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<p><a id="X87273420791F220E" name="X87273420791F220E"></a></p>
<div class="ChapSects"><a href="chap2.html#X87273420791F220E">2 <span class="Heading">Wedderburn decomposition</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X87273420791F220E">2.1 <span class="Heading">Wedderburn decomposition</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F1779ED8777F3E7">2.1-1 WedderburnDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8710F98A85F0DD29">2.1-2 WedderburnDecompositionInfo</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7D06959F7D444C55">2.2 <span class="Heading">Simple quotients</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8349114C83161C2D">2.2-1 SimpleAlgebraByCharacter</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X876FD2367E64462D">2.2-2 SimpleAlgebraByCharacterInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X812D667D7D913EB5">2.2-3 SimpleAlgebraByStrongSP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X858152C882129A0B">2.2-4 SimpleAlgebraByStrongSPInfo</a></span>
</div>
</div>
<h3>2 <span class="Heading">Wedderburn decomposition</span></h3>
<p><a id="X87273420791F220E" name="X87273420791F220E"></a></p>
<h4>2.1 <span class="Heading">Wedderburn decomposition</span></h4>
<p><a id="X7F1779ED8777F3E7" name="X7F1779ED8777F3E7"></a></p>
<h5>2.1-1 WedderburnDecomposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> WedderburnDecomposition</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>A list of simple algebras.</p>
<p>The input <var class="Arg">FG</var> should be a group algebra of a finite group G over the field F, where F is either an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals) or a finite field of characteristic coprime with the order of G.</p>
<p>The function returns the list of all <em>Wedderburn components</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of the group algebra <var class="Arg">FG</var>. If F is an abelian number field then each Wedderburn component is given as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>). If F is a finite field then the Wedderburn components are given as matrix algebras over finite fields.</p>
<table class="example">
<tr><td><pre>
gap> WedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) );
[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ),
( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ]
gap> WedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) );
[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),
<crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2> ]
gap> WedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) );
[ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ),
<crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40,
[ 1, 31 ]), CF(40) ) of a group of size 2> ]
</pre></td></tr></table>
<p>The previous examples show that if D_16 denotes the dihedral group of order 16 then the <em>Wedderburn decomposition</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of F_5 D_16, ℚ D_16 and ℚ (xi_5) D_16 are respectively</p>
<p class="pcenter">
\mathbb F_5 D_{16} = 4 \mathbb F_5 \oplus M_2( \mathbb F_5 ) \oplus M_2( \mathbb F_{25} ),
</p>
<p class="pcenter">
ℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus (K(\xi_8)/K,t),
</p>
<p>and</p>
<p class="pcenter">
ℚ (\xi_5) D_{16} = 4 ℚ (\xi_5) \oplus M_2( ℚ (\xi_5) ) \oplus (F(\xi_{40})/F,t),
</p>
<p>where (K(xi_8)/K,t) is a <em>cyclotomic algebra</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>) with the centre K=NF(8,[ 1, 7 ])= ℚ (sqrt2), (F(xi_40)/F,t) = ℚ (sqrt2,xi_5) is a cyclotomic algebra with centre F=NF(40,[ 1, 31 ]) and xi_n denotes a n-th root of unity.</p>
<p>Two more examples:</p>
<table class="example">
<tr><td><pre>
gap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );
[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),
<crossed product with center Rationals over CF(3) of a group of size 2>,
( CF(3)^[ 2, 2 ] ), <crossed product with center Rationals over CF(
3) of a group of size 2>, <crossed product with center NF(8,
[ 1, 7 ]) over AsField( NF(8,[ 1, 7 ]), CF(8) ) of a group of size 2>,
<crossed product with center Rationals over CF(12) of a group of size 4> ]
gap> WedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) );
[ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ),
( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ),
<crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24,
[ 1, 7 ]), CF(24) ) of a group of size 2>,
( <crossed product with center CF(3) over AsField( CF(3), CF(
12) ) of a group of size 2>^[ 2, 2 ] ) ]
</pre></td></tr></table>
<p>In some cases, in characteristic zero, some entries of the output of <code class="func">WedderburnDecomposition</code> do not provide full matrix algebras over a <em>cyclotomic algebra</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>), but "fractional matrix algebras". That entry is not an algebra that can be used as a <strong class="pkg">GAP</strong> object. Instead it is a pair formed by a rational giving the "size" of the matrices and a crossed product. See <a href="chap7.html#X87273420791F220E"><b>7.3</b></a> for a theoretical explanation of this phenomenon. In this case a warning message is displayed.</p>
<table class="example">
<tr><td><pre>
gap> QG:=GroupRing(Rationals,SmallGroup(240,89));
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecomposition(QG);
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!
