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[]\T1/ptm/m/n/10 O. Broche Cristo, A. Her-man, A. Kono-valov, A. Olivieri, G. O
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\T1/ptm/m/n/10 Van Gelder. \T1/ptm/m/it/10 Wed-derga --- Wed-der-burn De-com-po
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\T1/ptm/m/n/10 We all are very grate-ful to Steve Lin-ton for com-mu-ni-cat-ing
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\T1/ptm/m/n/10 care-ful test-ing \T1/phv/m/n/10 Wed-derga \T1/ptm/m/n/10 and us
e-ful sug-ges-tions. Also we ac-knowl-edge very much the mem-bers
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\T1/ptm/m/n/10 of the \T1/phv/m/n/10 GAP \T1/ptm/m/n/10 team: Thomas Breuer, Al
exan-der Hulpke, Frank Lübeck and many other col-leagues
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\T1/ptm/m/n/10 for help-ful com-ments and ad-vise. We would like also to thank 
Thomas Breuer for the code of
 []


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 [4]
Chapter 1.
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[]\T1/ptm/m/n/10.95 To use the \T1/phv/m/n/10.95 Wed-derga \T1/ptm/m/n/10.95 on
-line help it is nec-es-sary to in-stall the \T1/phv/m/n/10.95 GAP\T1/ptm/m/n/1
0.95 4 pack-age \T1/phv/m/n/10.95 GAP-
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[]\T1/phv/m/n/10.95 Wedderga \T1/ptm/m/n/10.95 is dis-tributed in stan-dard for
-mats (\T1/cmtt/m/n/10.95 tar.gz\T1/ptm/m/n/10.95 , \T1/cmtt/m/n/10.95 tar.bz2\
T1/ptm/m/n/10.95 , \T1/cmtt/m/n/10.95 -win.zip\T1/ptm/m/n/10.95 )
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\T1/ptm/m/n/10.95 and can be ob-tained from [][]\T1/cmtt/m/n/10.95 http://www.u
m.es/adelrio/wedderga.htm[][]\T1/ptm/m/n/10.95 , its
 []


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\T1/ptm/m/n/10.95 mir-ror [][]\T1/cmtt/m/n/10.95 http://www.cs.st-andrews.ac.uk
/~alexk/wedderga/[][] \T1/ptm/m/n/10.95 or the page
 []


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[][]\T1/cmtt/m/n/10.95 http://www.gap-system.org/Packages/wedderga.html[][] \T1
/ptm/m/n/10.95 at the \T1/phv/m/n/10.95 GAP \T1/ptm/m/n/10.95 web site. To in-s
tall
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\T1/ptm/m/n/10.95 The main func-tions of \T1/phv/m/n/10.95 Wed-derga \T1/ptm/m/
n/10.95 are \T1/cmtt/m/n/10.95 WedderburnDecomposition \T1/ptm/m/n/10.95 ([][]2
.1.1[][]) and
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[6]
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\T1/ptm/m/n/10.95 Instead of \T1/cmtt/m/n/10.95 WedderburnDecomposition \T1/ptm
/m/n/10.95 ([][]2.1.1[][]), that re-turns a list of \T1/phv/m/n/10.95 GAP \T1/p
tm/m/n/10.95 ob-jects,
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\T1/cmtt/m/n/10.95 WedderburnDecompositionInfo \T1/ptm/m/n/10.95 ([][]2.1.2[][]
) re-turns the nu-mer-i-cal de-scrip-tion of these ob-jects.
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[7]
Chapter 2.
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[8

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[]\T1/ptm/m/n/10.95 The next ex-am-ple shows the out-put of \T1/cmtt/m/n/10.95 
WedderburnDecompositionInfo \T1/ptm/m/n/10.95 for $\U/msb/m/n/10.95 Q\OML/ztmcm
/m/it/10.95 G$ \T1/ptm/m/n/10.95 and
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\U/msb/m/n/10.95 Q\OT1/ztmcm/m/n/10.95 (\OML/ztmcm/m/it/10.95 ^^X[]\OT1/ztmcm/m
/n/10.95 )\OML/ztmcm/m/it/10.95 G$\T1/ptm/m/n/10.95 , where $\OML/ztmcm/m/it/10
.95 G \OT1/ztmcm/m/n/10.95 = \OML/ztmcm/m/it/10.95 SmallGroup\OT1/ztmcm/m/n/10.
95 (48\OML/ztmcm/m/it/10.95 ; \OT1/ztmcm/m/n/10.95 15)$\T1/ptm/m/n/10.95 . The 
user can com-pare it with the out-put of
 []

[11]
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\T1/ptm/m/n/10.95 are not in-te-gers and so the cor-re-spod-ing \T1/ptm/m/it/10
.95 Wed-der-burn com-po-nents \T1/ptm/m/n/10.95 ([][]9.3[][]) are given as "fra
c-
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[12] [13] [14] [15]
Chapter 3.

