IMPRIMITIVE COMPLEX REFLECTION GROUPS G(m, p, n) Jin-yi Shi Est Chin Norml University, Shnghi nd Technische Universität Kiserslutern 1 Typeset by AMS-TEX
2 Jin-yi Shi 1. Preliminries. 1.1. V, n-dim spce/c. A reflection s on V : s GL(V ), o(s) <, codim V s = 1. A reflection group G on V is finite group generted by reflections on V. 1.2. A reflection group G in V is imprimitive, if (i) G cts on V irreducibly; (ii) V = V 1... V r with 0 V i V such tht G permutes {V i 1 i r}. For σ S n, denote by [( 1,..., n ) σ] the n n monomil mtrix with non-zero entries i in the (i,(i)σ)-positions.
Complex Reflection Groups 3 For p m (red p divides m ) in N, set { G(m,p,n) = [( 1,..., n ) σ] i C, m i =1,σ S n; ( j j) m/p = 1 } G(m,p,n) is the mtrix form of n imprimitive reflection group cting on V w.r.t. n orthonorml bsis e 1,e 2,...,e n.
4 Jin-yi Shi 1.3. Three tsks: (1) A reflection group G cn be presented by genertors nd reltions (not unique in generl). Clssify ll the presenttions for G (J. Shi). (2) Length function is n importnt tool in the study of reflection groups. Find explicit formule for the reflection length of elements of G (J. Shi). (3) Automorphisms of G is one of the min spects in the theory of reflection groups. Describe the utomorphism groups of G (J. Shi nd L. Wng). In this tlk, we only consider G = G(m,p,n).
Complex Reflection Groups 5 1.4. For reflection group G, presenttion of G by genertors & reltions (or presenttion in short) is by definition pir (S,P), where (1) S is finite set of reflection genertors for G with miniml possible crdinlity. (2) P is finite reltion set on S, nd ny other reltion on S is consequence of the reltions in P. 1.5. Two presenttions (S,P) nd (S,P ) for G re congruent, if bijection η : S S such tht for ny s,t S, ( ) s,t = η(s),η(t) Congruence to complex reflection groups is n nlogue of isomorphism to Coxeter systems (only concerning reltions involving t most two genertors).
6 Jin-yi Shi 2. Grphs ssocited to reflection sets. 2.1. two kinds of reflections in G(m,1,n): i < j in [n], ith jth {}}{{}}{ (i) s(i,j;k)=[(1,...,1, ζm k,1,...,1, ζm k,1,...,1) (i,j)], where (i,j) is the trnsposition of i nd j, nd ζ m := exp ( ) 2πi m. Cll s(i,j;k) reflection of type I. set s(j,i;k) = s(i,j; k). (ii) s(i;k) = [(1,...,1, k Z, m k. jth {}}{ ζ k m,1,...,1) 1] for some Cll s(i; k) reflection of type II, hving order m/gcd(m, k). All the reflections of type I lie in the subgroup G(m,m,n).
Complex Reflection Groups 7 2.2. To ny set X = {s(i h,j h ;k h ) h J} of reflections of G(m,1,n) of type I, we ssocite digrph Γ X = (V,E) s follows. Its node set is V = [n], nd its rrow set E consists of ll the pirs {i,j} with lbels k for ny s(i,j;k) X. Denote by Γ X the underlying grph of Γ X (replcing lbelled rrows by unlbelled edges). Γ X hs no loop but my hve multi-edges. Let Y = X {s(i;k)}. we define nother kind of grph Γ r Y, which is obtined from Γ X by rooting the node i, i.e., Γ r Y is rooted grph with the rooted node i. Sometimes we denote Γ r Y by ([n],e,i). Use nottion Γ Y for Γ X.
8 Jin-yi Shi Exmples 2.3. Let n = 6. (1) X = {s(1,2;4),s(3,4;2),s(4,6;0),s(3,4;3)}. Then Γ X is 1 2 3 2 4 6 5 4 3 0 Γ X is 1 2 3 4 6 5 (2) Let Y = X {s(6;3)}. Then Γ r Y is 1 2 3 4 6 5 Note: reflections of type I re represented by edges, rther thn nodes.