[ Rationals, Rationals, <crossed product with center Rationals over CF(
5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ),
( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ),
<crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12,
[ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>,
[ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ]
</pre></td></tr></table>
<p><a id="X8710F98A85F0DD29" name="X8710F98A85F0DD29"></a></p>
<h5>2.1-2 WedderburnDecompositionInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> WedderburnDecompositionInfo</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>A list with each entry a numerical description of a <em>cyclotomic algebra</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>).</p>
<p>The input <var class="Arg">FG</var> should be a group algebra of a finite group G over the field F, where F is either an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals) or a finite field of characteristic coprime to the order of G.</p>
<p>This function is a numerical counterpart of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>).</p>
<p>It returns a list formed by lists of lengths 2, 4 or 5.</p>
<p>The lists of length 2 are of the form</p>
<p class="pcenter">
[n,F],
</p>
<p>where n is a positive integer and F is a field. It represents the nx n matrix algebra M_n(F) over the field F.</p>
<p>The lists of length 4 are of the form</p>
<p class="pcenter">
[n,F,k,[d,\alpha,\beta]],
</p>
<p>where F is a field and n,k,d,alpha,beta are non-negative integers, satisfying the conditions mentioned in Section <a href="chap7.html#X84A142407B7565E0"><b>7.12</b></a>. It represents the nx n matrix algebra M_n(A) over the cyclic algebra</p>
<p class="pcenter">
A=F(\xi_k)[u | \xi_k^u = \xi_k^{\alpha}, u^d = \xi_k^{\beta}],
</p>
<p>where xi_k is a primitive k-th root of unity.</p>
<p>The lists of length 5 are of the form</p>
<p class="pcenter">
[n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ],
</p>
<p>where F is a field and n,k,d_i,alpha_i,beta_i,gamma_i,j are non-negative integers. It represents the nx n matrix algebra M_n(A) over the <em>cyclotomic algebra</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>)</p>
<p class="pcenter">
A = F(\xi_k)[g_1,\ldots,g_m \mid
\xi_k^{g_i} = \xi_k^{\alpha_i}, g_i^{d_i}=\xi_k^{\beta_i},
g_jg_i=\xi_k^{\gamma_{ij}} g_i g_j],
</p>
<p>where xi_k is a primitive k-th root of unity (see <a href="chap7.html#X84A142407B7565E0"><b>7.12</b></a>).</p>
<table class="example">
<tr><td><pre>
gap> WedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) );
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ] ]
gap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) );
[ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ],
[ 1, NF(40,[ 1, 31 ]), 8, [ 2, 7, 0 ] ] ]
</pre></td></tr></table>
<p>The interpretation of the previous example gives rise to the following <em>Wedderburn decompositions</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>), where D_16 is the dihedral group of order 16 and xi_5 is a primitive 5-th root of unity.</p>
<p class="pcenter">
ℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus M_2( ℚ (\sqrt{2})).
</p>
<p class="pcenter">
ℚ (\xi_5) D_{16} = 4 ℚ (\xi_5) \oplus
M_2( ℚ (\xi_5)) \oplus
M_2( ℚ (\xi_5,\sqrt{2})).
</p>
<table class="example">
<tr><td><pre>
gap> F:=FreeGroup("a","b");;a:=F.1;;b:=F.2;;rel:=[a^8,a^4*b^2,b^-1*a*b*a];;
gap> Q16:=F/rel;; QQ16:=GroupRing( Rationals, Q16 );;
gap> QS4:=GroupRing( Rationals, SymmetricGroup(4) );;
gap> WedderburnDecomposition(QQ16);
[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),
<crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2> ]
gap> WedderburnDecomposition( QS4 );
[ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ),
<crossed product with center Rationals over CF(3) of a group of size 2> ]
gap> WedderburnDecompositionInfo(QQ16);
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ] ]
gap> WedderburnDecompositionInfo(QS4);
[ [ 1, Rationals ], [ 1, Rationals ], [ 3, Rationals ], [ 3, Rationals ],
[ 1, Rationals, 3, [ 2, 2, 0 ] ] ]
</pre></td></tr></table>
<p>In the previous example we computed the Wedderburn decomposition of the rational group algebra ℚ Q_16 of the quaternion group of order 16 and the rational group algebra ℚ S_4 of the symmetric group on four letters. For the two group algebras we used both <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) and <code class="func">WedderburnDecompositionInfo</code>.</p>
<p>The output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) shows that</p>
<p class="pcenter">
ℚ Q_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus A,
</p>
<p class="pcenter">
ℚ S_{4} = 2 ℚ \oplus 2 M_3( ℚ ) \oplus B,
</p>
<p>where A and B are <em>crossed products</em> (<a href="chap7.html#X7FB21779832CE1CB"><b>7.6</b></a>) with coefficients in the cyclotomic fields ℚ (xi_8) and ℚ (xi_3) respectively. This output can be used as a <strong class="pkg">GAP</strong> object, but it does not give clear information on the structure of the algebras A and B.</p>
<p>The numerical information displayed by <code class="func">WedderburnDecompositionInfo</code> means that</p>
<p class="pcenter">
A = ℚ (\xi|\xi^8=1)[g | \xi^g = \xi^7 = \xi^{-1}, g^2 = \xi^4 = -1],
</p>
<p class="pcenter">
B = ℚ (\xi|\xi^3=1)[g | \xi^g = \xi^2 = \xi^{-1}, g^2 = 1].
