The invariants of the Clifford groups

Designs, Codes and Cryptography, 24, 99–122, 2001  C 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. The Invariants of the Clifford Groups GABRIELE NEBE∗ Abteilung Reine Mathematik der Universität Ulm, 89069 Ulm, Germany nebe@mathematik.uni-ulm.de E. M. RAINS rains@research.att.com Information Sciences Research, AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ 07932-0971, U.S.A. N. J. A. SLOANE njas@research.att.com Information Sciences Research, AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ 07932-0971, U.S.A. Communicated by: J. Key Received December 9, 1999; Revised September 18, 2000; Accepted September 26, 2000  3) is a subgroup of Abstract. The automorphism group of the Barnes-Wall lattice L m in dimension 2m (m = .O + (2m, 2). This group and its complex analogue Xm index 2 in a certain “Clifford group” C m of structure 21+2m + of structure (21+2m YZ 8 ).Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal + spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge’s 1996 result that the space of invariants for Cm of degree 2k is spanned by the complete weight enumerators of the codes C ⊗ F2m , where C ranges over all binary self-dual codes of length 2k; these are a basis  if m√ ≥ k − 1. We also give new constructions for L m and Cm : let M be the √ 2 2 Z[ 2]-lattice with Gram matrix √ . Then L m is the rational part of M ⊗m , and Cm = Aut(M ⊗m ). Also, 2 2 if C is a binary self-dual code not generated by vectors of weight 2, then Cm is precisely the automorphism group of the complete weight enumerator of C ⊗ F2m . There are analogues of all these results for the complex group Xm , with “doubly-even self-dual code” instead of “self-dual code.” Keywords: Clifford groups, Barnes-Wall lattices, spherical designs, invariants, self-dual codes 1. Introduction In 1959 Barnes and Wall [2] constructed a family of lattices in dimensions 2m , m = m 0, 1, 2, . . . . They distinguished two geometrically similar lattices L m ⊆ L ′m in R2 . The automorphism group† Gm = Aut(L m ) was investigated in a series of papers by Bolt, Room and Wall [8–10,50]. Gm is a subgroup of index 2 in a certain group Cm of structure 1+2m 2+ .O + (2m, 2). We follow Bolt et al. in calling Cm a Clifford group. This group and its complex analogue Xm are the subject of the present paper. These groups have appeared in several different contexts in recent years. In 1972 Broué and Enguehard [12] rediscovered the Barnes-Wall lattices and also determined their ∗ Most † More of this work was carried out during G. Nebe’s visit to AT&T Labs in the Summer of 1999. precisely, Gm = Aut(L m ) ∩ Aut(L ′m ) for all m, and Gm = Aut(L m ) unless m = 3. 100 NEBE ET AL. automorphism groups. In 1995, Calderbank, Cameron, Kantor and Seidel [13] used the Clifford groups to construct orthogonal spreads and Kerdock sets, and asked “is it possible to say something about [their] Molien series, such as the minimal degree of an invariant?.” Around the same time, Runge [39–42] (see also [20,36]) investigated these groups in connection with Siegel modular forms. Among other things, he established the remarkable result that the space of homogeneous invariants for Cm of degree 2k is spanned by the complete weight enumerators of the codes C ⊗F2 F2m , where C ranges over all binary selfdual (or type I) codes of length 2k, and the space of homogeneous invariants for Xm of degree 8k is spanned by the complete weight enumerators of the codes C ⊗F2 F2m , where C ranges over all binary doubly-even self-dual (or type II) codes of length 8k. One of our goals is to give a simpler proof of these two assertions, not involving Siegel modular forms (see Theorems 4.9 and 6.2). Around 1996, the Clifford groups also appeared in the study of fault-tolerant quantum computation and the construction of quantum error-correcting codes [4,15,16,29], and in the construction of optimal packings in Grassmannian spaces [14,17,44]. The story of the astonishing coincidence (involving the group C3 ) that led to [14,15] and [16] is told in [16]. (Other recent references that mention these groups are [23,30,51].) Independently, and slightly later, Sidelnikov [45–48] (see also [28]) came across the group Cm when studying spherical codes and designs. In particular, he showed that for m ≥ 3 the lowest degree harmonic invariant of Cm has degree 8, and hence that the orbit m under Cm of any point on a sphere in R2 is a spherical 7-design. (Venkov [49] had earlier shown that for m ≥ 3 the minimal vectors of the Barnes-Wall lattices form 7-designs.) In fact it is an immediate consequence of Runge’s results that for m ≥ 3 Cm has a unique harmonic invariant of degree 8 and no such invariant of degree 10 (see Corollary 4.13). The space of homogeneous invariants of degree 8 is spanned by the fourth power of the quadratic form and the complete weight enumerator of the code H8 ⊗F2 F2m , where H8 is the [8, 4, 4] Hamming code. An explicit formula for this complete weight enumerator is given in Theorem 4.14. Our proof of the real version of Runge’s theorem is given in Section 4 (Theorem 4.9), following two preliminary sections dealing with the group Cm and with generalized weight enumerators. In Section 5 we study the connection between the group Cm and lattices. √ √the Barnes-Wall We define the balanced Barnes-Wall lattice Mm to be the Z[ 2]-lattice 2L ′m + L m . Then Mm = M1⊗m (Lemma 5.2), which leads to a simple construction: the Barnes-Wall lattice is just the rational part of M1⊗m . Furthermore Cm = Aut(Mm ) (Proposition 5.3). Also, if C is any binary self-dual code that is not generated by vectors of weight 2, Cm = Aut(cwe(C ⊗ F2m )) (Corollary 5.7). The proof of this makes use of the fact that Cm is a maximal finite subgroup of G L(2m , R) (Theorem 5.6). Although there are partial results about the maximality of Cm in Kleidman and Liebeck [30], this result appears to be new. The proof does not use the classification of finite simple groups. The analogous results for the complex Clifford group Xm are given in Section 6. Theorem 6.2 is Runge’s theorem. Extending scalars, let Mm be the hermitian Z[ζ8 ]-lattice Z[ζ8 ]⊗Z[√2] Mm . Then Xm is the subgroup of U (2m , Q[ζ8 ]) preserving Mm (Proposition 6.4). Theorem 6.5 shows that, apart from the center, Xm is a maximal finite subgroup of U (2m , C), and Corollary 6.6 is the analogue of Corollary 5.7. 