ay [14, p. 347]. F or par t (b), we notice in (7 ) that e k ,l,m = 0 (in c haracteristic 0) only when k = m = 0 and l ≡ 1 mo d 3. T ypically then, G p falls under the fully non-singular case describ ed in Theorem 2.3 (a)–(iii).  Remarks . It may w ell b e in the previous Theorem that G p is a C -gro up for all primes. Certainly in specific cases, one can explicitly list the ‘doubtful’ pr imes; but there seems to b e little serv ed b y trying to do more here. No w w e hun t for con trary cases in whic h G p is not a string C -g r o up. Once a gain w e ma y assume p ≥ 5. Eviden tly w e should examine the groups G = [ ∞ , 3 k , ∞ ], with k ≥ 1, for whic h w e ha v e disc( V ) = ( k − 2 ) / 2 k +1 . By Theorem 3.1, bot h G p 0 and G p n − 1 are string C -groups; and disc( V 0 ) = disc( V n − 1 ) = − k / 2 k +1 . Finally , G p 0 ,n − 1 = [3 k ] p is the symmetric group o f or der ( k + 2)!, and disc( V 0 ,n − 1 ) = ( k + 2) / 2 k +1 . Lemma 3.2 S upp ose G = [3 k ] , acting a s usual on the ( k + 1 )–dimensiona l sp ac e V ; an d let p ≥ 5 . (a) If p ∤ ( k + 2) , then V is non-singular and G p ≤ O 1 ( V ) , with e quality only when O ( V ) = O (2 , 5 , − 1) or O (2 , 7 , +1) . (b) If p | ( k + 2) , then V is singular and G p is always a pr op er sub gr oup of b O 1 ( V ) . Pro of . Recall tha t G p ≤ O 1 ( V ) , b O 1 ( V ) in (a ), (b) resp ectiv ely . W e ha v e equalit y in (a) only when | O 1 ( V ) | = ( k + 2)!. But the highest p ow er of p dividing ( k + 2)! is p ν , where ν = ⌊ k + 2 p ⌋ + ⌊ k + 2 p 2 ⌋ + . . . < k + 2 p (1 + 1 p + . . . ) = k + 2 p − 1 , (see [5, Prop. 2.3.2]). T aking n = k + 1 in [14, § 3.1], w e find that the hig hest p ow er p µ dividing | O 1 ( V ) | for a non-singular space V has µ = ⌊ k 2 4 ⌋ . (Since p > 3 w e aga in ignore the 11 groups in (4).) Clearly , w e usually ha v e ν < µ , and it is easy b y insp ection to determine the t w o cases with G p = O 1 ( V ). The situation for (b) is similar. The splitting in (8) contin ues to hold in the presen t con text ( in whic h l = 0); from t his w e get µ = k + ⌊ ( k − 1) 2 4 ⌋ and ultimately no cases of equality at all.  No w we can sho w that G p is very oft en not a string C -group: Theorem 3.3 S upp ose that G has a string sub gr oup of the form [ . . . , ∞ , 3 k , ∞ , . . . ] , with k ≥ 1 . L e t p ≥ 5 . The n G p is not a string C -gr oup, exc ept p os sibly when p = 5 or 7 and k ≤ 1 for al l such string sub gr oups [ . . . , ∞ , 3 k , ∞ , . . . ] . Pro of . Since our inte n tion is to show that G p fails to b e a string C - group, w e ma y assume b y Prop o sition 1.1(b) that G = [ ∞ , 3 k , ∞ ]. If V 0 ,n − 1 is singular, then p | ( k + 2), so that p ∤ k ( k − 2), implying t ha t V , V 0 , V n − 1 are non-singular. By Theorem 2.3 (b)–(iii) and Lemma 3.2(b) (a pplied to V 0 ,n − 1 ), we hav e G p 0 ∩ G p n − 1 = b O 1 ( V 0 ,n − 1 ) 6 = G p 0 ,n − 1 . Th us G p is no t a string C -group. Supp ose t ha t V 0 ,n − 1 is non-singular, with p > 7 when