</p>
<p>Both A and B are quaternion algebras over its centre which is ℚ (xi+xi^-1) and the former is equal to ℚ (sqrt2) and ℚ respectively.</p>
<p>In B, one has (g+1)(g-1)=0, while g is neither 1 nor -1. This shows that B=M_2( ℚ ). However the relation g^2=-1 in A shows that</p>
<p class="pcenter">
A=ℚ (\sqrt{2})[i,g|i^2=g^2=-1,ig=-gi]
</p>
<p>and so A is a division algebra with centre ℚ (sqrt2), which is a subalgebra of the algebra of Hamiltonian quaternions. This could be deduced also using well known methods on cyclic algebras (see e.g. <a href="chapBib.html#biBR">[Rei03]</a>).</p>
<p>The next example shows the output of <code class="code">WedderburnDecompositionInfo</code> for ℚ G and ℚ (xi_3) G, where G=SmallGroup(48,15). The user can compare it with the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) for the same group in the previous section. Notice that the last entry of the <em>Wedderburn decomposition</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of ℚ G is not given as a matrix algebra of a cyclic algebra. However, the corresponding entry of ℚ (xi_3) G is a matrix algebra of a cyclic algebra.</p>
<table class="example">
<tr><td><pre>
gap> WedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) );
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
[ 2, Rationals ], [ 1, Rationals, 3, [ 2, 2, 0 ] ], [ 2, CF(3) ],
[ 1, Rationals, 6, [ 2, 5, 0 ] ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ],
[ 1, Rationals, 12, [ [ 2, 5, 9 ], [ 2, 7, 0 ] ], [ [ 9 ] ] ] ]
gap> WedderburnDecompositionInfo( GroupRing( CF(3), SmallGroup(48,15) ) );
[ [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 2, CF(3) ],
[ 2, CF(3), 3, [ 1, 1, 0 ] ], [ 2, CF(3) ], [ 2, CF(3) ],
[ 2, CF(3), 6, [ 1, 1, 0 ] ], [ 1, NF(24,[ 1, 7 ]), 8, [ 2, 7, 0 ] ],
[ 2, CF(3), 12, [ 2, 7, 0 ] ] ]
</pre></td></tr></table>
<p>In some cases some of the first entries of the output of <code class="func">WedderburnDecompositionInfo</code> are not integers and so the correspoding <em>Wedderburn components</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) are given as "fractional matrix algebras" of <em>cyclotomic algebras</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>). See <a href="chap7.html#X87273420791F220E"><b>7.3</b></a> for a theoretical explanation of this phenomenon. In that case a warning message will be displayed during the first call of <code class="code">WedderburnDecompositionInfo</code>.</p>
<table class="example">
<tr><td><pre>
gap> QG:=GroupRing(Rationals,SmallGroup(240,89));
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecompositionInfo(QG);
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ],
[ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ],
[ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ],
[ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ]
</pre></td></tr></table>
<p>The interpretation of the output in the previous example gives rise to the following <em>Wedderburn decomposition</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of ℚ G for G the small group [240,89]:</p>
<p class="pcenter">
ℚ G = 2 ℚ \oplus 2 M_4( ℚ ) \oplus
2 M_5( ℚ ) \oplus M_6( ℚ ) \oplus A \oplus B \oplus C
</p>
<p>where</p>
<p class="pcenter">
A = ℚ (\xi_{10})[u|\xi_{10}^u = \xi_{10}^3, u^4 = -1],
</p>
<p>B is an algebra of degree (4*2 )/2 = 4 which is <em>Brauer equivalent</em> (<a href="chap7.html#X7A24D5407F72C633"><b>7.5</b></a>) to</p>
<p class="pcenter">
B_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^{13},
u^4 = \xi_{60}^5,
\xi_{60}^v = \xi_{60}^{11}, v^2 = 1, vu=uv],
</p>
<p>and C is an algebra of degree (4*2)*3/4 = 6 which is <em>Brauer equivalent</em> (<a href="chap7.html#X7A24D5407F72C633"><b>7.5</b></a>) to</p>
<p class="pcenter">
C_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^7,
u^4 = \xi_{60}^5,
\xi_{60}^v = \xi_{60}^{31}, v^2 = 1, vu=uv].