101 THE INVARIANTS OF THE CLIFFORD GROUPS Bolt et al. [8–10,50] and Sidelnikov [45–47] also consider the group Cm( p) obtained by replacing 2 in the definition of Cm by an odd prime p. In the final section we give some analogous results for this group. In recent years many other kinds of self-dual codes have been studied by a number of authors. Nine such families were named and surveyed in [38]. In a sequel [35] to the present paper we will give a general definition of the “type” of a self-dual code which includes all these families as well as other self-dual codes over rings and modules. For each “type” we investigate the structure of the associated “Clifford-Weil group” (analogous to Cm and Xm for types I and II) and its ring of invariants. The results in this paper and in Part II can be regarded as providing a general setting for Gleason’s theorems [24,32,38] about the weight enumerator of a binary self-dual code (cf. the case m = 1 of Theorem 4.9), a doubly-even binary self-dual code (cf. the case m = 1 of Theorem 6.2) and a self-dual code over F p (cf. the case m = 1 of Theorem 7.1). They are also a kind of discrete analogue of a long series of theorems going back to Eichler (see for example [7,39,40,42]), stating that under certain conditions theta series of quadratic forms are bases for spaces of modular forms: here complete weight enumerators of generalized self-dual codes are bases for spaces of invariants of “Clifford-Weil groups.” 2. The Real Clifford Group C m This initial section defines the real Clifford group Cm . The extraspecial 2-group E(m) ∼ = 1+2m 2+ is a subgroup of the orthogonal group O(2m , R). If m = 1 then      1 0 0 1 ∼ , σ2 := E(1) := σ1 := = D8 0 −1 1 0 is the automorphism group of the 2-dimensional standard lattice. In general E(m) is the m-fold tensor power of E(1): E(m) := E(1)⊗m = E(1) ⊗ · · · ⊗ E(1) , and is generated by the tensor products of σ1 and σ2 with 2 × 2 identity matrices I2 . Definition 2.1. The real Clifford group Cm is the normalizer in O(2m , R) of the extraspecial 2-group E(m). The natural representation of E(m) is absolutely irreducible. So the centralizer of E(m) in the full orthogonal group is equal to {±I2m }, which is the center of E(m). Then Cm /E(m) embeds into the outer automorphism group of E(m). The quotient group E(m)/Z (E(m)) is isomorphic to a 2m-dimensional vector space over F2 . Since every outer automorphism has to respect the {+1, −1}-valued quadratic form ∼ E(m)/Z (E(m)) ∼ = F2m 2 → Z (E(m)) = F2 , x → x 2 , it follows easily that the outer automorphism group of E(m) is isomorphic to O + (2m, 2), the full orthogonal group of a quadratic form of Witt defect 0 over F2 (see e.g. [51]). 102 NEBE ET AL. 1+2m Since the group 2+ .O + (2m, 2) is a subgroup of O(2m , R) (cf. [10] or the explicit 1+2m .O + (2m, 2). The order of Cm is construction below), we find that Cm ∼ = 2+ 2m 2 +m+2 (2m − 1) m−1  j=1 (4 j − 1). To perform explicit calculations we need a convenient set of generators for Cm . THEOREM 2.2. Cm is generated by the following elements of O(2m , R) : (1) diag((−1)q(v)+a ), where q ranges over all {0, 1}-valued quadratic forms on F2m and a ∈ {0, 1}, m (2) AG L(m, 2), acting on R2 = ⊗m (R2 ) = R[F2m ] by permuting the basis vectors in F2m , and (3) the single matrix h ⊗ I2 ⊗ · · · ⊗ I2 where h := √1 2 1 1 1 −1 . Proof. Let H be the group generated by the elements in (1) and (2). First, H contains the extraspecial group E(m), since σ1 ⊗ I2m−1 and σ2 ⊗ I2m−1 are in H and their images under G L(m, 2) generate E(m). To see that H/E(m) is a maximal parabolic subgroup of O + (2m, 2), note that by [13] a 0 the elements a ∈ G L(m, 2) act on E(m)/Z (E(m)) ∼ = F2m 2 as 0 a −tr , and the elements diag((−1)q(v) ) act as 10 b1 , where b is the skew-symmetric matrix corresponding to the bilinear form bq (x, y) := q(x + y) − q(x) − q(y). Since h ⊗ I2m−1 ∈ G L(2m , Q) is not in H , the group generated by H and this element is Cm . COROLLARY 2.3. Cm is generated by σ1 ⊗ I2 ⊗ · · · ⊗ I2 , σ2 ⊗ I2 ⊗ · · · ⊗ I2 , h ⊗ I2 ⊗ · · · ⊗ I2 , G L(m, 2), diag((−1) where (v) ), is the particular quadratic form (ǫ1 , . . . , ǫm ) → ǫ1 ǫ2 ∈ {0, 1} on F2m . 3. Full Weight Enumerators and Complete Weight Enumerators We now introduce certain weight enumerators and show that they are invariant under the real Clifford group. Let C ≤ F2N be a linear code‡ of length N over the field F2 . For m ∈ N let C(m) := C ⊗F2 F2m be the extension of C to a code over the field with 2m elements. Let V be the group algebra V := R[F2m ] = ⊕ f ∈F2m Rx f . Regarding F2m ∼ = F2m as an m-dimensional vector space over F2 , we have a tensor decomposition V ∼ = ⊗m (R2 ). binary linear code C of length N is a subspace of F2N . If C ⊆ C ⊥ , C is self-orthogonal; if C = C ⊥ , C is self-dual [32,38]. ‡A 103 THE INVARIANTS OF THE CLIFFORD GROUPS In the same manner the group algebra R[C(m)] = ⊕c∈C(m) Rec embeds naturally into the group algebra R F2Nm ∼ = ⊗m (⊗ N (R2 )). = ⊗ N V = ⊗ N (⊗m (R2 )) ∼ Definition 3.1. The full weight enumerator of C(m) is the element fwe(C(m)) := c∈C(m) ec ∈ R[C(m)] ⊂ ⊗ N V. (This was called a generalized weight polynomial in [24] and an exact enumerator in [32, Chapter 5].) Fix a basis (a1 , . . . , am ) of F2m over m F2 . Then a codeword c ∈ C(m) is just an m-tuple mof codewords in C. The element c = i=1 ai ci corresponds to the m-tuple (c1 , . . . , cm ) ∈ C , which can also be regarded as an m × N -matrix M of which the rows are the elements ci ∈ C. LEMMA 3.2. Let fwem (C) := c1 ,...,cm ∈C ec1 ⊗ · · · ⊗ ecm ∈ ⊗m R[C] ⊂ ⊗m ⊗ N (R2 ). Then the isomorphism ⊗m R[C] ∼ =R[C(m)] induced by identifying an m-tuple (c1′ , . . . , m ′ m cm ) ∈ C with the codeword c := i=1 ai ci′ ∈ C(m) maps fwem (C) onto fwe(C(m)). m  (1) ′ Proof. Let c = i=1 ai ci′ = (c1 , . . . , c N ) ∈ C(m). If ci = mj=1 ǫ (i) j a j then ci = (ǫi , . . . , ǫi(N ) ) ∈ C. The generator ec of R[C(m)] is xc1 ⊗ · · · ⊗ xc N = yǫ (1) ⊗ · · · ⊗ yǫm(1) ⊗ · · · ⊗ yǫ (N ) ⊗ · · · ⊗ yǫm(N ) ∈ ⊗ N (⊗m (R2 )), 1 1 2 where R = R[F2 ] has a basis y0 , y1 . Under the identification above this element is mapped onto yǫ (1) ⊗ · · · ⊗ yǫ (N ) ⊗ · · · ⊗ yǫm(1) ⊗ · · · ⊗ yǫm(N ) ∈ ⊗m (⊗ N (R2 )), 1 1 which is the element ec1′ ⊗ · · · ⊗ ecm′ ∈ ⊗m R[C]. Definition 3.3. (Cf. [32, Chapter 5].) The complete weight enumerator of C(m) is the following homogeneous polynomial of degree N in 2m variables:  a f (c) cwe(C(m)) := xf ∈ R[x f | f ∈ F2m ] , c∈C(m) f ∈F2m where a f (c) is the number of components of c that are equal to f . Remark 3.4. The complete weight enumerator of C(m) is the projection under π of the full weight enumerator of C(m) to the symmetric power Sym N (V ), where π : ⊗ N V → R[x f | f ∈ F2m ] is the R-linear mapping defined by x f1 ⊗ · · · ⊗ x f N → x f1 · · · x f N : cwe(C(m)) = π(fwe(C(m))) ∈ Sym N (V ). 