The generalized Grothendieck-Witt ring of nonsingular Hermitian forms with positioning map is the cartesian product of the ordinary Grothendieck-Witt ring of nonsingular Hermitian forms and a map ring. We study the generalized Grothendieck-Witt ring by computing the map ring

Ju, Xian Meng

Okayama University Digital Information Repository

Computation of the Grothendieck-Witt ring of nonsingular Hermitian forms with positioning map

Okayama University Scientific Achievement Repository

Journal of the Faculty of Environmental Science and Technology,Okayama UniversityVoLlO, No.I, pp.29-39, February 2005Computation of the Grothendieck-Witt ring ofnonsingular Hermitian forms with positioning mapXianMeng JU*(Received November 30, 2004)The generalized Grothendieck-Witt ring of nonsingular Hermitian forms with positioning mapis the cartesian product of the ordinary Grothendieck-Witt ring of nonsingular Hermitian formsand a map ring. We study the generalized Grothendieck-Witt ring by computing the map ring.2000 Mathematics Subject Classification: Primary lIE81, 19G12.keywords: Grothendieck Witt ring, orbit, GAP.1 IntroductionThroughout this paper, let G be a finite group, R a commutative ring with 1, and e a finite G-set.The ordinary Grothendieck-Witt ring GWo(R,G) in A. Dress [10] was generalized to GWo(R,G,e) in [5,p. 7], [13, p. 506], [16, p. 2356], [7, pp. 59-61], and the classical one is obtained as the case e = 0, namelyGWo(R,G) = GWo(R,G,0). Let 6 2 denote the symmetric group of degree 2 and T the generator of 6 2 .The cartesian product e x e is afforded the G x 6 2-action such that g(t,t') = (gt,gt') and T(t,t') = (t',t)for 9 E G and t, t' E e. Let Mapc(e x e, R) (resp. MapCx62 (e x e, R)) denote the set of all G-invariant(resp. G x 6 2-invariant) maps from e x e to R. This set MaPc(e x e, R) (resp. MaPCx62 (e x e, R))has a canonical ring structure and isomorphic to Map(G\(e x e), R) (resp. Map((G x 6 2)\(e x e), R)),where G\(e x e) (resp. (G x 6 2 )\(e x e)) is the G-orbit space (resp. G x 6 2-orbit space) of ex e. WedefineM(G,e,R) = MaPCx62(e x e,R).If R is a principal ideal domain then by [7] GWo(R, G, e) is isomorphic to the cartesian product ofGWo(R, G) and M(G, e, R) as rings. Let RO(G) denote the real representation ring of G.Theorem 1.1. The abelian group Z[t] 0 GWo(Z, G, e) is isomorphic to1 1Z[2] 0 RO(G) EEl M(G, e, Z[2])'The Z-rank of GW0 (Z, G, e) is the sum of the number of isomorphism classes of finite dimensional irre-ducible real G-modules and the Z-rank of M(G,e,Z). Moreover, the torsion subgroup of GWo(Z,G, e) isannihilated by 4.Theorem 1.2. Let e be a finite G-set isomorphic to i~l C~l G/Hi) (ai EN). Then M(G,e,R) is anR-free module of rankt 22:(ail(G x 6 2 )\(G/Hi x G/Hi)I+ ai ; ai IG\(G/Hi x G/Hi)l)i=l+ 2: aiaj!G\(G/Hi x G/Hj )!,1::;i<j9where I(G x 6 2 )\(G/ Hi x G / Hi)1 stands for the number ofG x 6 2 -orbits in G / Hi X G/Hi, and IG\(G/Hi xG / H j )I stands for the number of G -orbits in G / Hi X G / H j .'Department of Environmental Synthesis and Analysis, Graduate School of Natural Science and Technology, OkayamaUniversity, Okayama, 700-8530, Japan2930 ]. Fac. Environ. Sci. and Tech., Okayama Univ.10 (1) 2005As applications of these theorems, we compute the Z-rank of GWo(Z,G,0) in the cases G = A5 , thealternating group of degree 5, and SL(2, Z5), the special linear group of degree 2 with coefficients in Z5'The results are described in Sections 2 and 3,2 The rank of for GWo(Z, G, 0) for G = A5A complete set RAs of representatives for conjugacy classes of subgroups of G = A5 iswhere {e} is the unit group, Cn (n = 2, 3, 5) are the cyclic groups of order n, generated by the elementsa2 = (1,2)(3,4), a3 = (1,2,3), a5 = (1,2,3,4,5),D2n are the dihedral groups of order 2n obtained by D 2n = Na(Cn ), and A4 is the alternating group onfour letters {1,2,3,4}.Figure 1: Conjugacy Class Subgroups of A5Lemma 2.1. In the case G = A5 , RO(G) is a free Z-module of rank 5.Proof. This follows from [8, Theorem 38.1] and Frobenius-Shur's Theorem, since A 5 = SL(2, Z5)/C2 . 0Let 0 be a finite G-set. A G x 6 2-orbit X in 0 x 0 is called 6 2 -effective if X contains precisely twoG-orbits, and 6 2 -indecomposable if X contains only one G-orbit.The Program 5.6 for G = A5 given in Section 5 provides the next result which was theoretically computedby M. Kubo [12].Lemma 2.2. Let G = A 5 and RAs = {Hi Ii = 1, 2, ... ,9} the complete set above of representatives forconjugacy classes of subgroups of A 5 . Then the number of G -orbits in G / Hi X G / H j is given as in Table 1and the number of G x 6 2 -orbits in G / Hi X G / Hi is given as in Table 2.The notation in the tables reads that, for example in the casethere exist two 6 2-indecomposable G x 6 2-orbits of orbit-length 20, four 6 2-indecomposable G x 6 2-orbitsof orbit-length 60 and one 6 2-decomposable G x 6 2-orbit of orbit-length 120.The next theorem immediately follows from Theorem 2.2.XianMeng JU et al. / Computation of the Grotheadieck- Witt ring of nonsingular Hermitian forms with positioning map 31{e} C2 C3 D4 C5 D6 DIO A4 A5{e} 6060 3060 2060 1560 1260 1060 660 560 160C2 230 + 1460 1060 330 + 660 660 230 + 460 230 + 260 130 + 260 130C3 220 +660 660 460 120 + 360 260 220 + 160 120D4 315 + 360 360 330 + 160 330 115 + 160 115C5 212 + 260 260 112 + 160 160 112D6 110 + 130 + 160 230 120 + 130 110DIO 16 + 130 130 16A4 15 + ho 15A5 11Table 1: The number of G-orbits in G/ Hi X G/ Hj where G = A5Hi orbits{e} 1660X1 + 2260X2C2 230X1 + 660X1 + 46Ox2C3 220X1 + 460X1 + 16ox2D4 115X1 + 115X2 + 36Ox1C5 212X1 + 26OX1D6 I 1Ox1 + 130X1 + 16Ox1DIO 16X1 + 13Ox1A4 15X1 + 12OX1A5 11X1Table 2: The number of G x 6 2-orbits in G/ Hi X G/ Hi where G = A5Theorem 2.3. Let G and'RAs = {Hi I i = 1, 2, ... ,9} be as above. Then the numbers I(G x ( 2 )\(G/Hi xG/Hi)I and IG\(G/Hi x G/Hj)1 appearing in Theorem 1.2 are given as in Tables 3 and 4, respectively.Table 3: I(G x ( 2 )\(G/Hi x G/Hi)1 where