k = 1 . Th us, p ∤ ( k + 2), and G p 0 ,n − 1 6 = O 1 ( V 0 ,n − 1 ) b y Lemma 3.2(a). If also p ∤ k , then w e hav e G p 0 ∩ G p n − 1 = O 1 ( V 0 ,n − 1 ) b y Theorem 2.3(a )–(iii), so that G p is not a string C -group. Finally , supp ose p | k . Thus , k ≥ 5, and this gives enough wiggle ro om to destro y p olytopality in a nother w a y . Let I = { 0 , 1 } , J = { 5 , . . . , n − 1 } . Recall, for example, that G p J denotes the subgroup generated b y the complemen ta r y set of reflections r 0 , . . . , r 4 ; these reflections leav e in v arian t the subspace V J spanned b y { b j | j 6∈ J } = { b 0 , . . . , b 4 } . But disc( V I ) ∼ (1 − k ) / 2 k , disc ( V J ) ∼ − 3 and disc( V I ∪ J ) ∼ 1 / 2, so tha t these subspaces are a ll non-singular subspaces of the non-singular space V . Hence, G p I = O 1 ( V I ) ≥ O 1 ( V I ∪ J ) and G p J = O 1 ( V J ) ≥ O 1 ( V I ∪ J ). Th us, if G p is a string C -group, w e must hav e O 1 ( V I ∪ J ) ≤ G p I ∩ G p J = G p I ∪ J = h r 2 , r 3 , r 4 i p ≃ S 4 . But O 1 (3 , p, 0) has order p ( p 2 − 1) > 24, for p ≥ 5.  W e can summ arize the results of this section as follow s: if G = [ p 1 , . . . , p n − 1 ] has eac h p erio d p j ∈ { 3 , ∞} , then except for a few small primes, we cannot exp ect G p to b e a string C -group if t w o of the p j ’s ar e ∞ ’s separated b y a string of 3 ’s. That is, o nly the gro ups [3 k , ∞ l , 3 m ] can give a string C - g roup, and they do for most primes p . 4 Lo cally t o roidal p ol yt o p es of ranks 5 or 6 The crystallographic string Coxe ter groups of spherical or Euclidean t ype, along with the asso ciated mo dular p olytop es, w ere describ ed in [14, § 5- 6]. When the gr oup is spherical with connected diagram on m + 1 no des, we o bta in (up to isomorphism) familar con v ex regular ( m + 1)-p olytop es. After cen t ral pro jection, suc h p olytop es can usefully b e view ed as regular spherical tessellations of the circumsphere S m . Lik ewis e, each Euclidean group E a cts as the f ull symm etry group of a certain regular tessellation of Euclidean space A m . Indeed, E m ust b e one of the Cox eter groups display ed 12 in T able 1, though p erhaps with generators specified in dual o rder. A r e gular ( m + 1)- tor oid P is the quotien t o f such a tessellation by a non-trivial normal subgroup L of translations in E . Th us ev ery toroid can b e view ed as finite, regular tessellation of the m -torus. W e refer to [12, 1D and 6D-E] fo r a complete classification; briefly , for eac h g roup E the distinct toroids are indexed by a typ e ve ctor q := ( q k , 0 m − k ) = ( q , . . . , q , 0 , . . . , 0), where q ≥ 2 and k = 1 , 2 or m . (F or G = [3 , 3 , 4 , 3], the case k = 4 is subsumed by the case k = 1.) Anyw a y , L is generated (as a normal subgroup of G ) by the translation t := t q 1 · · · t q k , where { t 1 , . . . , t m } is a standard set of generators for the full group T of translations in E . The mo dular toroids P ( E p ) described