115 Algebra package --- finite dimensional algebras

This package has been developped by Cédric Bonnafé to work with finite dimensional algebras under GAP3; it depends on the package "chevie".

Note that these programs have been mainly developed for working with Solomon descent algebras.

We start with a list of utility functions which are used in various places.

Subsections

  1. Digits
  2. ByDigits
  3. SignedCompositions
  4. SignedPartitions
  5. PiPart
  6. CyclotomicModP
  7. PiComponent
  8. PiSections
  9. PiPrimeSections
  10. PRank
  11. PBlocks
  12. Finite-dimensional algebras over fields
  13. Elements of finite dimensional algebras
  14. Operations for elements of finite dimensional algebras
  15. IsAlgebraElement for finite dimensional algebras
  16. IsAbelian for finite dimensional algebras
  17. IsAssociative for finite dimensional algebras
  18. AlgebraHomomorphismByLinearity
  19. SubAlgebra for finite-dimensional algebras
  20. CentralizerAlgebra
  21. Center for algebras
  22. Ideals
  23. QuotientAlgebra
  24. Radical for algebras
  25. RadicalPower
  26. LoewyLength
  27. CharTable for algebras
  28. CharacterDecomposition
  29. Idempotents for finite dimensional algebras
  30. LeftIndecomposableProjectives
  31. CartanMatrix
  32. PolynomialQuotientAlgebra
  33. GroupAlgebra
  34. Augmentation
  35. GrothendieckRing
  36. Degree for elements of Grothendieck rings
  37. Solomon algebras
  38. SolomonAlgebra
  39. Generalized Solomon algebras
  40. GeneralizedSolomonAlgebra
  41. SolomonHomomorphism
  42. ZeroHeckeAlgebra
  43. Performance

115.1 Digits

Digits(n [, basis])

returns the list of digits of the nonnegative integer n in basis basis (in basis 10 if no second argument is given). 

    gap> Digits(0); Digits(3); Digits(123); Digits(123,16);
    [  ]
    [ 3 ]
    [ 1, 2, 3 ]
    [ 7, 11 ]

115.2 ByDigits

ByDigits(l [, basis])

Does the converse of Digits, that is, computes an integer give the sequence of its digits (by default in basis 10; in basis basis if a second argument is given). 

    gap> ByDigits([2,3,4,5]);
    2345
    gap> ByDigits([2,3,4,5],100);
    2030405

115.3 SignedCompositions

SignedCompositions(n)

computes the set of signed compositions of n that is, the set of tuples of non-zero integers [i1,...ir] such that |i1| + ... + |ir| = n. Note that Length(SignedCompositions(n)) = 2*3^(n-1). 

    gap> SignedCompositions(3);
    [ [ -3 ], [ -2, -1 ], [ -2, 1 ], [ -1, -2 ], [ -1, -1, -1 ],
      [ -1, -1, 1 ], [ -1, 1, -1 ], [ -1, 1, 1 ], [ -1, 2 ], [ 1, -2 ],
      [ 1, -1, -1 ], [ 1, -1, 1 ], [ 1, 1, -1 ], [ 1, 1, 1 ], [ 1, 2 ],
      [ 2, -1 ], [ 2, 1 ], [ 3 ] ]

Note that the compositions of n are obtained by the function OrderedPartitions in GAP3.

115.4 SignedPartitions

SignedPartitions(n)

computes the set of signed partitions of n that is, the set of tuples of integers [i1,...,ir,j1,...,js] such that ik > 0, jk < 0, |i1| + ... + |ir| + |j1| + ... + |js|= n, i1 ≥ ... ≥ ir and |j1| ≥ ... ≥ |js|. 

    gap> SignedPartitions(3);
    [ [ -3 ], [ -2, -1 ], [ -1, -1, -1 ], [ 1, -2 ], [ 1, -1, -1 ],
      [ 1, 1, -1 ], [ 1, 1, 1 ], [ 2, -1 ], [ 2, 1 ], [ 3 ] ]

115.5 PiPart

PiPart(n,pi)

