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Clifford-Weil Groups for Finite Group Rings, Some Examples. 1 Introduction

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Clifford-Weil Groups for Finite Group Rings, Some Examples. 1 Introduction Cli®ord-Weil groups for ¯nite group rings, some examples. Annika GuntherÄ and Gabriele Nebe Lehrstuhl D furÄ Mathematik, RWTH Aachen University 52056 Aachen, Germany [email protected], [email protected] Abstract. Finite group rings carry a natural involution that de¯nes a form ring structure. We investigate the associated Cli®ord-Weil groups for the indecomposable representations of the groups of order 2; 3 and the symmetric group Sym3 over the ¯elds with 2 and 3 elements as well as suitable symmetrizations. An analogue of Kneser's neighboring method is introduced, to classify all self-dual codes in a given representation. 1 Introduction. Let G be a ¯nite group and K be a ¯nite ¯eld. Then the group algebra KG is a ¯nite K-algebra with a natural K-linear involution ¡1 : agg 7! agg : gX2G gX2G This de¯nes a form ring structure R²(KG) on KG where ² = §1 (see Section 2). A ¯nite representation ½ of R²(KG) consists of a ¯nite KG-module V together with a G- invariant non-degenerate K-bilinear form ¯ : V £ V ! K which is symmetric, if ² = 1 and skew-symmetric if ² = ¡1. We do not deal with Hermitian forms here, since in our examples K will be a prime ¯eld. In this language, a self-dual code C of length N for the representation ½ (for short, a code in ½) is a KG-submodule of V N that is self-dual with respect to N N N N N ¯ : V £ V ! K; ¯ ((x1; : : : ; xN ); (y1; : : : ; yN )) = ¯(xi; yi): Xi=1 The complete weight enumerator of a code C · V N is N cwe(C) := xci 2 C[xv j v 2 V ] cX2C Yi=1 and a homogeneous polynomial of degree N in jV j variables. In Section 2 we will give explicit generators for a ¯nite complex matrix group, the associated Cli®ord-Weil group C(½) such that cwe(C) is invariant under all variable substitutions de¯ned by elements in C(½), hence cwe(C) 2 Inv(C(½)) lies in the invariant ring Inv(C(½)). In fact the main results of [7] (Corollary 5.7.4 and 5.7.5) show that for a fairly general class of form rings Inv(C(½)) is generated as a vector space over C by the complete weight enumerators of self-dual codes in ½. We conjecture that this is true for arbitrary ¯nite form rings (cf. [7, 1 Conjecture 5.7.2]) and in particular also for R²(KG), but we do not know how to prove this for arbitrary ¯nite group rings KG. We denote the cyclic group of order n by Zn and the symmetric group of degree n by Symn. Moreover we let Fp be the ¯eld with p elements. 2 Rings with involution. Rings with involution de¯ne certain form rings as explained below. We will apply the theory developed in this section to group rings with the natural involution . Let R be a ring with 1 and J : R ! R; x 7! xJ an involution, i.e. a ring antiautomorphism of order 1 or 2. So (ab)J = bJ aJ and (aJ )J = a for all a; b 2 R. Moreover let ² 2 Z(R) be a central unit of R such that ²J ² = 1. As explained in [7, Lemma 1.4.5] this setting de¯nes a twisted ring (R; id; M = R) where the twist ¿ on M = R is de¯ned by ¿ : R ! R; a 7! aJ ²: The quadrupel R(R; J; ²) := (R; id; M = R; © = R=¤) is a form ring (see [7, De¯nition 1.7.1]) where ¤ := fm ¡ ¿(m) j m 2 Mg with mappings ff gg : M ! ©; m 7! m + ¤ and ¸ : © ! M; m + ¤ 7! m + ¿(m): The R-qmodule structure on © is given by (m + ¤)[x] = xJ mx + ¤ for all m 2 M; x 2 R: A representation of the form ring R(R; J; ²) is given by a left R-module V together with a non-degenerate biadditive form ¯ : V £ V ! A into some abelian group A such that ¯(v; rw) = ¯(rJ v; w) and ¯(v; w) = ¯(w; ²v) for all v; w 2 V; r 2 R: That ¯ is non-degenerate means that it induces an isomorphism ¯¤ : V ! Hom(V; A); v 7! (w 7! ¯(w; v)) which is then an isomorphism of R-left-modules, where Hom(V; A) is an R-left-module by (rf)(v) := f(rJ v) for all f 2 Hom(V; A); r 2 R; v 2 V: The corresponding homomorphisms ½M : R ! Bil(V; A); ½© : © ! Quad0(V; A) are given by ½M (m) : (v; w) 7! ¯(v; mw); ½©(m + ¤) : v 7! ¯(v; mv): Note that for all m 2 M and all v 2 V ¯(v; mv) = ¯(mJ v; v) = ¯(v; ²mJ v) = ¯(v; ¿(m)v) so ½© is well-de¯ned. 2 2.1 Symmetric idempotents An idempotent e2 = e 2 R is called symmetric, if eR =» eJ R as right R-modules. In this case there J J J are ue 2 eRe and ve 2 e Re such that ueve = e and veue = e . A set of representatives of the R¤-conjugacy classes of symmetric idempotents in R will be denoted by SymId(R). 