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[16

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[17]
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[18]
Chapter 4.

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[19

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\T1/ptm/m/n/10.95 The out-put of \T1/cmtt/m/n/10.95 PrimitiveCentralIdempotents
BySP \T1/ptm/m/n/10.95 con-tains the out-put of
 []

[21]
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[]\T1/ptm/m/n/10.95 The out-put is a com-plete set of or-thog-o-nal prim-i-tive
 idem-po-tents of the sim-ple al-ge-bra
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[22]
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[]\T1/ptm/m/n/10.95 The out-put is a com-plete set of or-thog-o-nal prim-i-tive
 idem-po-tents of the sim-ple al-ge-bra
 []

[23]
Chapter 5.
[24

] [25] [26] [27] [28] [29]
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[30]
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[]\T1/ptm/m/n/10.95 Similarly to magma rings, one can ob-tain the list of co-ef
-fi-cients and el-e-ments with
 []

[31]
Chapter 6.

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[32

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[33] [34] [35]
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[36] [37]
Chapter 7.
[38

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[41] [42] [43] [44]
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[]\T1/ptm/m/n/10.95 This func-tion is an al-ter-na-tive to \T1/cmtt/m/n/10.95 S
impleAlgebraByCharacterInfo(GroupRing(F,G),
 []

[45] [46] [47] [48]
Underfull \hbox (badness 10000) in paragraph at lines 2778--2784
[]\T1/cmtt/m/n/10.95 LocalIndicesOfRationalQuaternionAlgebra \T1/ptm/m/n/10.95 
first fac-tors the al-ge-bra as a ten-
 []


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\T1/ptm/m/n/10.95 sor prod-uct of ra-tio-nal quater-nion al-ge-bras, ob-tain-in
g suit-able pairs \T1/cmtt/m/n/10.95 a \T1/ptm/m/n/10.95 and \T1/cmtt/m/n/10.95
 b \T1/ptm/m/n/10.95 to which
 []


Underfull \hbox (badness 10000) in paragraph at lines 2822--2823
[]\T1/ptm/m/n/10.95 The in-put $\OML/ztmcm/m/it/10.95 A$ \T1/ptm/m/n/10.95 must
 be a quater-nion al-ge-bra over the ra-tio-nals, as out-put from
 []


Underfull \hbox (badness 10000) in paragraph at lines 2824--2825
[]\T1/ptm/m/n/10.95 The func-tion cal-cu-lates the ra-tio-nal Schur in-dex of t
he al-ge-bra us-ing
 []

[49]
Underfull \hbox (badness 5133) in paragraph at lines 2877--2879
[]\T1/ptm/m/n/10.95 The in-put must be list rep-re-sent-ing a cy-clo-tomic al-g
e-bra of length 5 whose Ga-
 []


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\T1/ptm/m/n/10.95 lois group has \T1/cmtt/m/n/10.95 2 \T1/ptm/m/n/10.95 gen-er-
a-tors. This is rep-re-sented in \T1/phv/m/n/10.95 Wed-derga \T1/ptm/m/n/10.95 
as a list of the form
 []

[50]
Underfull \hbox (badness 4699) in paragraph at lines 2912--2917
[]\T1/ptm/m/n/10.95 This func-tion con-verts a cyclic al-ge-bra given by a list
 \T1/cmtt/m/n/10.95 [F,F(E(n)),[E(n)^c]] \T1/ptm/m/n/10.95 to
 []


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\T1/ptm/m/n/10.95 an iso-mor-phic cyclic cy-clo-tomic al-ge-bra rep-re-sented a
s the list \T1/cmtt/m/n/10.95 [1,F,n,[a,b,c]]\T1/ptm/m/n/10.95 . []$\OML/ztmcm/
m/it/10.95 .$
 []


