`‣ WedderburnDecomposition` ( FG ) | ( attribute ) |

Returns: A list of simple algebras.

The input `FG` 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 \(G\).

The function returns the list of all *Wedderburn components* (9.3) of the group algebra `FG`. If \(F\) is an abelian number field then each Wedderburn component is given as a matrix algebra of a *cyclotomic algebra* (9.11). If \(F\) is a finite field then the Wedderburn components are given as matrix algebras over finite fields.

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> ]

The previous examples show that if \(D_{16}\) denotes the dihedral group of order \(16\) then the *Wedderburn decomposition* (9.3) of \(\mathbb F_5 D_{16}\), \(ℚ D_{16}\) and \(ℚ (\xi_5) D_{16}\) are respectively

\[ \mathbb F_5 D_{16} = 4 \mathbb F_5 \oplus M_2( \mathbb F_5 ) \oplus M_2( \mathbb F_{25} ), \]

\[ ℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus (K(\xi_8)/K,t), \]

and

\[ ℚ (\xi_5) D_{16} = 4 ℚ (\xi_5) \oplus M_2( ℚ (\xi_5) ) \oplus (F(\xi_{40})/F,t), \]

where \((K(\xi_8)/K,t)\) is a *cyclotomic algebra* (9.11) with the centre \(K=NF(8,[ 1, 7 ])= ℚ (\sqrt{2})\), \((F(\xi_{40})/F,t) = ℚ (\sqrt{2},\xi_5)\) is a cyclotomic algebra with centre \(F=NF(40,[ 1, 31 ])\) and \(\xi_n\) denotes a \(n\)-th root of unity.

Two more examples:

gap> WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) ); [ Rationals, Rationals, Rationals, Rationals, <crossed product with center Rationals over CF(3) of a group of size 2>, <crossed product with center Rationals over GaussianRationals of a group of \ size 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>, ( CF(3)^[ 2, 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 ] ), <crossed product with center CF(3) over AsField( CF(3), CF( 12) ) of a group of size 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>, ( CF(3)^[ 2, 2 ] ), ( CF(3)^[ 2, 2 ] ), ( <crossed product with center CF(3) over AsField( CF( 3), CF(12) ) of a group of size 2>^[ 2, 2 ] ) ]

In some cases, in characteristic zero, some entries of the output of `WedderburnDecomposition`

do not provide full matrix algebras over a *cyclotomic algebra* (9.11), but "fractional matrix algebras". That entry is not an algebra that can be used as a **GAP** object. Instead it is a pair formed by a rational giving the "size" of the matrices and a crossed product. See 9.3 for a theoretical explanation of this phenomenon. In this case a warning message is displayed.

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> ] ]

`‣ WedderburnDecompositionInfo` ( FG ) | ( attribute ) |

Returns: A list with each entry a numerical description of a *cyclotomic algebra* (9.11).

The input `FG` 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 \(G\).

This function is a numerical counterpart of `WedderburnDecomposition`

(2.1-1).

It returns a list formed by lists of lengths 2, 4 or 5.

The lists of length 2 are of the form

\[ [n,F], \]

where \(n\) is a positive integer and \(F\) is a field. It represents the \(n\times n\) matrix algebra \(M_n(F)\) over the field \(F\).

The lists of length 4 are of the form

\[ [n,F,k,[d,\alpha,\beta]], \]

where \(F\) is a field and \(n,k,d,\alpha,\beta\) are non-negative integers, satisfying the conditions mentioned in Section 9.12. It represents the \(n\times n\) matrix algebra \(M_n(A)\) over the cyclic algebra

\[ A=F(\xi_k)[u | \xi_k^u = \xi_k^{\alpha}, u^d = \xi_k^{\beta}], \]

where \(\xi_k\) is a primitive \(k\)-th root of unity.

The lists of length 5 are of the form

\[ [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ], \]

where \(F\) is a field and \(n,k,d_i,\alpha_i,\beta_i,\gamma_{i,j}\) are non-negative integers. It represents the \(n\times n\) matrix algebra \(M_n(A)\) over the *cyclotomic algebra* (9.11)

\[ A = F(\xi_k)[g_1,\ldots,g_m \mid \xi_k^{g_i} = \xi_k^{\alpha_i}, g_i^{d_i}=\xi_k^{\beta_i}, g_jg_i=\xi_k^{\gamma_{ij}} g_i g_j], \]

where \(\xi_k\) is a primitive \(k\)-th root of unity (see 9.12).

gap> WedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) ); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ] ] gap> WedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) ); [ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ], [ 2, NF(40,[ 1, 31 ]) ] ]

The interpretation of the previous example gives rise to the following *Wedderburn decompositions* (9.3), where \(D_{16}\) is the dihedral group of order 16 and \(\xi_5\) is a primitive \(5\)-th root of unity.

\[ ℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus M_2( ℚ (\sqrt{2})). \]

\[ ℚ (\xi_5) D_{16} = 4 ℚ (\xi_5) \oplus M_2( ℚ (\xi_5)) \oplus M_2( ℚ (\xi_5,\sqrt{2})). \]

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, <crossed product with center Rationals over CF( 3) of a group of size 2>, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ) ] 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 ], [ 2, Rationals ], [ 3, Rationals ], [ 3, Rationals ] ]

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 `WedderburnDecomposition`

(2.1-1) and `WedderburnDecompositionInfo`

.