2.2 The associated Cli®ord-Weil group In coding theory one is mainly interested in ¯nite alphabets V . We now assume that R is a ¯nite dimensional algebra over a ¯nite ¯eld K such that the restriction of J is the identity on K. For any representation ½ = (V; ¯) of the form ring R(R; J; ²) we may take the abelian group A to be ¤ ¤ the ¯eld K and ¯ : V ! V := HomK (V; K). Let p be the characteristic of K and trace : K ! Fp » 1 denote the trace from K into its prime ¯eld Fp. Identifying Fp = Z=pZ with p Z=Z · Q=Z the form ¯ : V £ V ! K de¯nes a biadditive form 1 ¯~ : V £ V ! Q=Z; ¯~(v; w) := trace(¯(v; w)) p which is again non-degenerate by the non-degeneracy of the trace form. jV j To de¯ne the associated Cli®ord-Weil group C(½) we index a basis (evjv 2 V ) of C by the elements of V . Then C(½) · GLjV j(C) is the ¯nite complex matrix group ¤ C(½) = hmr; dÁ; he;ue;ve : r 2 R ; Á 2 R; e = ueve 2 SymId(R)i where mr : bv 7! brv; dÁ : bv 7! exp(2¼i¯~(v; Áv))bv and 1 he;u ;v : bv 7! exp(2¼i¯~(w; vev))b : e e jeV j1=2 w+(1¡e)v wX2eV 2.3 Symmetrized weight enumerators. Very often certain elements of V share the same property (for instance they have the same Ham- ming weight). Then one might be interested in the symmetrized weight enumerators of the codes rather than the complete weight enumerators. One way to obtain the ring spanned by these symmetrized weight enumerators is of course to ¯rst calculate generators of the ring of complete weight enumerators and then apply the appropriate symmetrization. Since the ring spanned by the complete weight enumerators might be rather large, it is very helpful to have shortcuts to this procedure. This is only possible, if the action of the associated Cli®ord-Weil group commutes with the symmetrization. De¯nition 2.1. Let G · Sym(V ) be a group permuting the elements of V and X0; : : : ; Xn denote N the G-orbits on V . Then the G-symmetrized weight enumerator sweG(C) of a code C · V is the homogeneous polynomial in C[x0; : : : ; xn] of degree N, n ai(c) sweG(C) := xi cX2C Yi=0 where ai(c) := jfj 2 f1; : : : ; Ng j cj 2 Xigj for 0 · i · n. The V -Hamming weight enumerator of C is N¡wV (c) wV (c) hweV (C) := x y 2 C[x; y] cX2C 3 N where the V -weight of c = (c1; : : : ; cN ) 2 V is wV (c) := jfi 2 f1; : : : ; Ng j ci 6= 0gj: There are certain symmetrizations that commute with the action of the associated Cli®ord-Weil group, for instance if one takes G to be a subgroup of the central unitary group of R as de¯ned and proven below. Usually the symmetization yielding the V -Hamming weight enumerators does not commute with C(½) and one may not expect that in general the V -Hamming weight enumerators of self-dual codes in a given representation generate the invariant ring of a ¯nite group (see the end of Section 7.2 and [7, Section 5.8] for examples). De¯nition 2.2. Let (R; J) be a ring with involution. Then the central unitary group ZU(R; J) := fg 2 Z(R) j ggJ = gJ g = 1g: Theorem 2.3. Let ½ := (V; ¯) be a ¯nite representation of the form ring R(R; J) and U · ZU(R; J). Then ½(U) := hmu j u 2 Ui is in the center of C(½). ¤ ¤ Proof. Clearly ½(U) · C(½) commutes with the generators mr for r 2 R since U is central in R . For Á 2 ©, u 2 U and v 2 V we have ¯(uv; Áuv) = ¯(uv; uÁv) = ¯(uJ uv; Áv) = ¯(v; Áv) so mu commutes with dÁ. To see that mu commutes with the last type he;ue;ve of generators of C(½) one has to note that ueV = eV since u is a central unit and that ¯(uw; veuv) = ¯(w; vev) for all v; w 2 V; u 2 U. ¤ Remark 2.4. The theorem uses that ff gg is surjective in our situation.
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