104 NEBE ET AL. THEOREM 3.5. Let C be a self-dual code over F2 . (i) The Clifford group Cm preserves the full weight enumerator fwe(C(m)). (ii) The Clifford group Cm preserves the complete weight enumerator cwe(C(m)). Proof. Let N be the length of C, which is necessarily even. Then Cm acts on R[F2Nm ] = m ⊗ N (R2 ) diagonally. This action commutes with the projection π : ⊗ N V → R[x f | f ∈ F2m ]. So statement (ii) follows immediately from (i) by Remark 3.4. To prove (i) it is enough to consider the generators of Cm . The generators σ1 ⊗ I2 ⊗ · · · ⊗ I2 , σ2 ⊗ I2 ⊗ · · · ⊗ I2 and h ⊗ I2 ⊗ · · · ⊗ I2 of Corollary 2.3 are tensor products of the form x ⊗ I2m−1 . By Lemma 3.2 it is therefore enough to consider the case m = 1 for these generators. But then the matrix σ1 acts as σ1 ⊗ · · · ⊗ σ1 on ⊗ N (R2 ), mapping a codeword c = (c1 , . . . , c N ) ∈ C to c + 1 = (c1 + 1, . . . , c N + 1) ∈ C, where 1 is the all-ones vector. Since C is self-dual, 1 is in C and therefore σ1 only permutes the codewords and hence fixes fwe(C). Analogously, the matrix σ2 changes signs of the components of the codewords in the full weight enumerator: if c = (c1 , . . . , c N ), then xci is mapped to (−1)ci xci . Since the codewords in C have even weight, the tensor product xc1 ⊗ · · · ⊗ xc N is fixed by σ2 ⊗ · · · ⊗ σ2 . That h preserves the full weight enumerator follows from the MacWilliams identity [32, Chapter 5, Theorem 14]. The generator d := diag((−1) (v) ) = diag(1, 1, 1, −1)⊗ I2m−2 only occurs for m ≥ 2. By Lemma 3.2 it suffices to consider the case m = 2. Again by Lemma 3.2, we regard d as acting on pairs (c, c′ ) of codewords in C. Then d fixes or negates (xc1 ⊗· · ·⊗xc N )⊗(xc1′ ⊗· · ·⊗xc′N ), and negates it if and only if c and c′ intersect in an odd number of 1’s. This is impossible since C is self-dual, and so d also preserves fwe(C(m)). m The remaining generators in g ∈ G L(m, 2) permute mthe elements of F2 . The codewords c ∈ C(m) are precisely the elements of the form c = i=1 ai ci with ci  ∈ C and (a1 , . . . , am ) m m m . Since g acts linearly on F , mapping ai onto a fixed F2 -basis for F2 2 j=1 gi j a j , the word m m c is mapped to j=1 i=1 gi j a j ci which again is in C(m). Hence these generators also fix fwe(C(m)). 4. The Ring of Invariants of Cm In this section we establish Runge’s theorem that the complete weight enumerators of the codes C(m) generate the space of invariants for Cm . Definition 4.1. A polynomial p in 2m variables is called a Clifford invariant of genus m if it is an invariant for the real Clifford group Cm . Furthermore, p is called a parabolic invariant if it is invariant under the parabolic subgroup P generated by the elements of type (1) and (2) of Theorem 2.2, and a diagonal invariant if it is invariant under the group generated by the elements of type (1). The following is obvious: LEMMA 4.2. A polynomial p is a diagonal invariant if and only if all of its monomials are diagonal invariants. 105 THE INVARIANTS OF THE CLIFFORD GROUPS Let M be an m × N matrix over F2 . We can associate a monic monomial µ M ∈ R[x f | f ∈ F2m ] with such a matrix by taking the product of the variables associated with its columns. Clearly all monic monomials are of this form, and two matrices correspond to the same monic monomial if and only if there is a column permutation taking one to the other. LEMMA 4.3. A monic monomial µ M is a diagonal invariant if and only if the rows of M are orthogonal. Proof. It suffices to consider quadratic forms qi j with qi j (ǫ1 , . . . , ǫm ) = ǫi ǫ j (1 ≤ i ≤ j ≤ m); we easily check that the action of diag((−1)qi j ) is to multiply µ M by (−1)k , where k is the inner product of rows i and j of M; the lemma follows. For g ∈ G L(m, 2) ≤ AG L(m, 2) we have g(µ M ) = µgtr M , and b ∈ F2m ≤ AG L(m, 2) maps µ M onto µ M+b , where the matrix M + b has entries (M + b)i j = Mi j + bi . This implies that µ M is equivalent to µ M ′ under the action of AG L(m, 2) if and only if the binary codes M, 1 and M ′ , 1 are equivalent. We can thus define a parabolic invariant µm (C) for any self-orthogonal code C containing 1 and of dimension at most m + 1 by µm (C) := µM . M∈F2m×N M,1 = C We define µm (C) to be 0 if 1 ∈ C or dim(C) > m + 1. Since the invariants µm (C) are sums over orbits, we have: LEMMA 4.4. A basis for the space of parabolic invariants of degree N is given by polynomials of the form µm (C) where C ranges over the equivalence classes of binary self-orthogonal codes of length N containing 1 and of dimension ≤ m + 1. LEMMA 4.5. For any binary self-orthogonal code C containing 1, cwe(C(m)) = µm (D). 1∈D⊆C Proof. From the definition, cwe(C(m)) = µM , M where M ranges over m × N matrices with all rows in C. Let M be such a matrix. Then M uniquely determines a subcode D := M, 1 of C; we thus have cwe(C(m)) = 1∈D⊆C M,1=D µM = µm (D) 1∈D⊆C as required. THEOREM 4.6. A basis for the space of parabolic invariants is given by the polynomials cwe(C(m)), where C ranges over equivalence classes of self-orthogonal codes containing 1 and of dimension ≤ m + 1. 106 NEBE ET AL. Proof. The equations in Lemma 4.5 form a triangular system which we can solve for the polynomials µm (C). In particular, µm (C) is a linear combination of the cwe(D(m)) for subcodes 1 ∈ D ⊆ C. Let X P denote the linear transformation x → 1 |P| g∈P g·x where P is the parabolic subgroup of Cm ; that is, X P is the operation of averaging over the parabolic subgroup. LEMMA 4.7. For any binary self-orthogonal code C of even length N containing 1 and of dimension N /2 − r , X P (h ⊗ I2m−1 ) cwe(C(m)) = (2m 1 [(2m−r − 2r )cwe(C(m)) − 1) + 2−r cwe(C ′ (m))]. ′ (1) ′⊥ C⊂C ⊆C [C ′ :C] = 2 The final sum is over all self-orthogonal codes C ′ containing C to index 2. Proof. By the MacWilliams identity, we find that (h ⊗ I2m−1 )cwe(C(m)) = 2−r µM , where M ranges over m × N matrices such that the first row of M is in C ⊥ and the remaining rows are in C. For each code 1 ∈ D ⊆ C ⊥ , consider the partial sum over the terms with M, 1 = D. If D ⊆ C, the partial sum is just µm (D), so in particular is a parabolic invariant. The other possibility is that [D : D ∩ C] = 2. For a matrix M with M, 1 = D, define a vector v M ∈ F2m such that (v M )i = 1 if the ith row of M is not in C, and (v M )i = 0 otherwise. In particular, the partial sum we are considering is µM . M,1=D v M =(1,0,0, . . .) If D is not self-orthogonal then this sum is annihilated by averaging over the diagonal subgroup. Similarly, if we apply an element of AG L m (2) to this sum, this simply has the effect of changing v M . Thus, when D ⊆ D ⊥ , XP M,1=D v M =(1,0,0, . . .) 1  µm (D). µ M =  m v ∈ F2 : v = 0  Hence X P (h ⊗ I2m−1 ) cwe(C(m)) = 2−r 1∈ D⊆C µm (D) + 2−r 2m − 1 µm (D), 1∈D⊆C ⊥ [D:D∩C] = 2 107 THE INVARIANTS OF THE CLIFFORD GROUPS where the sums are restricted to self-orthogonal codes D. Introducing a variable C ′ = D, C into the second sum (note that since D ⊆ C ⊥ , C ′ ⊆ C ′⊥ precisely when D ⊆ D ⊥ ), this becomes X P (h ⊗ I2m−1 ) cwe(C(m)) = 2−r 1∈D⊆C µm (D) + 2−r 2m − 1 µm (D). C⊂C ′ ⊆C ′⊥ D⊆C ′ [C ′ :C] = 2 D⊂C Any given C ′ will, of course, contain each subcode of C exactly once, so we can remove the condition D ⊂ C as follows: X P (h ⊗ I2m−1 ) cwe(C(m)) = 2−r 1∈D⊂C µm (D) + − (22r − 1) =  2−r 2m − 1 µm (D) C⊂C ′ ⊆C ′⊥ 1∈D⊆C ′ [C ′ :C] = 2 2−r µm (D) 2m − 1 1∈D⊆C 1   m−r − 2r )cwe(C(m)) + 2−r (2 2m − 1     cwe(C ′ (m)) ,  ′⊥ C⊂C ′ ⊆C [C ′ :C] = 2 as required. LEMMA 4.8. Let V be a finite dimensional vector space, M a linear transformation on V , and P a partially ordered set. Suppose there exists a spanning set v p of V indexed by p ∈ P on which M acts triangularly; that is, Mv p = c pq vq , q≥ p for suitable coefficients c pq . Suppose furthermore that c pp = 1 if and only if p is maximal in P. Then the fixed subspace of M in V is spanned by the elements v p for p maximal. Proof. Since the matrix C = (c pq ) is triangular, there exists another triangular matrix D that conjugates C into Jordan canonical form. Setting wp = d pq vq , q≥ p (d pp = 0), we find Mw p = c′pq w q , q≥ p with c′pp = c pp and (M − c pp I )n w p = 0 for