Let n be an integer and π a set of prime numbers. Write n=n1n2 where no prime factor of n2 is in π and all prime factors of n1 are in π. Then n1 is called the π-part of n and n2 the π'-part of n. This function returns the π-part of n. The set π may be given as a list of primes, or as an integer in which case the set π is taken to be the list of prime factors of that integer. 

    gap> PiPart(720,2);
    16
    gap> PiPart(720,3);
    9
    gap> PiPart(720,6);
    144
    gap> PiPart(720,[2,3]);
    144

115.6 CyclotomicModP

CyclotomicModP(z,p)

p should be a prime and z a cyclotomic number which is p-integral (that is, z times some number prime to p is a cyclotomic integer). The function returns the reduction of z mod. p, an element of some extension 𝔽pr of the prime field 𝔽p. 

    gap> CyclotomicModP(E(7),3);
    Z(3^6)^104

115.7 PiComponent

PiComponent(G,g,pi)

Let g be an element of the finite group G and π a set of prime numbers. Write g=g1g2 where g1 and g2 are both powers of g, no prime factor of the order of g2 is in π and all prime factors of the order of g1 are in π. Then g1 is called the π-component of g and g2 the π'-component of n. This function returns the π-component of g. The set π may be given as a list of primes, or as an integer in which case the set π is taken to be the list of prime factors of that integer.

115.8 PiSections

PiSections(G,pi)

Let π be a set of prime numbers. Two conjugacy classes of the finite group G are said to belong to the same π-section if the π-components (see PiComponent) of elements of the two classes are conjugate. This function returns the partition of the set of conjugacy classes of G in π-sections, represented by the list of indices of conjugacy classes of G in each part. The set π may be given as a list of primes, or as an integer in which case the set π is taken to be the list of prime factors of that integer. 

    gap> W:=SymmetricGroup(5);
    Group( (1,5), (2,5), (3,5), (4,5) )
    gap> PiSections(W,2);
    [ [ 1, 4, 7 ], [ 2, 5 ], [ 3 ], [ 6 ] ]
    gap> PiSections(W,3);
    [ [ 1, 2, 3, 6, 7 ], [ 4, 5 ] ]
    gap> PiSections(W,6);
    [ [ 1, 7 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ] ]

115.9 PiPrimeSections

PiPrimeSections(G,pi)

Let π be a set of prime numbers. Two conjugacy classes of the finite group G are said to belong to the same π'-section if the π'-components (see PiComponent) of elements of the two classes are conjugate. This function returns the partition of the set of conjugacy classes of G in π'-sections, represented by the list of indices of conjugacy classes of G in each part. The set π may be given as a list of primes, or as an integer in which case the set π is taken to be the list of prime factors of that integer. 

    gap> W:=SymmetricGroup(5);
    Group( (1,5), (2,5), (3,5), (4,5) )
    gap> PiPrimeSections(W,2);
    [ [ 1, 2, 3, 6 ], [ 4, 5 ], [ 7 ] ]
    gap> PiPrimeSections(W,3);
    [ [ 1, 4 ], [ 2, 5 ], [ 3 ], [ 6 ], [ 7 ] ]
    gap> PiPrimeSections(W,6);
    [ [ 1, 2, 3, 4, 5, 6 ], [ 7 ] ]

115.10 PRank

PRank(G,p)

Let p be a prime. This function returns the p-rank of the finite group G, defined as the maximal rank of an elementary abelian p-subgroup of G. 

    gap> W:=SymmetricGroup(5);
    Group( (1,5), (2,5), (3,5), (4,5) )
    gap> PRank(W,2);
    2
    gap> PRank(W,3);
    1
    gap> PRank(W,7);
    0

115.11 PBlocks

PBlocks(G,p)

Let p be a prime. This function returns the partition of the irreducible characters of G in p-blocks, represented by the list of indices of irreducibles characters in each part. 

    gap> W:=SymmetricGroup(5);
    Group( (1,5), (2,5), (3,5), (4,5) )
    gap> PBlocks(W,2);
    [ [ 1, 2, 5, 6, 7 ], [ 3, 4 ] ]
    gap> PBlocks(W,3);
    [ [ 1, 3, 6 ], [ 2, 4, 5 ], [ 7 ] ]
    gap> PBlocks(W,7);
    [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ] ]

115.12 Finite-dimensional algebras over fields

Let K be a field and let A be a K-algebra of finite dimension d. In our implementation, A must be endowed with a basis X = (xi)i ∈ I, where I={i1,...,id}. Then A is represented by a record containing

the following fields:

A.field:
the field K.