in [14, § 6B] are sp ecial instances; with one exception, w e had there q = ( p, 0 , . . . , 0). In this Section, w e consider lo c al ly tor o i d al regular p olytop es, that is, p olytop es in whic h the facets and vertex -figures are globally spherical or toroidal, as describ ed ab ov e (with at least one kind toro ida l). The n -p olytop es of this kind hav e not y et b een fully classified, although quite a lot has b een disco v ered since Gr ¨ un baum first prop osed the problem in t he 1970’s (see [6]). What is kno wn rests on a broad range of ideas, including frequen t use of unitary reflection groups, and ‘t wisting’ and ‘mixing’ op erations on presen tations for string C -groups. W e refer to [9, 10, 11] for some of the original in ve stigations, or to [12, Chs. 7-12 ] for a detailed surve y of the pro j ect. As usual, we b egin our o wn inv estigation with a crystallographic linear Co xeter group G = h r 0 , . . . , r m i , but immediately discard degenerate cases in whic h t he underlying diagram ∆( G ) is disconnec ted. ( In suc h cases G p is reducible ; a nd P ( G p ) has the sort of ‘flatness’ described in [12, 4 E].) In [15] w e discussed a ll lo cally toroidal 4- p olytop es P ( G p ) whic h arise from our construc- tion. T urning to higher rank n > 4, w e observ e that an y spherical facet, or v ertex-figure, m ust b e of type { 3 n − 2 } , { 4 , 3 n − 3 } , { 3 n − 3 , 4 } or { 3 , 4 , 3 } ( n = 5 only). Likew ise, the required Euclidean section m ust hav e type { 4 , 3 n − 4 , 4 } or when n = 6, { 3 , 3 , 4 , 3 } or { 3 , 4 , 3 , 3 } . As described in [12 , Lemma 10A1], these constraints sev erely limit the p ossibilities: in rank 5, we hav e just G = [4 , 3 , 4 , 3] acting on h yp erb olic space H 4 ; and in rank 6 w e ha v e G = [4 , 3 , 3 , 4 , 3 ] , [3 , 4 , 3 , 3 , 3] or [3 , 3 , 4 , 3 , 3], all acting on H 5 . Th us w e may complete our discussion by examining the mo dular p olytop es whic h result from these groups in ranks 5 or 6. 4.1 Rank 5 : the group G = [4 , 3 , 4 , 3] W e may supp ose that G has diagram 2 • 1 • 1 • 2 • 2 • . Note that G p 0 ≃ F 4 is a C - group b y [14, 6.3]. It f o llo ws at once from Theorem 2 .4, and a lo ok at T able 1, that G p is a C -group for an y prime p ≥ 3. (Alternativ ely , since the v ertex-figures are spherical, w e can app eal to Theorem 2.1.) Thus P ( G p ) is a lo cally toroida l regular p olytop e of t yp e { 4 , 3 , 4 , 3 } . Its v ertex-figures are copies of the 24-cell { 3 , 4 , 3 } . The 13 facets are toroids { 4 , 3 , 4 } ( p, 0 , 0) , which one could construct b y iden tifying opp o site square faces o f a p × p × p cub e. (See [14, 6.4]; w e note that the f acet and v ertex num b er men tioned there should b e p n − 1 rather than p n .) Of course, b y flipping the diagram end-f o r-end, w e just as easily obtain the dual p olytop e of t yp e { 3 , 4 , 3 , 4 } . In order to iden tify G p w e consult the list of irreducible reflection groups in [14, T