sufficiently large n. In other words, each w p is in the Jordan block of M with eigenvalue c pp . But the vectors w p span V ; it follows 108 NEBE ET AL. that the Jordan blocks of M on V are spanned by the corresponding Jordan blocks of C. In particular, this is true for the block corresponding to 1. THEOREM 4.9. (Runge [42].) Fix integers k and m ≥ 1. The space of homogeneous invariants of degree 2k for the Clifford group Cm of genus m is spanned by cwe(C(m)), where C ranges over all binary self-dual codes of length 2k; this is a basis if m ≥ k − 1. Proof. Let p be a parabolic invariant. If p is a Clifford invariant then X P (h ⊗ I2m−1 ) p = p. By Lemma 4.7, the operator X P (h ⊗ I2m−1 ) acts triangularly on the vectors cwem (C) (ordered by inclusion); since 2m−r − 2r = 1 ⇒ r = 0, 2m − 1 the hypotheses of Lemma 4.8 are satisfied. The first claim then follows by Lemma 4.8 and Theorem 3.5. Linear independence for m ≥ k − 1 follows from Lemma 4.4. In fact a stronger result holds: THEOREM 4.10. For any binary self-orthogonal code C of even length N containing 1 and of dimension N /2 − r ,  1 (2m + 2i )−1 cwe(C ′ (m)), g · cwe(C(m)) = | Cm | g∈C C′ 1≤i≤r m where the sum on the right is over all self-dual codes C ′ containing C. To see that this is indeed stronger than Theorem 4.9, we observe that if p is an invariant for Cm then 1 | Cm | g·p = p. g∈Cm Since the space of parabolic invariants contains the space of invariants, the same is true of the span of 1 | Cm | g∈Cm g·p where p ranges over the parabolic invariants. By Theorem 4.10 each of these can be written as a linear combination of complete weight enumerators of self-dual codes, and thus Theorem 4.9 follows. Proof. For any self-orthogonal code C, let E m (C) := 1 | Cm | g∈Cm g · cwe(C(m)). 109 THE INVARIANTS OF THE CLIFFORD GROUPS Averaging both sides of equation 1 in Lemma 4.7 over Cm , we find  E m (C) =  1  m−r − 2r )E m (C) + 2−r (2 m (2 − 1)  C⊂C ′ ⊆C ′⊥ [C ′ :C] = 2    E m (C ′ ) ,  and solving for E m (C) gives E m (C) = (2r 1 − 1)(2m + 2r ) E m (C ′ ). C⊂C ′ ⊆C ′⊥ [C ′ : C] = 2 By induction on r (observing that the result follows from Theorem 3.5 when r = 0), we have  1 (2m + 2i )−1 r E m (C) = cwem (C ′′ ). (2 − 1) ′ ′⊥ ⊥ ′ ′′ ′′ 1≤i≤r C⊂C ⊆C C ⊂ C =C [C ′ :C] = 2 But each code C ′′ is counted 2r − 1 times (corresponding to the 1-dimensional subspaces of C ′′ /C); thus eliminating the sum over C ′ gives the desired result. Note that cwe(C(m))(x0 , . . . , x2m−1 −1 , x0 , . . . , x2m−1 −1 ) ∝ cwe(C(m − 1))(x0 , . . . , x2m−1 −1 ) . This gives a surjective map from the space of genus m complete weight enumerators to the space of genus m − 1 complete weight enumerators. By Theorem 4.9 it follows that this also gives a surjective map from the genus m invariants to the genus m − 1 invariants. (Runge’s Proof of Theorem 4.9 proceeds by first showing this map is surjective, using Siegel modular forms, and then arguing that this implies Theorem 4.9.) Since by Theorem 4.6 the parabolic invariants of degree N become linearly independent when m ≥ N2 − 1, we have: COROLLARY 4.11. Let m (t) be the Molien series of the Clifford group of genus m. As m tends to infinity, the series m (t) tend monotonically to ∞ N2k t 2k , k=0 where N2k is the number of equivalence classes of self-dual codes of length 2k. (For the definition of Molien series, see for example [5] or [32, Chapter 19].) Explicit calculations for m = 1, 2 show: COROLLARY 4.12. The initial terms of the Molien series of the Clifford group of genus m ≥ 1 are given by 1 + t 2 + t 4 + t 6 + 2t 8 + 2t 10 + O(t 12 ), where the next term is 2t 12 for m = 1, and 3t 12 for m > 1. 110 NEBE ET AL. Sidelnikov [46,47] showed that the lowest degree of a harmonic invariant of Cm is 8. Inspection of the above Molien series gives the following stronger result. COROLLARY 4.13. The smallest degree of a harmonic invariant of Cm is 8, and there is a unique harmonic invariant of degree 8. There are no harmonic invariants of degree 10. The two-dimensional space of homogeneous invariants for Cm of degree 8 is spanned by the fourth power of the quadratic form and by h m := cwe(H8 ⊗F2 F2m ), where H8 is the [8, 4, 4] binary Hamming code. We can give h m explicitly. THEOREM 4.14. Let G(m, k) denote the set of k-dimensional subspaces of F2m . Then hm = v∈F2m xv8 + 14 + 168 U ∈G(m,1) d∈F2m /U  xv4 v∈d+U  U ∈G(m,2) d∈F2m /U v∈d+U xv2 + 1344  xv . (2) U ∈G(m,3) d∈F2m /U v∈d+U  The second term on the right-hand side is equal to 14 {u,v} xu4 xv4 , where {u, v} runs through unordered pairs of elements of F2m . The total number of terms is 2m + 14  m where k        m m−1 m m−2 m m−3 2 + 168 2 + 1344 2 = 24m , 1 2 3 = |G(m, k)|. Proof. We will compute cwe(H8 ⊗ F2m ) (which is equal to cwe(H8 ⊗ F2m )). Let H8 be defined by the generator matrix 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 . 1 1 A codeword corresponds to a choice of (a, b, c, d) ∈ F2m , one for each row; from the columns of the generator matrix we find that the corresponding term of the weight enumerator is xd xc+d xb+d xb+c+d xa+d xa+c+d xa+b+d xa+b+c+d . This depends only on the affine space a, b, c + d. The four terms on the right-hand side of equation (2) correspond to dima, b, c = 0, 1, 2, 3; the coefficients are the number of ways of choosing (a, b, c, d) for a given affine space. If dima, b, c = 3, for example, there are 7 · 6 · 4 ways to choose a, b, c and 8 ways to choose d, giving the coefficient 8 · 7 · 6 · 4 = 1344. THE INVARIANTS OF THE CLIFFORD GROUPS 111 Remark 4.14. (1) The unique harmonic invariant of degree 8 integrates to zero over the sphere, and so must have zeros on the sphere. The orbit of any such point under Cm therefore forms a spherical 11-design, cf. [25]. This was already observed by Sidelnikov [48]. (2) The case m = 1: C1 is a dihedral group of order 16 with Molien series 1/(1 − λ2 )(1 − 8 λ ), as in Gleason’s theorem on the weight enumerators of binary self-dual codes [24], [32, Problem 3, p. 602], [38]. (3) The case m = 2: C2 has order 2304 and Molien series (1 − λ2 )(1 1 + λ18 . − λ8 )(1 − λ12 )(1 − λ24 ) (The reflection group [3, 4, 3], No. 28 on the Shephard-Todd list, cf. [5, p. 199], is a subgroup of C2 of index 2.) The unique harmonic invariants f 8 and f 12 (say) of degrees 8 and 12 are easily computed, and then one can find real points (x00 , x01 , x10 , x11 ) ∈ S 3 where both f 8 and f 12 vanish. Any orbit of such a point under C2 forms a spherical 15-design of size 2304 (cf. [25]). We conjecture that such points exist for all m ≥ 2. (4) The group C3 of order 5160960 has appeared in sufficiently many different contexts that it is worth placing its Molien series on record. It is p(λ)/q(λ), where p(λ) is the symmetric polynomial of degree 154 beginning 1 +λ8 + λ16 + 2λ20 + λ22 + 2λ24 + 3λ26 + 4λ28 +2λ30 + 5λ32 + 4λ34 + 7λ36 + 6λ38 + 7λ40 +8λ42 + 11λ44 + 9λ46 + 12λ48 + 13λ50 + 14λ52 +15λ54 + 17λ56 + 17λ58 + 20λ60 + 19λ62 +20λ64 + 20λ66 + 25λ68 + 22λ70 + 22λ72 +24λ74 + 25λ76 + · · · , and q(λ) = (1 − λ2 )(1 − λ12 )(1 − λ14 )(1 − λ16 )(1 − λ24 )2 (1 − λ30 )(1 − λ40 ) . 