</p>
<p>The precise description of B and C requires the usage of "ad hoc" arguments.</p>
<p><a id="X7D06959F7D444C55" name="X7D06959F7D444C55"></a></p>
<h4>2.2 <span class="Heading">Simple quotients</span></h4>
<p><a id="X8349114C83161C2D" name="X8349114C83161C2D"></a></p>
<h5>2.2-1 SimpleAlgebraByCharacter</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByCharacter</code>( <var class="Arg">FG, chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>A simple algebra.</p>
<p>The first input <var class="Arg">FG</var> should be a <em>semisimple group algebra</em> (<a href="chap7.html#X7FDD93FB79ADCC91"><b>7.2</b></a>) over a finite group G and the second input should be an irreducible character of G.</p>
<p>The output is a matrix algebra of a <em>cyclotomic algebras</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>) which is isomorphic to the unique <em>Wedderburn component</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) A of <var class="Arg">FG</var> such that chi(A)ne 0.</p>
<table class="example">
<tr><td><pre>
gap> A5 := AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> SimpleAlgebraByCharacter( GroupRing( Rationals , A5 ) , Irr( A5 ) [3] );
( NF(5,[ 1, 4 ])^[ 3, 3 ] )
gap> SimpleAlgebraByCharacter( GroupRing( GF(7) , A5 ) , Irr( A5 ) [3] );
( GF(7^2)^[ 3, 3 ] )
gap> G:=SmallGroup(128,100);
<pc group of size 128 with 7 generators>
gap> SimpleAlgebraByCharacter( GroupRing( Rationals , G ) , Irr(G)[19] );
<crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[ 1, 3 ]), CF(
8) ) of a group of size 2>
</pre></td></tr></table>
<p><a id="X876FD2367E64462D" name="X876FD2367E64462D"></a></p>
<h5>2.2-2 SimpleAlgebraByCharacterInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByCharacterInfo</code>( <var class="Arg">FG, chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>The numerical description of the output of <code class="func">SimpleAlgebraByCharacter</code> (<a href="chap2.html#X8349114C83161C2D"><b>2.2-1</b></a>).</p>
<p>The first input <var class="Arg">FG</var> is a <em>semisimple group algebra</em> (<a href="chap7.html#X7FDD93FB79ADCC91"><b>7.2</b></a>) over a finite group G and the second input is an irreducible character of G.</p>
<p>The output is the numerical description <a href="chap7.html#X84A142407B7565E0"><b>7.12</b></a> of the <em>cyclotomic algebra</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>) which is isomorphic to the unique <em>Wedderburn component</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) A of <var class="Arg">FG</var> such that chi(A)ne 0.</p>
<p>See <a href="chap7.html#X84A142407B7565E0"><b>7.12</b></a> for the interpretation of the numerical information given by the output.</p>
<table class="example">
<tr><td><pre>
gap> G:=SmallGroup(144,11);
<pc group of size 144 with 6 generators>
gap> QG:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 6 generators>
gap> SimpleAlgebraByCharacter( QG , Irr(G)[48] );
<crossed product with center NF(36,[ 1, 17 ]) over AsField( NF(36,
[ 1, 17 ]), CF(36) ) of a group of size 2>
gap> SimpleAlgebraByCharacterInfo( QG , Irr(G)[48] );
[ 1, NF(36,[ 1, 17 ]), 36, [ 2, 17, 18 ] ]
</pre></td></tr></table>
<p><a id="X812D667D7D913EB5" name="X812D667D7D913EB5"></a></p>
<h5>2.2-3 SimpleAlgebraByStrongSP</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSP</code>( <var class="Arg">QG, K, H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSPNC</code>( <var class="Arg">QG, K, H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSP</code>( <var class="Arg">FG, K, H, C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSPNC</code>( <var class="Arg">FG, K, H, C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>A simple algebra.</p>
<p>In the three-argument version the input must be formed by a <em>semisimple rational group algebra</em> <var class="Arg">QG</var> (see <a href="chap7.html#X7FDD93FB79ADCC91"><b>7.2</b></a>) and two subgroups <var class="Arg">K</var> and <var class="Arg">H</var> of G which form a <em>strong Shoda pair</em> (<a href="chap7.html#X81DAF5267D30C83A"><b>7.15</b></a>) of G.</p>
<p>The three-argument version returns the Wedderburn component (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of the rational group algebra <var class="Arg">QG</var> realized by the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>).</p>