able 1]. Since G p has an ab elian subgroup of order p 3 (generated b y translations in the facet), w e immediately rule out t he long shots A p 5 , B p 5 and D p 5 of o rders 6!, 2 5 5! and 2 4 5!, resp ectiv ely . Note also that the no de lab el 2 is a square mo d p if and only if p ≡ ± 1 (mo d 8). Our hand is now forced: P = P ( G p ) has automor phism group Γ( P ) =  O 1 (5 , p, 0) , if p ≡ ± 1 (mo d 8) O (5 , p, 0) , if p ≡ ± 3 (mo d 8) (10) F or an y prime p ≥ 3, O 1 (5 , p, 0) has order p 4 ( p 4 − 1)( p 2 − 1) and index tw o in O (5 , p, 0) (see [14, pp. 300-301 ]). The univ ersal lo cally toroidal p olytop es of rank 5 are completely describ ed in [12, 1 2B]. All but t hr ee a r e infinite. ( The three finite instances ha v e facets with t yp e v ector (2 , 0 , 0), (2 , 2 , 0) or (2 , 2 , 2); but clearly these do not o ccur when the mo dulus p is an o dd prime.) In short, P ( G p ), b eing finite, is nev er univ ersal for it s type. Using the fact that disc( V ) ∼ − 2, w e compute that the cen tral isometry − e ∈ G p , except when p ≡ − 1 (mo d 8). When − e ∈ G p , the p o lytop e P ( G p ) doubly co v ers t he quotient p olytop e P ( G p / {± e } ). The latter p olytop e still has the same facets and v ertex-figures as P ( G p ). T o v erify these claims, w e apply [12, 2E19 ], so w e must show that {± e } ∩ G p n − 1 G p 0 = { e } . Supp ose, o n the con trary , t hat − e ∈ G p n − 1 G p 0 . Since G p n − 1 = G p 4 = T ⋊ G p 0 , 4 , w e can assume − e = th , where h ∈ G p 0 ≃ F 4 and t is some transve ction. Th us h = − t − 1 , which has p erio d 2 p . This is already imp ossible if p > 3, since | F 4 | = 3 · 2 4 · 4!. Eve n when p = 3 it is easy to v erify the con tradiction directly . 4.2 Rank 6 : the groups [3 , 4 , 3 , 3 , 3] , [ 3 , 3 , 4 , 3 , 3 ] and [4 , 3 , 3 , 4 , 3] In rank 6 we mu st consider three closely related groups, b eginning with G = h r 0 , r 1 , r 2 , r 3 , r 4 , r 5 i ≃ [3 , 4 , 3 , 3 , 3] . W e may describ e a basic system (of ro ots) fo r G b y the diagram 1 • 1 • 2 • 2 • 2 • 2 • . (11) No w the subgroup H = h s 0 , . . . , s 5 i g enerated b y the reflections ( s 0 , s 1 , s 2 , s 3 , s 4 , s 5 ) := ( r 1 , r 0 , r 2 r 1 r 2 , r 3 , r 4 , r 5 ) (12) 14 has index 5 in G and is isomorphic to [3 , 3 , 4 , 3 , 3]. The basic system of ro ots attac hed to the s j ’s pro vides the dia gram 1 • 1 • 1 • 2 • 2 • 2 • (13) for H . Another subgroup K = h t 0 , . . . , t 5 i generated b y ( t 0 , t 1 , t 2 , t 3 , t 4 , t 5 ) := ( r 2 , r 1 , r 0 , r 3 r 2 r 1 r 2 r 3 , r 4 , r 5 ) (14) has index 10 in G , is isomorphic to [4 , 3 , 3 , 4 , 3] and has diagra m 2 • 1 • 1 • 1 • 2 • 2 • . (15) (See [12, 12A2]. Eac h group acts on H 5 with a simplicial fundamen tal doma in of finite v o lume. In [13], these indices w ere computed b y dissecting a simplex for H (or K ) in to copies of t he simplex for G .) Let us no w surv ey the three fa milies of lo cally t o roidal p o lytop es arising from reducing these