The output of `WedderburnDecomposition`

(2.1-1) shows that

\[ ℚ Q_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus A, \]

\[ ℚ S_{4} = 2 ℚ \oplus 2 M_3( ℚ ) \oplus B, \]

where \(A\) and \(B\) are *crossed products* (9.6) with coefficients in the cyclotomic fields \(ℚ (\xi_8)\) and \(ℚ (\xi_3)\) respectively. This output can be used as a **GAP** object, but it does not give clear information on the structure of the algebras \(A\) and \(B\).

The numerical information displayed by `WedderburnDecompositionInfo`

means that

\[ A = ℚ (\xi|\xi^8=1)[g | \xi^g = \xi^7 = \xi^{-1}, g^2 = \xi^4 = -1], \]

\[ B = ℚ (\xi|\xi^3=1)[g | \xi^g = \xi^2 = \xi^{-1}, g^2 = 1]. \]

Both \(A\) and \(B\) are quaternion algebras over its centre which is \(ℚ (\xi+\xi^{-1})\) and the former is equal to \(ℚ (\sqrt{2})\) and \(ℚ\) respectively.

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

\[ A=ℚ (\sqrt{2})[i,g|i^2=g^2=-1,ig=-gi] \]

and so \(A\) is a division algebra with centre \(ℚ (\sqrt{2})\), 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. [Rei03]).

The next example shows the output of `WedderburnDecompositionInfo`

for \(ℚ G\) and \(ℚ (\xi_3) G\), where \(G=SmallGroup(48,15)\). The user can compare it with the output of `WedderburnDecomposition`

(2.1-1) for the same group in the previous section. Notice that the last entry of the *Wedderburn decomposition* (9.3) 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.

gap> WedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) ); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 2, Rationals ], [ 2, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ], [ 2, CF(3) ], [ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ] ] 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) ], [ 2, CF(3) ], [ 2, NF(24,[ 1, 7 ]) ], [ 2, CF(3) ], [ 2, CF(3) ], [ 4, CF(3) ] ]

In some cases some of the first entries of the output of `WedderburnDecompositionInfo`

are not integers and so the correspoding *Wedderburn components* (9.3) are given as "fractional matrix algebras" of *cyclotomic algebras* (9.11). See 9.3 for a theoretical explanation of this phenomenon. In that case a warning message will be displayed during the first call of `WedderburnDecompositionInfo`

.

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 ] ] ]

The interpretation of the output in the previous example gives rise to the following *Wedderburn decomposition* (9.3) of \(ℚ G\) for \(G\) the small group \([240,89]\):

\[ ℚ G = 2 ℚ \oplus 2 M_4( ℚ ) \oplus 2 M_5( ℚ ) \oplus M_6( ℚ ) \oplus A \oplus B \oplus C \]

where

\[ A = ℚ (\xi_{10})[u|\xi_{10}^u = \xi_{10}^3, u^4 = -1], \]

\(B\) is an algebra of degree \((4*2 )/2 = 4 \) which is *Brauer equivalent* (9.5) to

\[ B_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^{13}, u^4 = \xi_{60}^5, \xi_{60}^v = \xi_{60}^{11}, v^2 = 1, vu=uv], \]

and \(C\) is an algebra of degree \((4*2)*3/4 = 6 \) which is *Brauer equivalent* (9.5) to

\[ C_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^7, u^4 = \xi_{60}^5, \xi_{60}^v = \xi_{60}^{31}, v^2 = 1, vu=uv]. \]

The precise description of \(B\) and \(C\) requires the usage of "ad hoc" arguments.

`‣ SimpleAlgebraByCharacter` ( FG, chi ) | ( operation ) |

Returns: A simple algebra.

The first input `FG` should be a *semisimple group algebra* (9.2) over a finite group \(G\) and the second input should be an irreducible character of \(G\).

The output is a matrix algebra of a *cyclotomic algebras* (9.11) which is isomorphic to the unique *Wedderburn component* (9.3) \(A\) of `FG` such that \(\chi(A)\ne 0\).