A.dimension:
the dimension of A.

A.multiplication:
this is a function which associates to (k,l) the coefficients of the product xik xil in the basis X (here, 1 ≤ k, l ≤ d). If the structure constants of A are known, then it is possible to record them in A.structureconstants: the entry A.structureconstants[k][l] is equal to A.multiplication(k,l). Once the function A.multiplication is defined, we can obtain the field A.structureconstants just by asking for FDAlgebraOps.structureconstants(A).

A.zero:
the zero element of A.

A.one:
the unity of A.

A.basisname:
a "name" for the basis X (for instance, A.basisname:="X").

A.parameters:
the parameter set I.

A.identification:
something characterizing A (this is used to test if two algebras are equal). For instance, if A=K[G] is the group algebra of G, we take A.identification:=["Group algebra",G,K];.

For convenience, the record A is often endowed with the following fields:

A.generators:
a list of generators of A.

A.basis:
the list of elements of X.

A.vectorspace:
the underlying vector space represented in GAP3 as Kd.

A.EltToVector:
the function sending an element of A to its image in A.vectorspace (i.e. a d-tuple of elements of K).

A.VectorToElt:
inverse function of A.EltToVector.

A.type:
for instance "Group algebra", or "Grothendieck ring"...

A.operations:
This is initialized to FDAlgebraOps which contains quite a few operations applicable to finite-dimensional algebras, like the following:

FDAlgebraOps.underlyingspace:
once A.dimension is defined, this function constructs the underlying space of A. It endows the record A with the fields A.basis, A.vectorspace, A.EltToVector, and A.VectorToElt.

FDAlgebraOps.structureconstants:
computes the structure constants of A and gathers them in A.structureconstants.

115.13 Elements of finite dimensional algebras

An element x of A is implemented as a record containing three fields:

x.algebra:
the algebra A

x.coefficients:
the list of pairs (ak,k) such that ak is a non-zero element of K and x=∑k=1d ak xik.

x.operations:
the operations record AlgebraEltOps defining the operations for finite dimensional algebra elements.

115.14 Operations for elements of finite dimensional algebras

The following operations are define for elements of a finite dimensional algebra A.

Print:
this function gives a way of printing elements of A. If A.print is defined, it is used. Otherwise, the element xi is printed using A.basisname and A.parameters: for instance, if A.basisname:="BASISNAME" and A.parameters:=[1..d], then xi is printed as BASISNAME(i).

\+ :
addition of elements of A.

\- :
subtraction of elements of A.

* :
multiplication of elements of A.

^ :
powers of elements of A (negative powers are allowed for invertible elements).

Coefficients(x):
the list of coefficients of x in Basis(A).

115.15 IsAlgebraElement for finite dimensional algebras

IsAlgebraElement(x)

This function retuirns true if x is an element of a finite dimensional algebra, false if it is another kind of object. 

    gap> q:=X(Rationals);; q.name:="q";;
    gap> A:=PolynomialQuotientAlgebra(q^2-q-1);;
    gap> IsAlgebraElement(Basis(A)[1]);
    true
    gap> IsAlgebraElement(1);
    false

115.16 IsAbelian for finite dimensional algebras

IsAbelian(A)

returns true if the algebra A is commutative and false otherwise. 

    gap> q:=X(Rationals);; q.name:="q";;
    gap> A:=PolynomialQuotientAlgebra(q^2-q-1);;
    gap> IsAbelian(A);
    true
    gap> B:=SolomonAlgebra(CoxeterGroup("A",2));;
    gap> IsAbelian(B);
    false

115.17 IsAssociative for finite dimensional algebras

IsAssociative(A)

returns true if the algebra A is associative and false otherwise. 