groups mo d p . Since H p 0 and K p 0 are C -gr o ups b y [1 4 , 6B], we conclude from Theo- rem 2.4 that H p and K p are string C -groups. Similarly , G p is a string C -g roup b ecause of Theorem 2.1. In eac h case the underlying space V is non- singular for an y prime p ≥ 3 and has disc( V ) = 2 · 0 − ( − 1) 2 (1 / 4) ∼ − 1 . Th us ǫ = +1 (indicating that the Witt index is 3). As in rank 5, we conclude, for e ac h of the three types, that t he p olytop e P has automorphism group Γ( P ) =  O 1 (6 , p, +1) , if p ≡ ± 1 (mo d 8) O (6 , p, +1 ) , if p ≡ ± 3 (mo d 8) (16) Of course, this gro up is differen tly generated in the three cases, as indicated ab o v e. W e observ e that the indices 5 and 10 in c har a cteristic 0 m ust collapse to 1 under reduction mo d p . F or any prime p ≥ 3, O 1 (6 , p, +1) has o r der p 6 ( p 4 − 1)( p 3 − 1)( p 2 − 1) and index tw o in O (6 , p, +1 ) (see [14, pp. 300-301]) . W e ha v e − e ∈ Γ( P ) exc ept wh en p ≡ − 1 (mo d 8). Remark . Some of the results established b elo w w ere already a nnounced in [12, 12C,D,E]. The P olyt o p es P = P ( G p ). By [14, 6.2] the vertex -figures of P ( G p ) are 5-cub es { 4 , 3 , 3 , 3 } . The fa cets of P ( G p ) are toroids { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) , eac h with 3 p 4 v ertices ( which corrects the 3 p n men tio ned in [14, 6.5]). The unive rsal p olytop e co v ering P ( G p ) is U G p := { { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) , { 4 , 3 , 3 , 3 } } . Recall that U G p is conjectured to b e finite only when p = 3 (see the discussion in [12, 12C1], restricted to the class of p olytop es under consideration here). As for p = 3, it is kno wn 15 that U G 3 has automorphism g roup Γ( U G 3 ) of order 3 | G 3 | = 3 | O (6 , 3 , +1) | = 7 2783360 . Using GAP w e find that Γ( U G 3 ) is a split extension of the additiv e cyclic group ( Z 3 , +) b y G 3 , that is, Γ( U G 3 ) = Z 3 ⋊ O (6 , 3 , +1 ) . (17) T o c hec k this directly w e exploit the spinor norm, whic h for p = 3 w e ma y view as a homomorphism θ : G 3 → {± 1 } = Z ∗ 3 . Using this w e define an action of G 3 on Z 3 b y g z := θ ( g ) det( g ) z , f or g ∈ G 3 , z ∈ Z 3 . W e obtain the semidirect pro duct Λ := Z 3 ⋊ G 3 , with iden tity (0 , e ) and ( y , g )( z , h ) = ( y + g z , g h ) , for all y , z ∈ Z 3 , g , h ∈ G 3 . Note that r i z = η i z , where η i =  − 1 , i ≤ 1 +1 , i > 1 . No w let ρ 0 := ( 1 , r 0 ) and ρ i := ( 0 , r i ), for i ≥ 1; in brief, ρ i = ( δ i 0 , r i ). It is then a ro utine matter to c hec k tha t t he ρ i ’s satisfy the standard relatio ns for the Co xeter g roup [3 , 4 , 3 , 3 , 3 ]. Indeed, ρ i ρ j = ( δ i 0 , r i )( δ j 0 , r j ) = ( δ i 0 + η i δ j 0 , r i r j ) , so that ρ 2 i = ( 0 , e ), for all i . Next w e get ( ρ i ρ j ) 2 = ( δ i 0 (1 + η i η j ) + δ j 0 ( η i + η j ) , ( r i r j ) 2 ) , so tha t ( ρ i ρ j ) 2 = (0 , e ), whenev er i < j − 1. Similarly , ρ i − 1 ρ i has the same p erio d as r i − 1 r i for eac