<p>In the four-argument version the first argument is a semisimple finite group algebra <var class="Arg">FG</var>, <var class="Arg">(K,H)</var> is a strong Shoda pair of G and the fourth input data is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or a representative of a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> (see <a href="chap7.html#X800D8C5087D79DC8"><b>7.17</b></a>).</p>
<p>The four-argument version returns the Wedderburn component (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of the finite group algebra <var class="Arg">FG</var> realized by the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>) and the cyclotomic class <var class="Arg">C</var> (or the cyclotomic class containing <var class="Arg">C</var>).</p>
<p>The versions ending in NC do not check if (<var class="Arg">K</var>,<var class="Arg">H</var>) is a strong Shoda pair of G. In the four-argument version it is also not checked whether <var class="Arg">C</var> is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or an integer coprime to the index of <var class="Arg">H</var> in <var class="Arg">K</var>.</p>
<table class="example">
<tr><td><pre>
gap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;
gap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;
gap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;
gap> QG:=GroupRing( Rationals, G );;
gap> FG:=GroupRing( GF(7), G );;
gap> SimpleAlgebraByStrongSP( QG, K, H );
<crossed product over CF(16) of a group of size 2>
gap> SimpleAlgebraByStrongSP( FG, K, H, [1,7] );
( GF(7)^[ 2, 2 ] )
gap> SimpleAlgebraByStrongSP( FG, K, H, 1 );
( GF(7)^[ 2, 2 ] )
</pre></td></tr></table>
<p><a id="X858152C882129A0B" name="X858152C882129A0B"></a></p>
<h5>2.2-4 SimpleAlgebraByStrongSPInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSPInfo</code>( <var class="Arg">QG, K, H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSPInfoNC</code>( <var class="Arg">QG, K, H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSPInfo</code>( <var class="Arg">FG, K, H, C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SimpleAlgebraByStrongSPInfoNC</code>( <var class="Arg">FG, K, H, C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>A numerical description of one simple algebra.</p>
<p>In the three-argument version the input must be formed by a <em>semisimple rational group algebra</em> (<a href="chap7.html#X7FDD93FB79ADCC91"><b>7.2</b></a>) <var class="Arg">QG</var> and two subgroups <var class="Arg">K</var> and <var class="Arg">H</var> of G which form a <em>strong Shoda pair</em> (<a href="chap7.html#X81DAF5267D30C83A"><b>7.15</b></a>) of G. It returns the numerical information describing the Wedderburn component (<a href="chap7.html#X84A142407B7565E0"><b>7.12</b></a>) of the rational group algebra <var class="Arg">QG</var> realized by a the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>).</p>
<p>In the four-argument version the first input is a semisimple finite group algebra <var class="Arg">FG</var>, <var class="Arg">(K,H)</var> is a strong Shoda pair of G and the fourth input data is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or a representative of a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> (<a href="chap7.html#X800D8C5087D79DC8"><b>7.17</b></a>). It returns a pair of positive integers [n,r] which represent the nx n matrix algebra over the field of order r which is isomorphic to the Wedderburn component of <var class="Arg">FG</var> realized by a the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>) and the cyclotomic class <var class="Arg">C</var> (or the cyclotomic class containing the integer <var class="Arg">C</var>).</p>
<p>The versions ending in NC do not check if (<var class="Arg">K</var>,<var class="Arg">H</var>) is a strong Shoda pair of G. In the four-argument version it is also not checked whether <var class="Arg">C</var> is either a generating q-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or an integer coprime with the index of <var class="Arg">H</var> in <var class="Arg">K</var>.</p>
<table class="example">
<tr><td><pre>
gap> F:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;
gap> G:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;
gap> K:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;
gap> QG:=GroupRing( Rationals, G );;
gap> FG:=GroupRing( GF(7), G );;
gap> SimpleAlgebraByStrongSP( QG, K, H );
<crossed product over CF(16) of a group of size 2>
gap> SimpleAlgebraByStrongSPInfo( QG, K, H );
[ 1, NF(16,[ 1, 7 ]), 16, [ [ 2, 7, 8 ] ], [ ] ]
gap> SimpleAlgebraByStrongSPInfo( FG, K, H, [1,7] );
[ 2, 7 ]
gap> SimpleAlgebraByStrongSPInfo( FG, K, H, 1 );
[ 2, 7 ]
</pre></td></tr></table>
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