1 (5) For completeness, we mention that the Molien series for E(1) is (1−λ2 )(1−λ 4 ) , with 2 2 2 2 basic invariants x0 + x1 and x0 x1 . For arbitrary m the Molien series for E(m) is   1 1 n2 + n − 2 n2 − n 1 , + + + 2n 2 (1 − λ)n (1 + λ)n (1 − λ2 )n/2 (1 + λ2 )n/2 where n = 2m . 5. Real Clifford Groups and Barnes-Wall-lattices In a series of papers [2,8–10,50], Barnes, Bolt, Room and Wall investigated a family of m lattices in Q2 (cf. also [12,18]). They distinguish two geometrically similar lattices L m ⊆  3 the automorphism groups Aut(L m ) = L ′m in each dimension 2m , for which if m = Aut(L ′m ) are subgroups Gm of index 2 in the real Clifford group Cm . When m = 3, L 3 and 112 NEBE ET AL. L ′3 are two versions of the root lattice E 8 , and G3 := Aut(L 3 ) ∩ Aut(L ′3 ) has index 270 in Aut(L 3 ) and index 2 in C3 . The lattices L m and L ′m can be defined in terms of an orthonormal basis b0 , . . . , b2m −1 of 2m R as follows. Let V := F2m and index the basis elements b0 , . . . , b2m −1 by the elements of 2m V . For each affine 2msubspace U ⊆ V let χU ∈ Q correspond to the characteristic function of U : χU := i=1 ǫi bi , where ǫi = 1 if i corresponds to an element of U and ǫi = 0 otherwise. Then L m (resp. L ′m ) is spanned by the set  ⌊(m−d+δ)/2⌋  2 χU | 0 ≤ d ≤ m, U is a d-dimensional affine subspace of V , where δ = 1 for L m and δ = 0 for L ′m .√ Extending scalars, we define the Z[ 2]-lattice √ Mm := 2L ′m + L m , which we call the balanced Barnes-Wall lattice. From the generating sets for L m and L ′m we have: √ m−d Remark 5.1. Mm is generated by the vectors 2 χU , where 0 ≤ d ≤ m and U runs through the affine subspaces of V of dimension d. LEMMA 5.2. For all m > 1, the lattice Mm is a tensor product: Mm = Mm−1 ⊗Z[√2] M1 = M1 ⊗Z[√2] M1 ⊗Z[√2] · · · ⊗Z[√2] M1 (with m factors). Proof. Write V = F2m = Vm−1 ⊕ V1 as the direct sum of an (m − 1)-dimensional vector space Vm−1 and a 1-dimensional space V1 = v, and arrange the basis vectors so that b0 , . . . , b2m−1 −1 correspond to the elements in Vm−1 and b2m−1 , . . . , b2m −1 to the elements in v + Vm−1 . √ m−d Let 2 χU be a generator for Mm , where U = a + U0 for a d-dimensional linear subspace U0 of V and a = am−1 + a1 ∈ Vm−1 ⊕ V1 . If U0 ≤ Vm−1 , then √ m−d √ √ m−1−d 2 χU = 2 χam−1 +U0 ⊗ 2χa1 ∈ Mm−1 ⊗Z[√2] M1 . Otherwise Um−1 := U0 ∩ Vm−1 has dimension d − 1 and U0 = Um−1 ∪ (vm−1 + v + Um−1 ) for some vm−1 ∈ Vm−1 . If vm−1 ∈ Um−1 , then √ m−1−(d−1) √ m−d 2 χU = 2 χam−1 +Um−1 ⊗ χV1 . If vm−1 ∈ Um−1 we have the identity √ √ m−1−d √ m−d 2 χU = 2 χam−1 +Um−1 +F2 vm−1 ⊗ 2χa1 √ m−1−(d−1) + 2 χam−1 +vm−1 +Um−1 ⊗ χV1 √ √ √ m−1−(d−1) χam−1 +vm−1 +Um−1 ⊗ 2χa1 . − 2 2 Hence Mm ⊆ Mm−1 ⊗Z[√2] M1 . The other inclusion follows more easily by similar arguments. 113 THE INVARIANTS OF THE CLIFFORD GROUPS In view of Lemma 5.2, we have the following simple and apparently √ new construction for the Barnes-Wall lattice L m . Namely, L m is the rational part of the Z[ 2]-lattice M1⊗m , √ √ where M1 is the Z[ 2]-lattice with Gram matrix √22 22 . For more about this construction see [34]. PROPOSITION 5.3. For all m ≥ 1, the automorphism group Aut(Mm ) (the subgroup of the orthogonal group O(2m , R) that preserves Mm ) is isomorphic to Cm . ′ m m−1 m−1 be a Z-basis Proof. Let (v1 , . . . , v2m )√ √for L m such that (2v1 , . . . , 2v2 √, v2 +1 , . . . , v2 ) m−1 , v2m−1 +1 , . . . , v2m ) is a Z[ 2v , . . . , 2v 2]-basis for M = is a Z-basis for L . Then ( 1 m m 2 √ √ √ ′ 2L m + L m . Hence Mm has a Z-basis ( 2v1 , . . . , 2v2m , 2v1 , . . . , 2v2m−1 , v2m−1 +1 , . . . , v2m ). Since the scalar products of the vi are integral, the Z-lattice Mm with respect to 12 the √ ′ √ on M is isometric to 2L m ⊥ L m√. In partrace form of the Z[ 2]-valued standard form m √ ticular, the automorphism group of the Z[ 2]-lattice Mm is √ the subgroup of Aut( 2L m ⊥ L ′m ) ∼ = Gm ≀ S2 that commutes with the multiplication by 2. Hence Aut(Mm ) contains Gm = Aut(L m ) ∩ Aut(L ′m ) as a subgroup of index at most two. Since h ⊗ I2m−1 ∈ Aut(M1 ) ⊗ Aut(Mm−1 ) ⊆ Aut(Mm ), by Lemma 5.2, [Aut(Mm ) : Gm ] = 2 and so Aut(Mm ) ∼ = Cm . LEMMA 5.4. If m ≥ 2, m ]) of the matrices in Cm acting on √ m Z[C √then the Z-span (denoted m the 2m -dimensional Z[ 2]-lattice Mm is Z[ 2]2 ×2 . Proof. We proceed by induction on m. Explicit calculations show that the lemma √ m−2is true m−2 for m = 2 and m√= 3. If m ≥ 4 then m −2 ≥ 2 and by induction Z[Cm−2 ] = Z[ 2]2 ×2 4×4 √ and Z[C2 ] = Z[ 2] . Since Mm = M2 ⊗Z[ 2] Mm−2 , the automorphism group of Mm contains C2 ⊗ Cm−2 . Hence Z √ 2 2m ×2m ⊇ Z[Cm ] ⊇ Z[Cm−2 ] ⊗Z[√2] Z[C2 ] = Z √ 2 2m ×2m . We now proceed to show that for m ≥ 2 the real Clifford group Cm is a maximal finite subgroup of G L(2m , R). For the investigation of possible normal subgroups of finite groups containing Cm , the notion of a primitive matrix group plays a central role. A matrix group G ≤ G L(V ) is called imprimitive if there is a nontrivial decomposition V = V1 ⊕ . . . ⊕ Vs of V into subspaces which are permuted under the action of G. G is called primitive if it is not imprimitive. If N is a normal subgroup of G then G permutes the isotypic components of V|N . So if G is primitive, the restriction of V to N is isotypic, i.e., is a multiple of an irreducible representation. In particular, since the image of an irreducible representation of an abelian group N is cyclic, all abelian normal subgroups of G are cyclic. LEMMA 5.5. Let m ≥ 2. Let G be a finite group with Cm ≤ G ≤ G L(2m , R) and let p be a prime. If p is odd, the maximal normal p-subgroup of G is trivial. The maximal normal 2-subgroup of G is either E(m) if G = Cm , or Z (E(m)) = −I2m  if G > Cm . 114 NEBE ET AL. Proof. We first observe that the only nontrivial normal subgroup of Cm that is properly contained in E(m) is Z (E(m)) = −I2m . Therefore, if U is a normal subgroup of G, U ∩ E(m) is one of 1, Z (E(m)) or E(m). The matrix group Cm and hence also G is primitive. In particular, all abelian normal subgroups of G are cyclic. Let p > 2 be a rational prime and U
G a normal p-subgroup of m G. The degree of the absolutely irreducible representations of U that occur in R2|U is a power m of p and divides 2 . So this degree is 1 and U is abelian, hence cyclic by the primitivity of G. Therefore the automorphism group of U does not contain E(m)/Z (E(m)). Since C G (U ) ∩ E(m) is a normal subgroup of Cm , it equals E(m) and hence E(m) centralizes U . Since E(m) is already absolutely irreducible, U consists of scalar matrices in G L(2m , R), and therefore U = 1. If p = 2 and G =  Cm , then U =  E(m), because Cm is the largest finite subgroup of G L(2m , R) that normalizes E(m). Since the normal 2-subgroups of G do not contain an abelian noncyclic characteristic subgroup, the possible normal 2-subgroups are classified in a theorem of P. Hall (cf. [27, p. 357]). In particular they do not contain Cm /Z (E(m)) as a subgroup of their automorphism groups, so again U commutes with E(m), and therefore consists only of scalar matrices. THEOREM 5.6. Let m ≥ 2. Then the real Clifford group Cm is a maximal finite subgroup of G L(2m , R). Proof. Let G be a finite subgroup of G L(2m , R) that properly contains Cm . By Lemma 5.5, all normal p-subgroups of G are central. By a theorem of Brauer, every representation of a finite group is realizable over a cyclotomic number field (cf. [43, §12.3]). In fact, since the natural representation of G is real, it is even true that G is conjugate to√a subgroup of G L(2m , K ) for some totally real abelian number field K containing Q[ 2] (cf. [19, Proposition 5.6]). Let K be a minimal such field and assume that G ≤ G L(2m , K ). Let R be the ring of integers of K . Then G fixes an RCm -lattice. By Lemma 5.4 all RCm -lattices are of the form I ⊗Z[√2] Mm for some fractional ideal I of R, so the group G fixes all √ RCm -lattices and hence also R ⊗Z[√2] Mm . So any choice of a Z[ 2]-basis for Mm gives rise to an embedding G ֒→ G L(2m , R), by which we may regard G as a group of matrices. √ Without loss of generality √ we may assume that G = Aut(R ⊗Z[ 2] Mm ). Then the Galois group Ŵ := Gal(K /Q[ 2]) acts on G√by acting componentwise on the matrices. Seeking a contradiction, we assume K =  Q[ 2]. It is enough to show that there is a nontrivial element σ ∈ Ŵ that acts trivially on G, because then the matrices in G have their entries in the fixed field of σ , contradicting the minimality of K . Assume first that there is an odd prime p ramified √ in K /Q, and let ℘ be a prime ideal of R that lies over p. Then p is also ramified in K /Q[ 2] and therefore the action of the ramification group, the stabilizer in Ŵ of ℘, on R/℘ is not faithful, hence the first inertia group √   Ŵ℘ := σ ∈ Gal(K /Q 2 ) | σ (x) ≡ x (mod ℘) for all x ∈ R is nontrivial (see e.g. [22, Corollary III.4.2]). Since G ℘ := {g ∈ G | g ≡ I2m (mod ℘)} is a normal p-subgroup of G, G ℘ = 1 by Lemma 5.5. Therefore all the elements in Ŵ℘ act trivially on G, which is what we were seeking to prove. −1 So 2 is the only ramified prime in K , which implies that √ K = Q[ζ2a + ζ2a ] for some a ≥ 3, where ζt = exp(2πi/t). If a = 3, then K = Q[ 2], G = Aut(Mm ) = Cm and THE INVARIANTS OF THE CLIFFORD GROUPS 115 we are done. So assume a > 3 and let ℘ be the prime ideal of R over 2 (generated by −1 (1 − ζ2a )(1 − ζ2−1 a )) and let σ ∈ Ŵ be the Galois automorphism defined by σ (ζ2a + ζ2a ) = a−1 a−1 −1 2 ζ22a +1 + ζ2−2 = −(ζ2a + ζ2−1 =  σ and a a ). Then id = σ ζ2a + ζ2−1 − σ ζ2a + ζ2−1 = 2 ζ2a + ζ2−1 ∈ 2℘. a a a Therefore σ ∈ Ŵ2℘ . Since the subgroup G 2℘ := {g ∈ G | g ≡ I2m (mod 2℘)} of G is trivial (cf. [3, Hilfssatz 1]) one concludes that σ acts trivially on G, and thus G is in fact defined over Q[ζ2a−1 + ζ2−1 a−1 ]. The theorem follows by induction. COROLLARY 5.7. Let m ≥ 1 and let C be a self-dual code over F2 that is not generated by vectors of weight 2. Then Cm = Aut O(2m ,R) (cwe(C(m)). Proof. The proof for the case m = 1 will be postponed to Section 6. Assume m ≥ 2. We first show that the parabolic subgroup H ≤ Cm acts irreducibly on the Lie algebra Lie(O(2m , R)), the set of real 2m × 2m matrices X such that X = −X tr . The group m AGL(m, 2) acts 2-transitively on our standard basis b0 , . . . , b2m −1 for R2 . A basis for Lie(O(2m , R)) is given by the matrices bi j := bi ⊗ b j − b j ⊗ bi for 0 ≤ i < j ≤ 2m − 1. Since AGL(m, 2) acts transitively on the bi j , a basis for the endomorphism ring End AG L(m,2) (Lie(O(2m , R))) is given by the orbits of the stabilizer of b01 . Representatives for these orbits are b01 , b02 , b23 and b24 . But the generator corresponding to the quadratic form q(v1 , . . . , vm ) := v22 negates b2 and fixes b0 and b4 , and therefore does not commute with the endomorphism corresponding to b02 or b24 . Similarly the endomorphism corresponding to b23 is ruled out by q(v1 , . . . , vm ) := v1 v2 . Let G := Aut O(2m ,R) (cwe(C(m)). Then G is a closed subgroup of O(2m , R) and hence is a Lie group (cf. [37, Theorem 3.4]). Since G contains Cm it acts irreducibly on Lie(O(2m , R)). Assume that G =  Cm . Then G is infinite by Theorem 5.6 and therefore G contains SO m (2 , R). However, the ring of invariants of S O(2m , R) is generated by the quadratic form 2m −1 2 i=0 x bi . The only binary self-dual codes C that produce such complete weight enumerators are direct sums of copies of the code {00, 11}. 6. The Complex Clifford Groups and Doubly-Even Codes There are analogues for the complex Clifford group Xm for most of the above results. (Z a will denote a cyclic group of order a.) Definition 6.1. The complex Clifford group Xm is the normalizer in U (2m , Q[ζ8 ]) of the central product E(m)Y Z 4 . As in the real case, one concludes that 1+2m 1+2m Xm ∼ Y Z 8 .O(2m + 1, 2) Y Z 8 .Sp(2m, 2) ∼ = 2+ = 2+ (cf. [33, Cor. 8.4]). 116 NEBE ET AL. The analogue of Theorem 4.9 is the following, which can be proved in essentially the same way. THEOREM 6.2. (Runge [42].) Fix integers N and m ≥ 1. The space of homogeneous invariants of degree N for the complex Clifford group Xm is spanned by cwe(C(m)), where C ranges over all binary doubly-even self-dual codes of length N . (In particular, when N is not a multiple of 8, the invariant space is empty.) The analogues of Theorem 4.10 and Proposition 5.3 are: THEOREM 6.3. For any doubly-even binary code C of length N ≡ 0(8) containing 1 and of dimension N /2 − r ,  1 (2m + 2i )−1 cwe(C ′ (m)), g · cwe(C(m)) = |Xm | g∈X ′ C 0≤i<r m where the sum is over all doubly-even self-dual codes C ′ containing C. PROPOSITION 6.4. Let Mm := Z[ζ8 ] ⊗Z[√2] Mm . Then the subgroup of U (2m , Q[ζ8 ]) preserving Mm is precisely Xm . We omit the proofs. For the analogue of Lemma 5.4, observe that the matrices in Xm generate a maximal order. Even for m = 1 the Z-span of the matrices in X1 acting on M1 is the maximal order Z[ζ8 ]2×2 . m m Hence the induction argument used to prove Lemma 5.4 shows that Z[Xm ] = Z[ζ8 ]2 ×2 . Therefore the analogue of Theorem 5.6 holds even for m = 1: THEOREM 6.5. Let m ≥ 1 and let G be a finite group such that Xm ≤ G ≤ U (2m , C). Then there exists a root of unity ζ such that G = Xm , ζ I2m . Proof. As in the proof of Theorem 5.6, we may assume that G is contained in U (2m , K ) for some abelian number field K containing ζ8 . Let R be the ring of integers in K and T the group of roots of unity in R. Then T Xm is the normalizer in U (2m , K ) of T E(m) (cf. [33, Cor. 8.4]). As before, the RXm -lattices in the natural module are of the form I ⊗ Z[ζ8 ] Mm , where I is a fractional ideal of R. Since G fixes one of these lattices, it also fixes R ⊗ Z[ζ8 ] Mm . As in the proof of Theorem 5.6, we write the elements of G as matrices with respect to a basis for Mm and assume that G is the full (unitary) automorphism group of R ⊗ Z[ζ8 ] Mm . Then the Galois group Ŵ := Gal(K /Q[ζ8 ]) acts on G. Assume that G =  T Xm . Then T E(m) is not normal in G. As in Lemma 5.5 one shows that the maximal normal p-subgroup of G is central for all primes p. Let ℘ be a prime ideal in R that ramifies in K /Q[ζ8 ], and let σ be an element of the inertia group Ŵ℘ . Then for all g ∈ G, the image g σ satisfies a(g) := g −1 g σ ∈ G ℘ := {g ∈ G | g ≡ I2m (mod ℘)}. Since G ℘ is a normal p-subgroup, where p is the rational prime divisible by ℘, it is