    gap> q:=X(Rationals);; q.name:="q";;
    gap> A:=PolynomialQuotientAlgebra(q^2-q-1);;
    gap> IsAssociative(A);
    true

115.18 AlgebraHomomorphismByLinearity

AlgebraHomomorphismByLinearity(A,B[,l])

returns the linear map from A to B that sends A.basis to the list l (if omitted to B.basis). If this is not an homomorphism of algebras, the function returns an error. 

    gap> q:=X(Rationals);; q.name:="q";;
    gap> A:=PolynomialQuotientAlgebra(q^4);;
    gap> hom:=AlgebraHomomorphismByLinearity(A,Rationals,[1,0,0,0]);
    function ( element ) ... end
    gap> hom(A.class(q^4+q^3+1));
    1
    gap> hom2:=AlgebraHomomorphismByLinearity(A,Rationals,[1,1,1,1]);
    Error, This is not a morphism of algebras in
    AlgebraHomomorphismByLinearity( A, Rationals, [ 1, 1, 1, 1 ] ) called from
    main loop

115.19 SubAlgebra for finite-dimensional algebras

SubAlgebra(A,l)

returns the sub-algebra B of A generated by the list l. The elements of B are written as elements of A. 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",4));
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> B:=SubAlgebra(A,[A.xbasis(23),A.xbasis(34)]);
    SubAlgebra(SolomonAlgebra(CoxeterGroup("B",4),Rationals),
    [ X(23), X(34) ])
    gap> Dimension(B);
    6
    gap> IsAbelian(B);
    false
    gap> B.basis;
    [ X(1234), X(23), X(34), X(2)-X(4), X(3)+X(4), X(0) ]

115.20 CentralizerAlgebra

CentralizerAlgebra(A,l)

returns the sub-algebra B of A of elements commuting with all the elements in the list l. The elements of B are written as elements of A. 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",4));
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> B:=CentralizerAlgebra(A,[A.xbasis(23),A.xbasis(34)]);
    Centralizer(SolomonAlgebra(CoxeterGroup("B",4),Rationals),
    [ X(23), X(34) ])
    gap> Dimension(B);
    10
    gap> IsAbelian(B);
    false

115.21 Center for algebras

Centre(A)

returns the center B of the algebra A. The elements of B are written as elements of A. 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",4));
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> B:=Centre(A);
    Centre(SolomonAlgebra(CoxeterGroup("B",4),Rationals))
    gap> Dimension(B);
    8
    gap> IsAbelian(B);
    true

115.22 Ideals

If l is an element, or a list of elements of the algebra A, then LeftIdeal(A,l) (resp. RightIdeal(A,l), resp. TwoSidedIdeal(A,l)) returns the left (resp. right, resp. two-sided) ideal of A generated by l. The result is a record containing the following fields:

.parent:
the algebra A

.generators:
the list l

.basis:
a K-basis of the ideal

.dimension:
the dimension of the ideal

LeftTraces(A,I), RightTraces(A,I):
the character afforded by the left (or right) ideal I (written as a list of traces of elements of the A.basis). 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",4));
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> I:=LeftIdeal(A,[A.xbasis(234)]);
    LeftIdeal(SolomonAlgebra(CoxeterGroup("B",4),Rationals),[ X(234) ])
    gap> I.basis;
    [ X(234), X(23)+X(34), X(24), X(2)+X(4), X(3), X(0) ]
    gap> Dimension(I);
    6
    gap> LeftTraces(A,I);
    [ 6, 18, 40, 50, 42, 64, 112, 112, 100, 136, 100, 192, 224, 224, 224,
    384 ]

115.23 QuotientAlgebra

QuotientAlgebra(A,I)

A is a finite dimensional algebra, and I a two-sided ideal of A. The function returns the algebra A/I. It is also allowed than I be an element of A or a list of elements of A, in which case it is understood as the two-sided ideal generated by I.

115.24 Radical for algebras

Radical(A)

If the record A is endowed with the field A.radical (containing the radical of A) or with the field A.Radical (a function for computing the radical of A), then Radical(A) returns the radical of A (as a two-sided ideal of A). At this time, this function is available only in characteristic zero: it works for group algebras, Grothendiek rings, Solomon algebras and generalized Solomon algebras.