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> chi4:=Filtered(Irr(G),x->Degree(x)=4);; gap> List(chi4,x->SimpleAlgebraByCharacter(GroupRing(Rationals,G),x)); [ ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8, [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ), ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8, [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ), ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8, [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ), ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8, [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ) ]

`‣ SimpleAlgebraByCharacterInfo` ( FG, chi ) | ( operation ) |

Returns: The numerical description of the output of `SimpleAlgebraByCharacter`

(2.2-1).

The first input `FG` is a *semisimple group algebra* (9.2) over a finite group \(G\) and the second input is an irreducible character of \(G\).

The output is the numerical description 9.12 of the *cyclotomic algebra* (9.11) which is isomorphic to the unique *Wedderburn component* (9.3) \(A\) of `FG` such that \(\chi(A)\ne 0\).

See 9.12 for the interpretation of the numerical information given by the output.

gap> G:=SmallGroup(128,100); <pc group of size 128 with 7 generators> gap> QG:=GroupRing(Rationals,G); <algebra-with-one over Rationals, with 7 generators> gap> chi4:=Filtered(Irr(G),x->Degree(x)=4);; gap> List(chi4,x->SimpleAlgebraByCharacterInfo(QG,x)); [ [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ], [ 4, NF(8,[ 1, 3 ]) ] ]

`‣ SimpleAlgebraByStrongSP` ( QG, K, H ) | ( operation ) |

`‣ SimpleAlgebraByStrongSPNC` ( QG, K, H ) | ( operation ) |

`‣ SimpleAlgebraByStrongSP` ( FG, K, H, C ) | ( operation ) |

`‣ SimpleAlgebraByStrongSPNC` ( FG, K, H, C ) | ( operation ) |

Returns: A simple algebra.

In the three-argument version the input must be formed by a *semisimple rational group algebra* `QG` (see 9.2) and two subgroups `K` and `H` of \(G\) which form a *strong Shoda pair* (9.15) of \(G\).

The three-argument version returns the Wedderburn component (9.3) of the rational group algebra `QG` realized by the strong Shoda pair (`K`,`H`).

In the four-argument version the first argument is a semisimple finite group algebra `FG`, `(K,H)` is a strong Shoda pair of \(G\) and the fourth input data is either a generating \(q\)-cyclotomic class modulo the index of `H` in `K` or a representative of a generating \(q\)-cyclotomic class modulo the index of `H` in `K` (see 9.19).

The four-argument version returns the Wedderburn component (9.3) of the finite group algebra `FG` realized by the strong Shoda pair (`K`,`H`) and the cyclotomic class `C` (or the cyclotomic class containing `C`).

The versions ending in NC do not check if (`K`,`H`) is a strong Shoda pair of \(G\). In the four-argument version it is also not checked whether `C` is either a generating \(q\)-cyclotomic class modulo the index of `H` in `K` or an integer coprime to the index of `H` in `K`.

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 ] )

`‣ SimpleAlgebraByStrongSPInfo` ( QG, K, H ) | ( operation ) |

`‣ SimpleAlgebraByStrongSPInfoNC` ( QG, K, H ) | ( operation ) |

`‣ SimpleAlgebraByStrongSPInfo` ( FG, K, H, C ) | ( operation ) |

`‣ SimpleAlgebraByStrongSPInfoNC` ( FG, K, H, C ) | ( operation ) |

Returns: A numerical description of one simple algebra.

In the three-argument version the input must be formed by a *semisimple rational group algebra* (9.2) `QG` and two subgroups `K` and `H` of \(G\) which form a *strong Shoda pair* (9.15) of \(G\). It returns the numerical information describing the Wedderburn component (9.12) of the rational group algebra `QG` realized by a the strong Shoda pair (`K`,`H`).

In the four-argument version the first input is a semisimple finite group algebra `FG`, `(K,H)` is a strong Shoda pair of \(G\) and the fourth input data is either a generating \(q\)-cyclotomic class modulo the index of `H` in `K` or a representative of a generating \(q\)-cyclotomic class modulo the index of `H` in `K` (9.19). It returns a pair of positive integers \([n,r]\) which represent the \(n\times n\) matrix algebra over the field of order \(r\) which is isomorphic to the Wedderburn component of `FG` realized by a the strong Shoda pair (`K`,`H`) and the cyclotomic class `C` (or the cyclotomic class containing the integer `C`).

The versions ending in NC do not check if (`K`,`H`) is a strong Shoda pair of \(G\). In the four-argument version it is also not checked whether `C` is either a generating \(q\)-cyclotomic class modulo the index of `H` in `K` or an integer coprime with the index of `H` in `K`.

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 ]

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