h i . In pa r t icular, we note that ( ρ 0 ρ 1 ) 3 = ( 0 , e ), since 3 = 0 . Last of all we must verify the required extra r elation for the facet, namely ( ρ 4 σ τ σ ) 3 = (0 , e ) , where σ := ρ 3 ρ 2 ρ 1 ρ 2 ρ 3 , τ := ρ 0 ρ 1 ρ 2 ρ 1 ρ 0 ; see [14 , 6.5]. Here c hec k first tha t σ = (0 , s ) and τ = (0 , t ), where s := r 3 r 2 r 1 r 2 r 3 , t := r 0 r 1 r 2 r 1 r 0 , then observ e that r 4 sts has p erio d 3 in G 3 . Finally , w e observ e that the Petrie eleme n t h := r 0 r 1 r 2 r 3 r 4 r 5 of G has c haracteristic p olynomial x 6 − x 4 − x 3 − x 2 + 1, so that h 13 = 6 h 5 + 9 h 4 + 6 h 3 + 3 h 2 − 3 h − 5 e and hence h 13 = e in G 3 . The corresp onding elemen t of Λ is π := ρ 0 ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 = (1 , h ) . Since π j = ( j, h j ), w e hav e π 13 = (1 , e ); th us the ρ i ’s generate Λ, and π has p erio d 39. Since Γ( U G 3 ) and Λ hav e equal o rders, we conclude that the t w o gro ups a r e isomorphic (and that Λ is a string C -gro up with resp ect to the ρ i ’s). Observ e that w e obtain the mo dular p o lytop e P ( G 3 ) from U G 3 b y iden titfying v ertices separated b y 13 steps along Petrie p olygons of U G 3 . The P olyt o p es P = P ( K p ). The facets of P ( K p ) are toro ids { 4 , 3 , 3 , 4 } ( p, 0 , 0 , 0) , while the v ertex-figures are toroids { 3 , 3 , 4 , 3 } ( p, 0 , 0 , 0) of another t ype. Consulting [12, pp. 466- 4 67], w e note that the univ ersal p olytop e U K p := { { 4 , 3 , 3 , 4 } ( p, 0 , 0 , 0) , { 3 , 3 , 4 , 3 } ( p, 0 , 0 , 0) } 16 is conjectured to exist for all primes p ≥ 3 and to b e infinite f o r p > 3; it is known to b e finite for p = 3 (and p = 2, which again is outside our discussion). Our construction o f P ( K p ) establishes the existence pa r t of this conjecture for all primes p ≥ 3. In fact, restricting ourselv es to p = 3 f or the momen t, Γ( U K 3 ) also has the same order as Λ = h ρ 0 , . . . , ρ 5 i [12, T a ble 12E1]. T aking a cue fr o m (14), w e let ( τ 0 , τ 1 , τ 2 , τ 3 , τ 4 , τ 5 ) := ( ρ 2 , ρ 1 , ρ 0 , ρ 3 ρ 2 ρ 1 ρ 2 ρ 3 , ρ 4 , ρ 5 ) . It is routine to ch ec k that the τ i ’s satisfy the defining relations f or U K 3 . Next consider the new P etrie elemen t π 1 := τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 = ( − 1 , h 1 ) , where h 1 := t 0 t 1 t 2 t 3 t 4 t 5 . Here h 1 has c haracteristic p olynomial x 6 − x 5 − x 4 − x 2 − x + 1, so that h 13 1 = 6 0 h 5 1 + 48 h 4 1 + 24 h 3 1 + 42 h 2 1 + 15 h 1 − 34 e , and hence h 13 1 ≡ − e mo d 3. Th us π 13 1 = ( − 1 , − e ), and b oth π 1 and h 1 ha v e p erio d 26 . Moreo v er, w e find that the subgroup generated by the τ i ’s contains the crucial elemen t ( τ 1 π 13 1 ) 2 = (1 , − t 1 )(1 , − t 1 ) = ( − 1 , e ) . Th us, Γ( U K 3 ) ≃ Λ = h τ 0 , . . . , τ 5 i . The P olyt o p es P = P ( H p ). F or eac h prime p ≥ 3, the p olytop e P ( H p ) inherits self-dualit y fr om the h yperb olic tessellation { 3 , 3 , 4 , 3 , 3 } . T o v erify this claim, w e first