115.25 RadicalPower

RadicalPower(A,n)

returns (when possible) the n-th power of the two-sided ideal Radical(A).

115.26 LoewyLength

LoewyLength(A)

returns (when possible) the Loewy length of A that is, the smallest natural number n >= 1 such that the n-th power of the two-sided ideal Radical(A) vanishes. 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",4));
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> R:=Radical(A);
    TwoSidedIdeal(SolomonAlgebra(CoxeterGroup("B",4),Rationals),
    [ X(13)-X(14), X(23)-X(34), X(2)-X(3), X(2)-X(4) ])
    gap> Dimension(R);
    4
    gap> LoewyLength(A);
    2

115.27 CharTable for algebras

CharTable(A)

For certain algebras, the function CharTable may be applied. It returns the character table of the algebra K ⊗K A: different ways of printing are used according to the type of the algebra. If A is a group algebra in characteristic zero, then CharTable(A) returns the character table of A.group. This function is available whenever K is of characteristic zero for group algebras, Grothendieck rings, Solomon algebras and generalized Solomon algebras. 

    gap> A:=GrothendieckRing(SymmetricGroup(4));
    GrothendieckRing(Group( (1,4), (2,4), (3,4) ),Rationals)
    gap> CharTable(A);

         X.1 X.2 X.3 X.4 X.5

    MU.1   1   1   2   3   3
    MU.2   1  -1   .  -1   1
    MU.3   1   1   2  -1  -1
    MU.4   1   1  -1   .   .
    MU.5   1  -1   .   1  -1

    gap> B:=SolomonAlgebra(CoxeterGroup("B",2));
    SolomonAlgebra(CoxeterGroup("B",2),Rationals)
    gap> CharTable(B);

         1
         2  1  2  0

    12   1  .  .  .
     1   1  2  .  .
     2   1  .  2  .
     0   1  4  4  8

115.28 CharacterDecomposition

CharacterDecomposition(A,char)

Given a list char of elements of K (indexed by A.basis), then CharacterDecomposition(A,char) returns the decomposition of char into a sum of irreducible characters of A, if possible. 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",3));
    SolomonAlgebra(CoxeterGroup("B",3),Rationals)
    gap> I:=LeftIdeal(A,[A.xbasis(13)]);
    LeftIdeal(SolomonAlgebra(CoxeterGroup("B",3),Rationals),[ X(13) ])
    gap> I.basis;
    [ X(13), X(1), X(3), X(0) ]
    gap> LeftTraces(A,I);
    [ 4, 12, 20, 12, 32, 28, 28, 48 ]
    gap> CharTable(A);

          1
          2  1   1  2
          3  2   3  3   1   2   0

    123   1  .   .  .   .   .   .
     12   1  2   .  .   .   .   .
     13   1  .   2  .   .   .   .
     23   1  .   .  2   .   .   .
      1   1  4   4  .   8   .   .
      2   1  2   2  4   .   4   .
      0   1  6  12  8  24  24  48

    gap> CharacterDecomposition(A,LeftTraces(A,I));
    [ 0, 0, 1, 0, 1, 1, 1 ]

115.29 Idempotents for finite dimensional algebras

Idempotents(A)

returns a complete set of orthogonal primitive idempotents of A. This is defined currently for Solomon algebras, quotient by polynomial algebras, group algebras and Grothendieck rings. 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",2));
    SolomonAlgebra(CoxeterGroup("B",2),Rationals)
    gap> e:=Idempotents(A);
    [ X(12)-1/2*X(1)-1/2*X(2)+3/8*X(0), 1/2*X(1)-1/4*X(0),
      1/2*X(2)-1/4*X(0), 1/8*X(0) ]
    gap> Sum(e)=A.one;
    true
    gap> List(e, i-> i^2-i);
    [ 0*X(12), 0*X(12), 0*X(12), 0*X(12) ]
    gap> l:=[[1,2],[1,3],[1,4],[2,1],[2,3],[2,4],[3,1],[3,2],[3,4]];;
    gap> Set(List(l, i-> e[i[1]]*e[i[2]]));
    [ 0*X(12) ]