Complex Reflection Groups 9 2.4. We described the congruence clsses of presenttions (c.c.p. in short) for two specil fmilies of imprimitive complex reflection groups G(m,1,n) nd G(m,m,n) in terms of grphs. Theorem 2.5. The mp (S,P) Γ r S induces bijection from the set of c.c.p. s of G(m,1,n) to the set of isom. clsses of rooted trees with n nodes. Theorem 2.6. The mp (S,P) Γ S induces bijection from the set of c.c.p. s of G(m,m,n) to the set of isom. clsses of connected grphs with n nodes nd n edges (or equivlently with n nodes nd exctly one circle).
10 Jin-yi Shi Exmples 2.7. Let n = 4. (1) There re 4 isomorphic clsses of rooted trees of 4 nodes: Hence G(m,1,4) hs 4 congruence clsses of presenttions. (2) There re 5 isomorphic clsses of connected grphs with 4 nodes nd exctly one circle: === === Hence G(m,m,4) hs 5 congruence clsses of presenttions.
Complex Reflection Groups 11 Now we consider the imprimitive complex reflection group G(m,p,n) for ny m,p,n N with p m (red p divides m ) nd 1 < p < m. Lemm 2.8. The genertor set S in presenttion (S,P) of the group G(m,p,n) consists of n reflections of type I nd one reflection of order m/p nd type II. Moreover, the grph Γ S is connected with exctly one circle. 2.9. Assume tht X is reflection set of G(m,p,n) with Γ X connected nd contining exctly one circle, sy the edges of the circle re { h, h+1 }, 1 h r (the subscripts re modulo r) for some integer 2 r n. Then X contins the reflections s( h, h+1 ;k h ) with some integers k h for ny 1 h r (the subscripts re modulo r). Denote by δ(x) := r h=1 k h.
12 Jin-yi Shi Now we cn chrcterize reflection set of G(m,p,n) to be the genertor set of presenttion s follows. Theorem 2.10. Let X be subset of G(m,p,n) consisting of n reflections of type I nd one reflection of order m/p nd type II such tht the grph Γ X is connected. Then X is the genertor set in presenttion of G(m, p, n) if nd only if gcd{p,δ(x)} = 1.
Complex Reflection Groups 13 2.11. Define the following sets: Σ(m,p,n): the set of ll S which form the genertor set in some presenttion of G(m,p,n). Λ(m,p): the set of ll d N such tht d m nd gcd{d,p} = 1. Γ(m,p,n): the set of ll the connected rooted grphs with n nodes nd n edges. Γ 1 (m,p,n): the set of ll the rooted grphs in Γ(m,p,n) ech contins two-nodes circle. Γ 2 (m,p,n): the complement of Γ 1 (m,p,n) in Γ(m,p,n). Γ(m,p,n), resp., Γ i (m,p,n): the set of the isomorphism clsses in the set Γ(m,p,n), resp., Γ i (m,p,n) for i = 1,2. Σ(m,p,n): set of congruence clsses in Σ(m,p,n).
14 Jin-yi Shi Now we describe ll the congruence clsses of presenttions for G(m,p,n) in terms of rooted grphs. Theorem 2.12. (1) The mp ψ : S Γ r S from Σ(m, p, n) to Γ(m, p, n) induces surjection ψ: Σ(m,p,n) Γ(m,p,n). (2) Let Σ i (m,p,n) = ψ 1 ( Γ i (m,p,n)) for i = 1,2. Then the mp ψ gives rise to bijection: Σ 2 (m,p,n) Γ 2 (m,p,n); lso, S (Γ r S,gcd{m,δ(S)}) induces bijection: Σ 1 (m,p,n) Γ 1 (m,p,n) Λ(m,p).
Complex Reflection Groups 15 Exmple 2.13. Let n = 4, m = 6 nd p = 2. Then Λ(6, 2) = {1, 3}. 13 isomorphic clsses of rooted connected grphs with 4 nodes nd exctly one circle, 9 of them contin twonodes circle. So G(6,2,4) hs 22 = 9 2 + 4 congruence clsses of presenttions. === === === === === ===
16 Jin-yi Shi 3. The reltion set of presenttion for G(m,p,n). 3.1. Let S = {s,t h 1 h n} be in Σ(m,p,n), where s = s(;k); ll the t h s re of type I; is the rooted node of Γ r S.