use (12) to establish a basic system of r o ots c i for the s i , namely c 0 := b 1 , c 1 := b 0 , c 2 := r 2 ( b 1 ) = b 1 + b 2 , c 3 := b 3 , c 4 := b 4 , c 5 := b 5 . (The Gra m matrix [ c i · c j ] for these is enco ded in diagr am (13).) Ov er a suitable extension of the field Z p , we may no w define an isometry w on V b y mapping c i 7→ α i c 5 − i , where α i = 1 / √ 2 for i ≤ 2, α i = √ 2 for i > 2. Th us, w 2 = e , w s j w = s 5 − j , a nd w induces a p ol a rity in the p olytop e P ( H p ). The facets of P ( H p ) are toroids { 3 , 3 , 4 , 3 } ( p, 0 , 0 , 0) ; its v ertex-figures are the dua l t oroids { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) . Consulting [1 2, 12D3], w e note that t he self-dual unive rsal p olytop e U H p := { { 3 , 3 , 4 , 3 } ( p, 0 , 0 , 0) , { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) } is conjectured to exist for all primes p ≥ 3 and to b e infinite for p > 3, but is actually kno wn t o b e finite only for p = 3 (and p = 2). Our construction of P ( H p ) ag ain establishes the existence part of the conjecture for all primes p ≥ 3. Unexpectedly , considering our previous lo ok at U G 3 and U K 3 , w e hav e that U H 3 is a 9-fold co ver of P ( H 3 ) ([12, T able 12D1]). After constructing U H 3 , w e will see t ha t the g roup Λ from ab ov e reapp ears here as the automorphism group for a no n -self-dual 3-f o ld cov er of P ( H 3 ); see (21) b elo w. T o start the construction w e use the automorphism induced on H 3 b y w to extend the earlier action of H 3 ( = G 3 ) on ( Z 3 , +) to an action on Z 3 ⊕ Z 3 : g ( y 1 , y 2 ) := ( θ ( w g w ) det( g ) y 1 , θ ( g ) det( g ) y 2 ) , 17 for all y 1 , y 2 ∈ Z 3 , g ∈ H 3 . In the semidirect pro duct Σ := ( Z 3 ⊕ Z 3 ) ⋊ H 3 , with m ultiplicatio n giv en b y ( y 1 , y 2 , g ) · ( z 1 , z 2 , h ) = ( y 1 + θ ( w g w ) det( g ) z 1 , y 2 + θ ( g ) de t( g ) z 2 , g h ) , w e define σ i := ( δ i 4 , δ i 1 , s i ) , 0 ≤ i ≤ 5 . It is a straigh tforw ard calculation to c hec k that these σ i satisfy the defining relations fo r the automorphism group Γ( U H 3 ). The w ork is halve d b y first noting that the map δ : Σ → Σ ( y 1 , y 2 , g ) 7→ ( y 2 , y 1 , w g w ) (18) defines an in v oluto ry auto morphism whic h transp oses eac h pair σ i , σ 5 − i . (Th us δ m ust induce the standard p ola rit y o n the self-dual unive rsal p olytop e U H 3 .) It remains only to c hec k that the σ i ’s g enerate Σ, since then Σ and Γ( U H 3 ), having equal orders, mus t b e isomorphic. So consider π 2 := σ 0 σ 1 σ 2 σ 3 σ 4 σ 5 = ( − 1 , − 1 , h 2 ) , where h 2 := s 0 s 1 s 2 s 3 s 4 s 5 has p erio d 26 and satisfies h 13 2 = − e in H 3 . Then fr om γ 0 := σ 0 π 13 2 = ( − 1 , 1 , − s 0 ) and dually γ 5 := σ 5 π − 13 2 = ( 1 , − 1 , − s 5 ) we obtain γ 2 0 = (0 , − 1 , e ) , γ 2 5 = ( − 1 , 0 , e ) , (19) so that Σ = h σ 0 , . . . , σ 5 i ≃ Γ( U H 3 ). F urthermore, it is clear that the pro jection ϕ : Σ → Λ ( y 1 , y 2 , g ) 7→ ( y 2 , g ) (20) yields y et anot her set of generators σ i = ϕ ( σ i ) for the group Λ. Thus Λ = h σ 0 , . . . , σ 5 i is t he automorphism group for an intermediate p olytop e P ( Λ ), still of type { { 3 , 3 , 4 , 3 } (3 , 0 , 0 , 0) , { 3 , 4 , 3 , 3 } (3 , 0 , 0 , 0) } , but now a 3-fold cov er of P ( H 3 ). F ro m ( 1 9) w e get that ϕ ( γ 2 0 ) and ϕ ( γ 2 5 ) ha v e differen t p erio ds in Λ, so tha t P (Λ) is not self-dual. Eviden tly the other pro jection ϕ ∗ : ( y 1 , y 2 , g ) 7→ ( y 1 , g ) yields the automorphism group of the dual p olytop e P (Λ) ∗ . The situation is summarized here: P (Λ) 3:1 $ $ I I I I I I I I I U H 3 3:1 ϕ ; ; w w w w w w w w w 3:1 ϕ ∗ # # G G G G G G G G G P ( H 3 ) P (Λ) ∗ 3:1 : : u u u u u u u u u (21) 18 As interesting as the results in this Section are, it seems that in order to mak e further progress with the conjectures in [12 , § 12 C,D,E] concerning lo cally toroidal p o lytop es, w e m ust relax our restriction to a prime mo dulus p in fav our of a more g eneral (comp osite) mo dulus s . This necessitates a somewhat different plan of a ttac k, whic h w e shall pursue in [16]. References [1] E. Artin, Geometric Algebra, Inters cience, New Y ork, 1957. [2] P . Cohn, Alg ebra V olume 2 , second ed., John Wiley & Sons, New Y ork, 1989. [3] H.S.M. Co xeter, Regular P olytop es, third ed., Dov er, New Y ork, 197 3. [4] The GAP Group, GAP—Groups, algorithms, and programming, v ersion 4.3 (2002) , h ttp://www.gap-system .org , 2002. [5] L.C. Gro v e, Gro ups and Characters, John Wiley & Sons, New Y ork, 1997. [6] B. Gr ¨ un baum, Regula r ity of graphs, complexes and designs, in: Probl` emes com binatoires et th´ eorie des graphes, Collo q. In ternat. C.N.R.S. No. 260, Orsa y (1977), pp. 191–197. [7] J.E. Humphreys, Reflection Gro ups and Co xeter Groups, Cambridge Univ ersity Press, Cam br idg e, 1990. [8] P . McMullen, The groups of the regular star-p o lytop es, Canad. J. Math. 5 0 (1998), 426–448. [9] P . McMullen and E. Sc h ult e, Hermitian fo rms and lo cally toroidal regular p olytop es, Adv. in Math. 82 (199 0), 88–1 2 5. [10] P . McMullen and E. Sch ulte, Higher to r oidal regular p o lytop es, Adv. in Math. 117 (1996), 1 7 –51. [11] P . McMullen and E. Sc h ulte, Twisted groups and lo cally toroidal regular p olytop es, T r a ns. Amer. Math. So c. 348 (1996), 1 3 73–1410 . [12] P . McMullen and E. Sc h ulte, Abstract Regula r Poly top es, in: Encyclop edia o f Math. Appl., v ol. 9 2, Cambridge Univers it y Press, Cambridge, 200 2. [13] B. Monson, Simplicial Quadratic F orms, Canad. J. Math. 35 (198 3), 1 01–116. [14] B. Monson and E. Sc hulte, Reflection groups and p olytop es ov er finite fields, I, Adv. in Appl. Math. 33 (2 0 04), 290 –317. [15] B. Monson and E. Sc h ulte, Reflection groups and po lyto p es o v er finite fields, I I, Adv. in Appl. Math. 38 (2 0 07), 327 – 356. [16] B. Monson and E. Sch ulte, L o cally toroidal p olytop es a nd mo dular linear groups, in preparation. 19