G = A5{e} C2 C3 D4 C5 D6 DIO A4 A5{e} 60 30 20 15 12 10 6 5 1C2 16 10 9 6 6 4 3 1C3 8 6 4 4 2 3 1D4 6 3 4 3 2 1C5 4 2 2 1 1D6 3 2 2 1DIO 2 1 1A4 2 1A5 1Table 4: IG\(G/Hi x G/Hj)1 where G = A532 J. Fac. Environ. Sci. and Tech., Okayama Univ.l0 (1) 20053 The rank of for GWo(Z,G, 8) for G = 8L(2,Z5)We recall basic properties of 8L(2, Z5), the special linear group of degree 2 on Z5' A complete set'RSL (2,Zs) of representatives for conjugacy classes of subgroups of 8L(2, Z5) is given bywhere {e} is unit group, en (n = 2, 3, 4, 5, 6, 10) are the cyclic groups of order n, Q4m (n = 2, 3, 5) arethe quaternion groups of order 4m, namelyQ4m = (a,b Ibm = a2,b2m = e,a-1ba = b- 1 ),andFigure 2: Conjugacy Class 8ubgroups of 8L(2, Z5)Lemma 3.1. In the case G = 8L(2, Z5), RO(G) is a free Z-module of rank 9.Proof. This follows from [8, Theorem 38.1] and Frobenius-8hur's Theorem. oLemma 3.2. Let G = 8L(2, Z5) and 'RSL(2,Zs) = {Hi I i = 1, 2, ... , 12} the complete set above ofrepresentatives for conjugacy classes of subgroups of8L(2, Z5)' Then the number ofG-orbits in G / Hi xG / Hjis given as in Table 5 and the number of G x 62 -orbits in G / Hi X G / Hi is given as in Table 6.The next theorem immediately follows from Theorem 3.2.Theorem 3.3. Let G and'RsL (2,Zs) = {Hi I i = 1,2, ... , 12} be as above. Then the numbers I(G x( 2)\(G/Hi x G/Hi)I and /G\(G/Hi x G/Hj)1 appearing in Theorem 1.2 are given as in Tables 7 and 8,respectively.4 Proofs of Theorems 1.1 and 1.2The ordinary Grothendieck-Witt ring GWo(Z, G) is defined in [9, p. 742], and the Grothendieck-Wittring GWo(R,G,8) is defined in [5], [13], [16, §4] and [7, pp. 59-61].Proof of Theorem 1.1. If R is a principal ideal domain then by Theorem 1 of [7]GWo(R,G,8) ~ GWo(R,G) (fJ M(G,8,R).By definition, M(G, 8, R) = MapCx62(8 x 8, R) is a free R-module. Thus Theorem 1.1 follows from [9,Theorem 3]. 0XianMeng JU et al. / Computation of the Grothendieck- Witt ring of nonsingular Hermitian forms with positioning map 33{e} C2 C3 C4 C5 C6{e} 120120 60120 40120 30120 24120 20120C2 6060 20120 3060 12120 2060C3 4 40 + 12120 10120 8120 2 40 + 6 120C4 2 30 + 1460 6 120 1060C5 4 24 + 4 120 4 120C6 220 + 660Q8ClDQ12Q201r -1(A4)8L(2, Z5)Q8 ClD Q12 Q20 1r -1 (A4) 8L(2, Z5)15120 12120 10120 6 120 5 120 11201560 1260 1060 660 5 60 1 605 120 4 120 1 40 + 3 120 2 120 2 40 + 1120 1 403 30 + 660 660 2 30 + 4 60 2 30 + 2 60 1 30 + 2 60 1 303 120 224 + 2 120 2 120 b4 + 1120 1120 1 245 60 4 60 1 20 + 3 60 2 60 220 + 1 60 1 203 15 + 3 60 3 60 3 30 + 1 60 3 30 115 + 1 60 h52 12 + 2 60 260 112 + 1 60 1 60 112l lD + bo + 1 60 2 30 1 20 + 1 30 l lD1 6 + 1 30 1 30 1 61 5 + 1 20 1 511Table 5: The number of G-orbits in G/ Hi X G/ Hj where G = 8L(2, Z5)Hi orbits{e} 2120Xl + 59120x2C2 1660X1 + 226Ox2C3 2 40X1 + 1 4ox2 + 6120x2C4 2 30X1 + 660Xl + 4 6Ox2C5 224Xl + 124x2 + 2120x2C6 220Xl + 4 60X1 + 16ox2Q8 115Xl + 115X2 + 3 6OX1ClD 2 12X1 + 2 6OX1Q12 l lDx1 + 1 30X1 + 1 6Ox1Q20 1 6x1 + 1 3OX11r -1 (A4) 1 5X1 + bOXl8L(2, Z5) 