115.30 LeftIndecomposableProjectives

LeftIndecomposableProjectives

returns the list of left ideals Ae, where e runs over the list Idempotents(A). 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",3));
    SolomonAlgebra(CoxeterGroup("B",3),Rationals)
    gap> proj:=LeftIndecomposableProjectives(A);;
    gap> List(proj,Dimension);
    [ 2, 1, 1, 1, 1, 1, 1 ]

115.31 CartanMatrix

CartanMatrix(A)

returns the Cartan matrix of A that is, the matrix dim Hom(P,Q), where P and Q run over the list LeftIndecomposableProjectives(A). 

    gap> A:=SolomonAlgebra(CoxeterGroup("B",4));
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> CartanMatrix(A);

           1
           2        1  2     1  1
           3  1  2  2  3     2  3  1  2
           4  3  3  3  4  2  4  4  2  4  1  0

    1234   1  .  .  .  .  .  .  .  .  .  .  .
      13   1  1  .  .  .  .  .  .  .  .  .  .
      23   1  .  1  .  .  .  .  .  .  .  .  .
     123   .  .  .  1  .  .  .  .  .  .  .  .
     234   .  .  .  .  1  .  .  .  .  .  .  .
       2   .  .  .  1  1  1  .  .  .  .  .  .
     124   .  .  .  .  .  .  1  .  .  .  .  .
     134   .  .  .  .  .  .  .  1  .  .  .  .
      12   .  .  .  .  .  .  .  .  1  .  .  .
      24   .  .  .  .  .  .  .  .  .  1  .  .
       1   .  .  .  .  .  .  .  .  .  .  1  .
       0   .  .  .  .  .  .  .  .  .  .  .  1

115.32 PolynomialQuotientAlgebra

An example - quotient by polynomial algebras

PolynomialQuotientAlgebra(P)

Given a polynomial P with coefficients in K, A:=PolynomialQuotientAlgebra(P) returns the algebra A=K[X]/(P(X)). Note that the class of a polynomial Q is printed as Class(Q) and that A is endowed with the field A.class: this function sends a polynomial Q to its image in A. 

    gap> q:=X(Rationals);; q.name:="q";;
    gap> P:=1+2*q+q^3;;
    gap> A:=PolynomialQuotientAlgebra(P);
    Rationals[q]/(q^3 + 2*q + 1)
    gap> x:=A.basis[3];
    Class(q^2)
    gap> x^2;
    Class(-2*q^2 - q)
    gap> 3*x - A.one;
    Class(3*q^2 - 1)
    gap> A.class(q^6);
    Class(4*q^2 + 4*q + 1)

115.33 GroupAlgebra

GroupAlgebra(G,K)

returns the group algebra K[G] of the finite group G over K. If K is not given, then the program takes for K the field of rational numbers. The i-th element in the list of elements of G is printed by default as e(i). This function endows G with G.law containing the multiplication table of G.

115.34 Augmentation

Augmentation(x)

returns the image of the element x of K[G] under the augmentation morphism. 

    gap> G:=SL(3,2);;
    gap> A:=GroupAlgebra(G);
    GroupAlgebra(SL(3,2),Rationals)
    gap> A.dimension;
    168
    gap> A.basis[5]*A.basis[123];
    e(87)
    gap> (A.basis[3]-A.basis[12])^2;
    e(55) - e(59) - e(148) + e(158)
    gap> Augmentation(last);
    0

115.35 GrothendieckRing

GrothendieckRing(G,K)

returns the Grothendieck ring K ⊗ Z Irr G. The i-th irreducible ordinary character is printed as X(i). This function endows G with G.tensorproducts containing the table of tensor products of irreducible ordinary characters of G.

115.36 Degree for elements of Grothendieck rings

Degree(x)

returns the image of the element x of GrothendieckRing(G,K) under the morphism of algebras sending a character to its degree (viewed as an element of K). 

    gap> G:=SymmetricGroup(4);
    Group( (1,4), (2,4), (3,4) )
    gap> Display(CharTable(G));

         2  3  2  3  .  2
         3  1  .  .  1  .