Complex Reflection Groups 17 3.2. The following reltions hold: (A) s m/p = 1; (B) t 2 i = 1 for 1 i n; (C) t i t j = t j t i if the edges e(t i ) nd e(t j ) hve no common end node; (D) t i t j t i = t j t i t j if the edges e(t i ) nd e(t j ) hve exctly one common end node; (E) st i st i = t i st i s if is n end node of e(t i ); (F) st i = t i s if is not n end node of e(t i ); (G) (t i t j ) m/d = 1 if t i t j with e(t i ) nd e(t j ) hving two common end nodes, where d = gcd{m,δ(s)}; (H) t i t j t l t j = t j t l t j t i for ny triple X = {t i,t j,t l } S with Γ X hving brnching node (I) s t i t j t i = t i t j t i s, if e(t i ) nd e(t j ) hve exctly one common end node ;
18 Jin-yi Shi Cll s hj := t h t h+1...t j 1 t j t j 1...t h pth reflection in Γ r S : (J) (s 1j s j+1,r ) t h t h+1 t j x= t t m gcd{m,δ(s)} = 1 for p < j < q. 0 1 1 r r r 1 p q t t p+1 j 1 p+1 t j q q 1 (K) ss 1j s j+1,r = s 1j s j+1,r s for p < j < q j+1 t j+1 (L) (s j+1,r s 1j ) p 1 = s δ(s) s 1j s δ(s) s j+1,r for j p < j < q. x= t t 0 1 1 r r r 1 p q t t p+1 j 1 p+1 t j q q 1 j+1 t j+1 j
Complex Reflection Groups 19 (M) For p < j < q () us 1j u vs j+1,r v = vs j+1,r v us 1j u, (b) us 1j s j+1,r us 1j s j+1,r =s 1j s j+1,r us 1j s j+1,r u, (c) vs 1j s j+1,r vs 1j s j+1,r =s 1j s j+1,r vs 1j s j+1,r v, x= t u t 0 1 1 r r r 1 q t t p+1 j 1 p+1 t j Cll ll the reltions (A)-(M) bove the bsic reltions on S. p q q 1 j+1 t j+1 j v
20 Jin-yi Shi Then we hve. Theorem 3.3. Let S Σ(m,p,n) nd let P S be the set of ll the bsic reltions on S. Then (S,P S ) forms presenttion of G(m,p,n). Remrk 3.4. There re too much bsic reltions on S in generl. We cn get much smller subset P S from P S such tht (S,P S ) still forms presenttion of G(m,p,n). Under the ssumption of reltions (A) (F), we cn reduce the size of reltion set (J) by replcing it by (J ), the ltter consists of ny single reltion in (J). Similr for (K), (L) nd (M). The size of the reltion sets (I) nd (J) cn lso be reduced.
Complex Reflection Groups 21 3.4. Two kinds of presenttions hve simpler reltion sets: (i) Γ r S is string: === (ii) Γ S is circle:
22 Jin-yi Shi 4. Reflection length. 4.1. T, the set of ll the reflections in G(m,p,n). Any w G(m,p,n) hs n expression w = s 1 s 2 s r with s i T. Denote by l T (w) the smllest possible r mong ll such expressions. Cll l T (w) the reflection length of w. l T (w) on G(m,p,n) is presenttion-free. We hve l T (w) l S (w) for ny presenttion (S,P) of G(m, p, n). Except for the cse of G(m,1,n) with one specil presenttion (see 5.1), so fr we hve no close formul of the length function l S (w) on G(m,p,n) with 1 < p m, where (S,P) is ny presenttion of G(m,p,n).