central. Therefore the map g → a(g) is a homomorphism of G into an abelian group, and hence the commutator subgroup G ′ is fixed THE INVARIANTS OF THE CLIFFORD GROUPS 117 under σ . Since any abelian extension K of Q that properly contains Q[ζ8 ] is ramified at some finite prime of Q[ζ8 ], we conclude that G ′ ⊆ Aut(Mm ). Since E(m)Y Z 8 ≤ Aut(Mm )′ Y Z 8 is characteristic in Aut(Mm ) and therefore also in G ′ Y Z 8 , the group T E(m) is normal in G, which is a contradiction. COROLLARY 6.6. Assume m ≥ 1 and let C be a binary self-dual doubly-even code of length N . Then AutU (2m ,C) (cwe(C ⊗ F2m )) = Xm , ζ N I2m . Remark 6.7. (1) The case m = 1: X1 is a unitary reflection group (No. 9 on the Shephard-Todd list) of order 192 with Molien series 1/(1 − λ8 )(1 − λ24 ), as in Gleason’s theorem on the weight enumerators of doubly-even binary self-dual codes [24], [32, p. 602, Theorem 3c], [38]. (2) The case m = 2: X2 has order 92160 and Molien series (1 − 1 + λ32 . − λ24 )2 (1 − λ40 ) λ8 )(1 This has a reflection subgroup of index 2, No. 31 on the Shephard-Todd list. (3) The case m = 3: X3 has order 743178240, and the Molien series can be written as p(λ8 )/q(λ), where p(λ) is the symmetric polynomial of degree 44 beginning 1 + λ3 + 3λ4 + 3λ5 + 6λ6 + 8λ7 + 12λ8 + 18λ9 + 25λ10 + 29λ11 + 40λ12 + 50λ13 + 58λ14 + 69λ15 + 80λ16 + 85λ17 + 96λ18 + 104λ19 + 107λ20 + 109λ21 + 112λ22 + · · · and q(λ) = (1 − λ8 )(1 − λ16 )(1 − λ24 )2 (1 − λ40 )(1 − λ56 )(1 − λ72 )(1 − λ120 ). Runge [40] gives the Molien series for the commutator subgroup H3 = X3′ , of index 2 in X3 . The Molien series for X3 consists of the terms in the series for H3 that have exponents divisible by 4. Oura [36] has computed the Molien series for H4 = X4′ , and that for X4 can be obtained from it in the same way. Other related Molien series can be found in [1]. Proof of Corollary 5.7, case m = 1. Let C be a self-dual binary code of length n with Hamming weight enumerator hweC (x, y). We will show that if C is not generated by vectors of weight 2 then Aut O(2) (hweC ) = C1 . Certainly G := Aut O(2) (hweC ) contains C1 = D16 ; we must show it is no larger. The only closed subgroups of O(2) containing D16 are the dihedral groups D16k for k ≥ 1 and O(2) itself. So if the result is false then G contains a rotation   cos(θ) sin(θ) ρ(θ) = −sin(θ) cos(θ) where θ is not a multiple of π/4. Consider the shadow S(C) of C [38]; that is, the set of vectors v ∈ Fn2 such that wt(v + w) ≡ wt(v) (mod 4), for all w ∈ C. 118 NEBE ET AL. The weight enumerator of S(C) is given by S(x, y) = 2−n/2 hweC (x + y, i(x − y)). Then ρ(θ) ∈ G if and only if S(x, y) = S(eiθ x, e−iθ y), or in other words if and only if for all v ∈ S(C), (n − 2wt(v))θ is a multiple of 2π . Now, pick a vector v0 ∈ S(C), and consider the polynomial W (x, y, z, w) given by x n−wt(v0 )−wt((1+v0 )∩v) y wt((1+v0 )∩v) z wt(v0 )−wt(v0 ∩v) w wt(v0 ∩v) . v∈C This has the following symmetries: W (x, i y, z, −iw) = W (x, y, z, w), √ √ √ √ W ((x + y)/ 2, (x − y)/ 2, (z + w)/ 2, (z − w)/ 2) = W (x, y, z, w). Furthermore, since S(C) = v0 + C, ρ(θ) ∈ G if and only if W (eiθ x, e−iθ y, e−iθ z, eiθ w) = W (x, y, z, w). To each of these symmetries we associate a 2 × 2 unitary matrix U such that (x, y) is transformed according to U and (z, w) according to U . The first two symmetries generate the complex group X1 , which is maximally finite in PU (2) by Theorem 6.5. On the other hand, we can check directly that  iθ  e 0 ∈ / X1 , 0 e−iθ even up to scalar multiplication. Thus the three symmetries topologically generate PU (2); and hence W is invariant under any unitary matrix of determinant ±1. Since hweC (x, y) = W (x, y, x, y), it follows that G = O(2). But then hweC (x, y) = (x 2 + y 2 )n/2 , implying that C is generated by vectors of weight 2. This completes the proof of Corollary 5.7. 7. Clifford Groups for p > 2 Given an odd prime p, there again is a natural representation of the extraspecial p-group 1+2m E p (m) ∼ of exponent p, this time in U ( p m , C); to be precise, E p (1) is generated = p+ by transforms X : vx → vx+1 , and Z : vx → exp(2πi x/ p)vx , x ∈ Z/ pZ, and E p (m) is the m-th tensor power of E p (1). The Clifford group Cm( p) is then defined to be the normalizer in U ( p m , Q[ζap ]) of E p (m), where a = gcd{ p + 1, 4}. As above, one finds that 1+2m Cm( p) ∼ .Sp(2m, p) = Z a × p+ (cf. e.g. [51]). THE INVARIANTS OF THE CLIFFORD GROUPS 119 As before, the invariants of these Clifford groups are given by codes: THEOREM 7.1. Fix integers N and m ≥ 1. The space of invariants of degree N for the Clifford group Cm( p) is spanned by cwe(C(m)), where C ranges over all self-dual codes over F p of length N containing 1. THEOREM 7.2. For any self-orthogonal code C over F p of length N containing 1 and of dimension N /2 − r ,  1 ( p m + pi )−1 cwe(C ′ (m)), g · cwe(C(m)) =  ( p)  Cm  ′ ( p) C 0≤i<r g∈Cm where the sum is over all self-dual codes C ′ containing C (and in particular is 0 if no such code exists). Regarding maximal finiteness, the arguments we used for p = 2 to prove Theorem 5.6 do not carry over to odd primes, since the groups Cm( p) do not span a maximal order. Lindsey ( p) [31] showed by group theoretic arguments that C1 is a maximal finite subgroup of S L( p, C) (cf. [6] for p = 3, [11] for p = 5). For p m = 9, the theorem below follows from [21] and [26]. THEOREM 7.3. Let p > 2 be a prime and m ≥ 1. If G is a finite group with Cm( p) ≤ G ≤ G L( p m , C), there exists a root of unity ζ such that   G = Cm( p) , ζ I pm . Proof. As before we may assume that G is contained in U ( p m , K ) for some abelian number field K containing ζ p . Let L denote the set of rational primes l satisfying the following four properties: (i) G is l-adically integral, (ii) l is unramified in K , (iii) |G| < |PGL( p m , l)|, (iv) l splits completely in K . Since all but finitely many primes satisfy conditions (i)-(iii), and infinitely many primes satisfy (iv) (by the Čebotarev Density Theorem), it follows that the set L is infinite. Fix a prime l over l ∈ L. Since G is l-adically integral, we can reduce it mod l, obtaining a representation of G in G L( p m , l). Since p is ramified in K , l = p, so this representation is faithful on the extraspecial group. Since the extraspecial group acts irreducibly, the representation is in fact faithful on the entire Clifford group. Thus G mod l contains the normalizer of an extraspecial group, but modulo scalars is strictly contained in PGL( p m , l) (by condition (iii)). It follows from the main theorem of [30] that for p m ≥ 13 G mod l and Cm( p) mod l coincide as subgroups of P G L( p m , l). For p m < 13 this already follows from the references in the paragraph preceding the theorem. Fix a coset S of Cm( p) in G. For each prime l|l with l ∈ L, the above argument implies that we can choose an element g ∈ S such that g ∝ 1 (mod l). As there are infinitely many such primes, at least one such g must get chosen infinitely often. But then we must actually have g ∝ 1 in K , and since g has finite order, g = ζ S for some root of unity ζ S . Since this holds for all cosets S, G is generated by Cm( p) together with the roots of unity ζ S , proving the theorem. 