11xlTable 6: The number of G x 6 2-orbits in G/ Hi X G / Hi where G = 8L(2, Z5)Table 7: I(G x ( 2 )\(G/Hi x G/Hi)1 where G = 8L(2,Z5)34 J. Fac. Environ. Sci. and Tech., Okayama Univ.lO (1) 2005{e} C2 C3 C4 C5 C6 Qs C10 Q12 Q20 7r -1 (A4 ) SL(2, Z5){e} 120 60 40 30 24 20 15 12 10 6 5 1C2 60 20 30 12 20 15 12 10 6 5 1C3 16 10 8 8 5 4 4 2 3 1C4 16 6 10 9 6 6 4 3 1C5 8 4 3 4 2 2 1 1C6 8 5 4 4 2 3 1Qs 6 3 4 3 2 1C10 4 2 2 1 1Q12 3 2 2 1Q20 2 1 17r '1(A4 ) 2 1SL(2, Z5) 1Table 8: IG\(GIHi x GIHj)1 where G = SL(2,Z5)Proof of Theorem 1.2. SinceMapax62((01 II O2 ) x (01 II O2 ),R)~ Mapax62(01 x 01,R) EEl Mapax62(02 x O2 , R) EEl Mapa (01 x O2 ,R),we obtaint tMapax62((.II 0 i ) x ell 0 i ),R).=1 .=1t~ EBMapax62(0 i x 0 i ,R) EEl EB Mapa(0 i x 0 j ,R).i=l l~i<j~tThe next Lemma follows from this.t aiLemma 4.1. 1f 0 ~ .II (.II G I Hi) (ai EN) then.=1 J=1t ai (a~-ai)/2M(G,0,R) ~~{~M(G,GIHi,R)EEl ~ Mapa(GIHi x GIHi,R)}EEl EB rEB; Mapa (GIHi x GIHj,R)}.1~i<j9 k=lLet Ra = {Hi 11 :S i :S t} be a complete set of representatives for conjugacy classes of subgroups of G.By the Lemma above, Mapa x62 (0 x 0) is determined by the numbers of elements in G \ (GI Hi X G I H j )and (G x ( 2 ) \ (GIHi x GIHi ), where Hi and H j run over Ra. 05 Appendix: Algorithms and ProgrammingsIn the Appendix, we give the algorithms and programmings for Lemma 2.2 and Lemma 3.2.Lemma 5.1. Let Hand K be subgroups of G. Then the following hold.(1) GIH x GIK has the G-orbit decompositionII G· (eH,gK).HgKEH\a/K(2) Each G-orbit G· (H, gH) is transposed to the G-orbit G . (H, g-1 H) by the generator T of 6 2 .XianMeng JU et al. / Computation of the Grothendieck-- Witt ring of nonsingular Hermitian forms with positioning map 35(3) A G-orbit G· (H,gH) is itself a G x 6 2-orbit if and only if HgH = Hg-l H.This lemma is well known but we give the proof for the reader's convenience.Proof. (1) Each G-orbit in G/H x G/K has a representative of the form (eH,gK). The points (eH,aK)and (eH, bK), where a and bEG, lie in a same G-orbit if and only if HaK = HbK. Thus we getG/H x G/K = II G· (eH,gK).HgKEH\G/K(2) The claim follows from r(eH, gK) = (gH, eK) and G . (eH, gH) = G . (eH, g-1 H).(3) Since (G x ( 2). (eH,gH) = G· (eH,gH) uG· (eH,g-IH),(G x ( 2 ), (eH,gH) =G· (eH,gH)holds if and only ifG· (eH,gH) = G· (eH,g-IH), in other words HgH = Hg- 1H.oProposition 5.2. One has the formulae:(1) IG\(G/H x G/K)I = IH\G/KI,(2) I(G x ( 2)\(G/H x G/H)I = I{HgH E H\G/H I HgH = Hg- 1H}II{HgH E H\G/H I HgH ¥- Hg-IH}I+ 2 .Proof. This immediately follows from Lemma 5.1. oWe use GAP (Groups, Algorithms, and Programming) to compute the number of orbits IG\(G/Hi xG/Hj)1 and I(G x (2)\(G/Hi x G/Hi)1 for G = A5 and SL(2,Z5) together with Proposition 5.2 (seeAlgorithm 5.3 and Program 5.4).Algorithm 5.3. Let G be a finite group, RG = {Hi I i = 1, 2, ... , t} a complete set of representatives forconjugacy classes of subgroups ofG. M(G) will be the matrix (IHi\G/Hjl) (namely (IG\(G/Hi xG/Hj)I)),and