Complex Reflection Groups 23 4.2. Given m,p,r P with p m. Let C = [[c 1,c 2,...,c r ]] be multi-set of r integers. P = {P 1,...,P l } prtition of [r]. Cll E [r] (C,m)-perfect if h E c h 0 (mod m); nd (C,m,p)-semi-perfect, if h E c h 0 (mod p) nd h E c h 0 (mod m). Cll P (C,m)-dmissible if P j is (C,m)-perfect for ny j [l]; nd (C,m,p)-semi-dmissible if P j is either (C,m)- perfect or (C,m,p)-semi-perfect for ny j [l].
24 Jin-yi Shi Let Λ(C;m) (resp., Λ(C;m,p)) be the set of ll the (C, m)-dmissible (resp., (C, m, p)-semidmissible) prtitions of [r]. When Λ(C;m) (resp., Λ(C;m,p) ), denote by t(p) (resp., u(p)) the number of (C,m)-perfect (resp., (C, m, p)-semi-perfect) blocks of P for ny P Λ(C;m) (resp., P Λ(C;m,p)), nd define t(c,m) = mx{t(p) P Λ(C;m)}. Define v(p) = 2t(P) + u(p) for ny P Λ(C;m,p). Define v(c,m,p) = mx{v(p) P Λ(C;m,p)} if Λ(C;m,p).
Complex Reflection Groups 25 4.3. For w = [ζ 1 m,...,ζ n m σ] G(m,p,n), write: σ = (i 11,i 12,...,i 1m1 )...(i r1,i r2,...,i rmr ) with j [r] m j = n. Denote (i) r(w) = r. Let I j = {i j1,i j2,...,i jmj } for j [r]. Then I(w) = {I 1,...,I r } is prtition of [n] determined by w. Let c j = k I j k nd let C(w) = [[c 1,c 2,...,c r ]]. Denote Λ(w;m,p) := Λ(C(w);m,p). For w G(m,p,n), we lwys hve Λ(w;m,p). Denote (ii) t(w) := t(c(w),m) if p = m nd (iii) v(w) = v(c(w), m, p) if p m. (iv) t 0 (w) = #{j [r] c j 0 (mod m)}.
26 Jin-yi Shi Theorem 4.4. l T (w) = n t 0 (w). for ny w G(m,1,n). Theorem 4.5. l T (w) = n + r(w) 2t(w) for ny w G(m,m,n). Theorem 4.6. Let m,p,n P be with p m. Then l T (w) = n + r(w) v(w) for ny w G(m,p,n). When w G(m, 1, n), we hve t 0 (w) = v(w) r(w); when w G(m, m, n), we hve v(w) = 2t(w). So Theorems 4.4 4.5 re specil cses of Theorem 4.6.
Complex Reflection Groups 27 4.7. For ny y,w G(m,p,n), denote by y w nd cll w covers y (or y is covered by w), if yw 1 is reflection with l T (w) = l T (y) + 1. The reflection order on G(m,p,n) is the trnsitive closure of the covering reltions.
28 Jin-yi Shi 4.8. For ny cyclic permuttion σ S n, the set B(σ) = {τ S n τ σ} cn be described in terms of circle non-intersecting prtitions. Put the nodes 1,2,...,k on circle clockwise. Prtition these k nodes into h blocks X 1,...,X h with X j, j [h], such tht the convex hulls X j, j [h], of these blocks re pirwise disjoint. The prtition X = {X 1,...,X h } is clled circle non-intersecting prtition of [k]. Reding the nodes of ech X j clockwise long the boundry of X j, we get cyclic permuttion τ j. Then set τ(x) = τ 1 τ 2 τ h.
Complex Reflection Groups 29 Exmple 4.9. Let σ = (1,2,3,4,5,6,7,8,9,10,11) S n. Tke prtition X of [11] s in Figure 1. Then τ(x) B(σ) is (1,2,9,10)(4,5,8)(6,7)(3)(11). 1 2 11 3 10 4 9 5 8 7 Figure 1. 6 τ σ if nd only if τ = τ(x) for some circle non-intersecting prtition X of [11].
30 Jin-yi Shi The reltion x y cn lso be described combintorilly in the group G(m,1,n). Theorem 4.10. Let w = [ζ 1 m,...,ζ n m σ] G(m,1,n) be with σ = (1,2,...,r) cyclic permuttion nd j = 0 for j > r. C r. (1) If j [r] j 0 (mod m), then B(w) = (2) If j [r] j 0 (mod m), then B(w) = (r + 1) C r, where C r = 1 r+1 ( 2r r ), the rth Ctln number.