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[]\T1/ptm/m/n/10.95 The in-put must be a quater-nion al-ge-bra whose cen-ter is
 an abelian num-ber field $\OML/ztmcm/m/it/10.95 F$\T1/ptm/m/n/10.95 ,
 []


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\T1/ptm/m/n/10.95 pre-sented as in the out-put from \T1/cmtt/m/n/10.95 Quaterni
onAlgebra( F, a, b )\T1/ptm/m/n/10.95 , with $\OML/ztmcm/m/it/10.95 a$\T1/ptm/m
/n/10.95 , $\OML/ztmcm/m/it/10.95 b$ \T1/ptm/m/n/10.95 in $\OML/ztmcm/m/it/10.9
5 F$\T1/ptm/m/n/10.95 . It
 []


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\T1/ptm/m/n/10.95 re-turns a list \T1/cmtt/m/n/10.95 [F,F(ER(a)),[b]] \T1/ptm/m
/n/10.95 rep-re-sent-ing the cyclic al-ge-bra iso-mor-phic to $\OML/ztmcm/m/it/
10.95 A$\T1/ptm/m/n/10.95 . []$\OML/ztmcm/m/it/10.95 .$
 []

[51] [52]
Chapter 8.
[53

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Chapter 9.
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/m/n/10.95 -\T1/cmtt/m/n/10.95 Alg\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 To\T1/p
tm/m/n/10.95 -\T1/cmtt/m/n/10.95 Cyclic\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Cy
clotomicAlg\T1/ptm/m/n/10.95 ,
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[]\T1/cmtt/m/n/10.95 Convert\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Cyclic\T1/ptm
/m/n/10.95 -\T1/cmtt/m/n/10.95 Cyclotomic\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 
Alg\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 To\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.9
5 CyclicAlg\T1/ptm/m/n/10.95 ,
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[]\T1/cmtt/m/n/10.95 Defining\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Character\T1
/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Of\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Cycl
otomic\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Algebra\T1/ptm/m/n/10.95 ,
 []


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[]\T1/cmtt/m/n/10.95 IsComplete\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Set\T1/ptm
/m/n/10.95 -\T1/cmtt/m/n/10.95 Of\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Orthogon
al\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Idempotents\T1/ptm/m/n/10.95 ,
 []


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[]\T1/cmtt/m/n/10.95 IsRational\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Quaternion
\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Algebra\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10
.95 A\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Division\T1/ptm/m/n/10.95 -
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[]\T1/cmtt/m/n/10.95 IsSemisimple\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Zero\T1/
ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Characteristic\T1/ptm/m/n/10.95 -\T1/cmtt/m/n
/10.95 Group\T1/ptm/m/n/10.95 -
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[]\T1/cmtt/m/n/10.95 Local\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Indices\T1/ptm/
m/n/10.95 -\T1/cmtt/m/n/10.95 Of\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Cyclic\T1
/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Cyclotomic\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10
.95 Algebra\T1/ptm/m/n/10.95 ,
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[]\T1/cmtt/m/n/10.95 Local\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Indices\T1/ptm/
m/n/10.95 -\T1/cmtt/m/n/10.95 Of\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Rational\
T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Quaternion\T1/ptm/m/n/10.95 -
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[]\T1/cmtt/m/n/10.95 Local\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Indices\T1/ptm/
m/n/10.95 -\T1/cmtt/m/n/10.95 Of\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Tensor\T1
/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Product\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95
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[]\T1/cmtt/m/n/10.95 Primitive\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Central\T1/
ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Idempotents\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10
.95 By\T1/ptm/m/n/10.95 -
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[]\T1/cmtt/m/n/10.95 Primitive\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Central\T1/
ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Idempotents\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10
.95 By\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 StrongSP\T1/ptm/m/n/10.95 ,
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[]\T1/cmtt/m/n/10.95 Simple\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Component\T1/p
tm/m/n/10.95 -\T1/cmtt/m/n/10.95 By\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Charac
ter\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 As\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.9
5 S\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 C\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95
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[]\T1/cmtt/m/n/10.95 Simple\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Component\T1/p
tm/m/n/10.95 -\T1/cmtt/m/n/10.95 Of\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Group\
T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Ring\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 
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[]\T1/cmtt/m/n/10.95 Wedderburn\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Decomposit
ion\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 With\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10
.95 Div\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Alg\T1/ptm/m/n/10.95 -
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