L(G x ( 2 ) will be the list of I(G x ( 2)\(G/Hi x G/Hi)l.Input: GOutput: M(G), L(G x ( 2 )RG := {Hi 11 ~ i ~ t}For 1 ~ i ~ t DoFor 1 ~ j ~ t DoM(G)ij:= IHi\G/Hj IEnd DoEnd DoFor 1 < i < t DoL(Gx ( 2)i:= I{HigHi E Hi\G/Hi I HigHi = Hig-1Hi}!I{HigHi E Hi\G/Hi I HigHi ¥- Hig- 1Hdl+ 2End DoWe compute L(G x ( 2 )i by using this algorithm with the function "GS2orbitnumber" in Program 5.4.Program 5.4.36 J. Fac. Environ. Sci. and Tech., Okayama Univ.lO (1) 2005G := SL( 2, Integers mod 5 );# Define the group SL(2,5).# G := AlternatingGroup( 5 );# Define the group A_5.CCS := ConjugacyClassesSubgroups( G );# the set of all conjugacy classes of element in G.CCSR := List( CCS, H -) Representative(H) );# the complete list [H_i] of representatives#for conjugacy classes of subgroups of G.N_G := List( CCSR, LC -) List(CCSR, RC -)Length( DoubleCosets( G, LC, RC ) ) ) );# the matrix of the number of double coset [H_i\G/H_j]#(namely G-orbits in (G/H_i * G/H_j)).N_GS2 := List( CCSR, ccsr -) GS20rbitnumber ( G, ccsr ) );# the list of the number of (G*S_2)-orbits in#(G/H_i * G/H_i).# ccsr: the element of the complete set of representatives#for conjugacy classes of subgroups of G.# OUTPUT: the number of (G*S_2)-orbits in (G/H_i * G/H_i).GS20rbitnumber := function( G, ccsr )local g; m, Temp, DC;Temp := [];m := 0;for g in G doDC := DoubleCoset( ccsr, g, ccsr );if not ( DC in Temp ) thenAdd( Temp, DC );if ( g~-l in DC )then m := m + 1;else m := m + 1/2;fi;fi;od;return m;end;But use Program 5.4, we are able to only compute the number of orbit space. For more information (forexample orbit-length), the next algorithm is useful.Algorithm 5.5. Let G and 'Rc = {Hi I i = 1,2, ... , t} be as above. M(G) is the upper triangularmatrix where (i,j)-entry is the set of G-orbits contained in G/Hi x G/Hj , £(G x ( 2 ) is the list of the setof G x 6 2-orbits contained in G / Hi X G / Hi'Input: GOutput: M(G), £(G x ( 2 )'Rc := {Hi 11 ~ i ~ t}For 1 ~ i ~ t DoFor i ~ j ~ t DoS :=0For g E G DoT:= the G-orbit of (eHi,gHj )If T ~ S Then S := S U {T}XianMeng JU et al. / Computation of the Grothendieck- Witt ring of nonsingular Hermitian forms with positioning map 37End DoM(G)ij := SEnd DoEnd DoFor 1 ::; i ::; t DoS :=0For {(Xj,Yj)} E M(G)ii DoT:= {(Xj,Yj)} U {(Yj,Xj)}If T ~ S Then S := S U {T}End Do£(G x ( 2 )i := SEnd DoIn this Algorithm, we need check whether T ~ S for provided S = {Si} and T, namely check whether Tcoincides with an element of S. Since T and Si are G-orbits, Tn Si ¥- 0 ¢:} T = Si- We use this propertyin programming the algorithm with GAP.Program 5.6.G := SL( 2, Integers mod 5 );# Define the group SL(2,5).