Complex Reflection Groups 31 5. Auto. group Aut(m,p,n) of G(m,p,n). Assume m > 2 nd (p,n) (m,2) (i.e. G(m,p,n) is not Coxeter). 5.1. G(m,p,n) hs genertor set S 0 : (i) {s 0,s 1,s i i [n 1]} if 1 < p < m; (ii) {s 0,s i i [n 1]} if p = 1; (iii) {s 1,s i i [n 1]} if p = m, where s 0 = s(1;p), s 1 = s(1,2; 1) nd s i = s(i,i + 1;0). G(m,1,n) s s s s 0 1 2 n 1 G(m,m,n) s 1 s 1 s 2 n 1 s G(m,p,n) s s s 1 2 n 1 s 0 s 1
32 Jin-yi Shi 5.2. By n utomorphism φ of reflection group G, it mens tht φ is n utomorphism of the group G s n bstrct group which sends reflections of G to reflections. 5.3. Two presenttions (S,P), (S,P ) of G(m,p,n) re clled strongly congruent, if there exists bijective mp η : S S such tht P = η(p), where η(p) is obtined from P by substituting ny s S by η(s). strongly congruent = = congruent. A strongly congruent mp η cn be extended uniquely to n utomorphism of G.
Complex Reflection Groups 33 5.4. Let τ g : h ghg 1 be the inner utomorphism of G(m, p, n). Let Int(m,p,n) = {τ g g G(m,p,n)}. 5.5. Set Φ(m) := {i [m 1] gcd(i,m) = 1}. For ny k Φ(m) nd ny mtrix w = ( ij ), define If then ψ k (w) = ( k ij ). w = [ζ 1 m,...,ζ n m σ] G(m,p,n), ψ k (w) = [ζ k 1 m,...,ζk n m σ] G(m,p,n). We hve ψ k Aut(m,p,n). Define Ψ(m) := {ψ k k Φ(m)}.
34 Jin-yi Shi 5.6. Let λ Aut(m,p,n), 1 < p m, be determined by λ(s 0 ) = s 1 0, λ(s 1) = s 1, λ(s 1 ) = s 1 nd λ(s i ) = s i for 1 < i < n s 1 s 0 1 2 s 2 3 n 1 n s n 1 s 1 λ s 1 1 s 0 1 2 s 1 s 2 3 n 1 n s n 1
Complex Reflection Groups 35 Let λ Aut(3,3,3) be determined by λ (s 1 ) = s(2,3; 1), λ (s i ) = s i for i = 1,2. s 1 1 2 s 2 3 s 1 λ s 1 s(2,3; 1) 1 2 3 s 2
36 Jin-yi Shi Let η Aut(4,2,2) be determined by (η(s 0 ),η(s 1 ),η(s 1 )) = (s 1,s 0,s 1 ). s 0 s 1 1 2 η s 1 s 1 1 2 s 1 s0
Complex Reflection Groups 37 Theorem 5.7. (1) If gcd(p,n) = 1, then Aut(m,p,n) = Int(m,p,n) Ψ(m); (2) If gcd(p,n) > 1 nd (m,p,n) (3,3,3),(4,2,2), then Aut(m,p,n) = Int(m,p,n),Ψ(m),λ ; (3) Aut(3,3,3) = τ s1,λ,λ. (4) Aut(4,2,2) = ψ 3,λ,η. Theorem 5.8. The order of Aut(m,p,n) is m n 1 n!φ(m) if (m,p,n) (3,3,3),(4,2,2), 432 if (m,p,n) = (3,3,3), 48 if (m,p,n) = (4,2,2).
38 Jin-yi Shi Some structurl properties of Aut(m, p, n) re studied. For exmple, we show tht the center Z(Aut(m,p,n)) of Aut(m,p,n) is trivil if n > 2; while Z(Aut(m,p,2)) contins 2 gcd(m,2) elements.