120 NEBE ET AL. Remark 7.4. It is worth pointing out that the proof of the main theorem in [30] relies heavily on the classification of finite simple groups, which is why we preferred to use our alternative arguments when proving Theorem 5.6. References 1. E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, IEEE Trans. Inform. Theory, Vol. 45 (1999) pp. 1194–1205. 2. E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Australian Math. Soc., Vol. 1 (1959) pp. 47–63. 3. H.-J. Bartels, Zur Galoiskohomologie definiter arithmetischer Gruppen, J. Reine Angew. Math., Vol. 298 (1978) pp. 89–97. 4. C. H. Bennett, D. DiVincenzo, J. A. Smolin and W. K. Wootters, Mixed state entanglement and quantum error correction, Phys. Rev. A, Vol. 54 (1996) pp. 3824–3851. 5. D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge University Press (1993). 6. H. F. Blichfeldt, Finite Collineation Groups, University of Chicago Press, Chicago (1917). 7. S. Böcherer, Siegel modular forms and theta series, in Theta functions (Bowdoin 1987), Proc. Sympos. Pure Math., 49, Part 2, Amer. Math. Soc., Providence, RI, (1989), pp. 3–17. 8. B. Bolt, The Clifford collineation, transform and similarity groups III: Generators and involutions, J. Australian Math. Soc., Vol. 2 (1961) pp. 334–344. 9. B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups I, J. Australian Math. Soc., Vol. 2 (1961) pp. 60–79. 10. B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., Vol. 2 (1961) pp. 80–96. 11. R. Brauer, Über endliche lineare Gruppen von Primzahlgrad, Math. Annalen, Vol. 169 (1967) pp. 73–96 12. M. Broué and M. Enguehard, Une famille infinie de formes quadratiques entières; leurs groupes d’automorphismes, Ann. scient. Éc. Norm. Sup., 4e série, 6 (1973), 17–52. Summary in C. R. Acad. Sc. Paris, Vol. 274 (1972) pp. 19–22. 13. A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, Z4 -Kerdock codes, orthogonal spreads and extremal Euclidean line-sets, Proc. London Math. Soc., Vol. 75 (1997) pp. 436–480. 14. A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor and N. J. A. Sloane, A group-theoretic framework for the construction of packings in Grassmannian spaces, J. Algebraic Combin., Vol. 9 (1999) pp. 129–140. 15. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction orthogonal geometry, Phys. Rev. Letters, Vol. 78 (1997) pp. 405–409. 16. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over G F(4), IEEE Trans. Inform. Theory, Vol. 44 (1998) pp. 1369–1387. 17. J. H. Conway, R. H. Hardin and N. J. A. Sloane, Packing lines, planes, etc.: packings in Grassmannian space, Experimental Math., Vol. 5 (1996) pp. 139–159. 18. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer-Verlag, New York (1998). 19. A. W. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Mathematics, Vol. 102 (1975) pp. 291–325. 20. W. Duke, On codes and Siegel modular forms, Intern. Math. Res. Notices, Vol. 5 (1993) pp. 125–136. 21. W. Feit, On finite linear groups in dimension at most 10, in Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Academic Press, New York (1976) pp. 397–407. 22. A. Fröhlich and M. J. Taylor, Algebraic Number Theory, Cambridge Univ. Press (1991). 23. S. P. Glasby, On the faithful representations, of degree 2n , of certain extensions of 2-groups by orthogonal and symplectic groups. J. Australian Math. Soc. Ser. A, Vol. 58 (1995) pp. 232–247. 24. A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, in Actes, Congrés International de Mathématiques (Nice, 1970), Gauthiers-Villars, Paris, Vol. 3 (1971) pp. 211–215. THE INVARIANTS OF THE CLIFFORD GROUPS 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 121 J.-M. Goethals and J. J. Seidel, The football, Nieuw Archief voor Wiskunde, Vol. 29 (1981) pp. 50–58. Reprinted in Geometry and Combinatorics: Selected Works of J. J. Seidel, (D. G. Corneil and R. Mathon, ed.) Academic Press (1991) pp. 363–371. W. C. Huffman and D. B. Wales, Linear groups of degree nine with no elements of order seven. J. Algebra, Vol. 51 (1978) pp. 149–163. B. Huppert, Endliche Gruppen I, Springer-Verlag (1967). L. S. Kazarin, On certain groups defined by Sidelnikov (in Russian), Mat. Sb., Vol. 189 (No. 7, 1998) pp. 131–144; English translation in Sb. Math., Vol. 189 (1998) pp. 1087–1100. A. Y. Kitaev, Quantum computations: algorithms and error correction (in Russian), Uspekhi Mat. Nauk., Vol. 52 (No. 6, 1997) pp. 53–112; English translation in Russian Math. Surveys, Vol. 52 (1997) pp. 1191–1249. P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge Univ. Press (1988). J. H. Lindsey II, Finite linear groups of prime degree, Math. Annalen, Vol. 189 (1970) pp. 47–59. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes North-Holland, Amsterdam (1977). G. Nebe, Finite quaternionic matrix groups, Representation Theory, Vol. 2 (1998) pp. 106–223. G. Nebe, E. M. Rains and N. J. A. Sloane, A simple construction for the Barnes-Wall lattices, in Forney Festschrift, (R. Blahut, eds.) to appear (2001). G. Nebe, E. M. Rains and N. J. A. Sloane, Generalized self-dual codes and Clifford-Weil groups, preprint. M. Oura, The dimension formula for the ring of code polynomials in genus 4, Osaka J. Math., Vol. 34 (1997) pp. 53–72. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, San Diego (1994). E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, ed. V. Pless and W. C. Huffman, Elsevier, Amsterdam (1998) pp. 177–294. B. Runge, On Siegel modular forms I, J. Reine Angew. Math., Vol. 436 (1993) pp. 57–85. B. Runge, On Siegel modular forms II, Nagoya Math. J., Vol. 138 (1995) pp. 179–197. B. Runge, The Schottky ideal, in Abelian Varieties (Egloffstein, 1993), de Gruyter, Berlin (1995) pp. 251–272. B. Runge, Codes and Siegel modular forms, Discrete Math., Vol. 148 (1996) pp. 175–204. J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag (1977). P. W. Shor and N. J. A. Sloane, A family of optimal packings in Grassmannian manifolds, J. Algebraic Combin., Vol. 7 (1998) pp. 157–163. V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform., (1997), Vol. 33 pp. 33 35–54 (1997); English translation in Problems Inform. Transmission, Vol. 33 (1997) pp. 29–44. V. M. Sidelnikov, On a finite group of matrices generating orbit codes on the Euclidean sphere, in Proceedings IEEE Internat. Sympos. Inform. Theory, Ulm, 1997, IEEE Press (1997) p. 436. V. M. Sidelnikov, Spherical 7-designs in 2n -dimensional Euclidean space, J. Algebraic Combin., Vol. 10 (1999) pp. 279–288. V. M. Sidelnikov, Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial (in Russian), Algebra i Analiz, Vol. 11 (No. 4, 1999) pp. 183–203. B. Venkov, Réseaux et “designs” sphériques, in Réseaux euclidiens, “designs” sphériques et groupes, L’Enseignement Mathématiques Monographie Vol. 37 (Martinet, J. ed.), to appear (2001). G. E. Wall, On Clifford collineation, transform and similarity groups IV: an application to quadratic forms, Nagoya Math. J., Vol. 21 (1962) pp. 199–222. D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mtn. J. Math., Vol. 2 (1972) pp. 159–168.