# G := AlternatingGroup( 5 );CCS := ConjugacyClassesSubgroups( G );# the set of all conjugacy classes of element in G.CCSR := List( CCS, H -> Representative( H ) );# the complete list [H_i] of representatives#for conjugacy classes of subgroups of G.CH := List( CCSR, ccsr -> RightCosets( G, ccsr ) );# the list of G/H_i.CHsize := Length( CH );# the length of CH.G_Orbit := List( [1 .. CHsize], k -> allorbit( G, CH, k ) );# the upper triangular matrix of G-orbits in (G/H_i * G/H_j).G_OS := List( G_Orbit, A -> List( A, B -> List( B, C ->Length(C) ) ) );# the upper triangular matrix of G-orbit-length in#(G/H_i * G/H_j).NG_OS := List( G_OS, Gos -> List( Gos, gos -> Length(gos) ) );# the upper triangular matrix of the number of G-orbits in# (G/H_i * G/H_j).GS2_0rbit:=List([1 .. CHsize],i->S2Gorbit(G_Orbit[i] [i]));# the list of (G*S_2)-orbits in (G/H_i * G/H_i).GS2_0S := List( GS2_0rbit, Gs -> List( Gs, gs ->Length (gs) ) );# the list of (G*S_2)-orbit-length in (G/H_i * G/H_i).NGS2_0S :=List(GS2_0S,s->Length(s));# the list of the number of (G*S_2)-orbits in (G/H_i * G/H_i).# ot: an orbit,# OT: a list of orbits.# OUTPUT: if ot in OT, then return true.orbitin := function( ot, OT )local r;for r in OT do38 ]. Fac. Environ. Sci. and Tech., Okayama Univ.l0 (1) 2005if ot[l] in r thenreturn true;fi;od;return false;end;# cosetl: a coset of left part in (G/H_i * G/H_j),# cosetr: a coset of right part in (G/H_i * G/H_j).# OUTPUT: G-orbit G(aH, bK).Gorbit := function( G, cosetl, cosetr )local g, Temp, Result;Result := [];for g in G doTemp := [ cosetl*g, cosetr*g];if not (Temp in Result) thenAdd( Result, Temp );it;od;return Result;end;# GOH: G/H where H is a subgroup of G,# GOK: G/K where K is a subgroup of G.# OUTPUT: the list of G-orbit in (G/H * G/K).Gorbitspace := function( G, GOH, GOK )local gK, Result, Temp;Result := [];for gK in GOK doTemp := Gorbit( G, GOH[l] , gK );if not orbitin( Temp, Result ) thenAdd( ResuJt, Temp );fi;od;return Result;end;# i: i-th row.# OUTPUT: a row of the upper triangular matrix of G-orbit#space G\{G/H_i * G/H_j).allorbit := function( G, CH, i )local j, Result;Result := [];for j in [i .. Length{ CH )] doResult[j] := Gorbitspace{ G, CH[i] , CH[j] );od;return Result;end;# GODC: G-orbit in (G/H_i * G/H_i).# OUTPUT: (G*S_2)-orbits in (G/H_i * G/H_i).S2Gorbit:=function{GODC)XianMeng et al. / Computation of the Grothendieck- Witt ring of nonsingular Hermitian forms with positioning maplocal i, s, Result, Temp;s:=Length( GODC );Result: =[] ;for i in [l .. s] doTemp := List ( [1.. s], j -> [ GODC [i] [j] [2] ,GODC [i] [j] [1] ] );if not orbitin( Temp, Result ) thenAdd( Result, GODC[i] );fi;od;return Result;end;References[1] The GAP Group, GAP - Groups, Algorithms, and Programming, version 4.3., http://www.gap-system.org/, 2002.[2] A. Bak, K-Theory of Forms, Annals of Mathematics Studies 98, Princeton Univ. Press, Princeton,1981.[3] A. Bak, Induction for finite groups revisited, J. Pure and Applied Alg. 104 (1995), 235-241.[4] A. Bak and M. Morimoto, Equivariant surgery and applications, in: Proceedings of Conference onTopology in Hawaii 1990 (ed. K. H. 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Computation of the Grothendieck-Witt ring of nonsingular Hermitian forms with positioning map

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