           1a 2a 2b 3a 4a
        2P 1a 1a 1a 3a 2b
        3P 1a 2a 2b 1a 4a

    X.1     1  1  1  1  1
    X.2     1 -1  1  1 -1
    X.3     2  .  2 -1  .
    X.4     3 -1 -1  .  1
    X.5     3  1 -1  . -1

    gap> A:=GrothendieckRing(G);
    GrothendieckRing(Group( (1,4), (2,4), (3,4) ),Rationals)
    gap> A.basis[4]*A.basis[5];
    X(2) + X(3) + X(4) + X(5)
    gap> Degree(last);
    9

115.37 Solomon algebras

Let (W,S) be a finite Coxeter group. If w is an element of W, let R(w)={s ∈ S | l(ws) > l(w)}. If I is a subset of S, we set YI={w ∈ W | R(w)=I}, XI={w ∈ W | R(w) ⊃ I}.

Note that XI is the set of minimal length left coset representatives of W/WI. Now, let yI=∑w ∈ YI w, xI=∑w ∈ XI w.

They are elements of the group algebra ZW of W over Z. Now, let

Σ(W) = ⊕I ⊂ S ℤ yI = ⊕I ⊂ S ℤ xI.
This is a sub-Z-module of ZW. In fact, Solomon proved that it is a sub-algebra of ZW. Now, let K(W) be the Grothendieck ring of W and let θ:Σ(W)→ K(W) be the map defined by θ(xI) = IndWIW 1. Solomon proved that this is an homomorphism of algebras. We call it the Solomon homomorphism.

115.38 SolomonAlgebra

SolomonAlgebra(W,K)

returns the Solomon descent algebra of the finite Coxeter group (W,S) over K. If S=[s1,...,sr], the element xI corresponding to the subset I=[s1,s2,s4] of S is printed as X(124). Note that A:=SolomonAlgebra(W,K) is endowed with the following fields:

A.group:
the group W

A.basis:
the basis (xI)I ⊂ S.

A.xbasis:
the function sending the subset I (written as a number: for instance 124 for [s1,s2,s4]) to xI.

A.ybasis:
the function sending the subset I to yI.

A.injection:
the injection of A in the group algebra of W, obtained by calling SolomonAlgebraOps.injection(A).

Note that SolomonAlgebra(W,K) endows W with the field W.solomon which is a record containing the following fields:

W.solomon.subsets:
the set of subsets of S

W.solomon.conjugacy:
conjugacy classes of parabolic subgroups of W (a conjugacy class is represented by the list of the positions, in W.solomon.subsets, of the subsets I of S such that WI lies in this conjugacy class).

W.solomon.mackey:
essentially the structure constants of the Solomon algebra over the rationals. 

    gap> W:=CoxeterGroup("B",4);
    CoxeterGroup("B",4)
    gap> A:=SolomonAlgebra(W);
    SolomonAlgebra(CoxeterGroup("B",4),Rationals)
    gap> X:=A.xbasis;;
    gap> X(123)*X(24);
    2*X(2) + 2*X(4)
    gap> SolomonAlgebraOps.injection(A)(X(123));
    e(1) + e(2) + e(3) + e(8) + e(19) + e(45) + e(161) + e(361)
    gap> W.solomon.subsets;
    [ [ 1, 2, 3, 4 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ],
      [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ], [ 1 ], [ 2 ],
      [ 3 ], [ 4 ], [  ] ]
    gap> W.solomon.conjugacy;
    [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7, 8 ], [ 9, 11 ], [ 10 ],
      [ 12 ], [ 13, 14, 15 ], [ 16 ] ]

115.39 Generalized Solomon algebras

In this subsection, we refer to the paper BH05.

If n is a non-zero natural number, we denote by Wn the Weyl group of type Bn and by W-n the Weyl group of type An-1 (isomorphic to the symmetric group of degree n). If C=[i1,...,ir] is a signed composition of n, we denote by WC the subgroup of Wn equal to WC = Wi1 x ... x Wir. This is a subgroup generated by reflections (it is not in general a parabolic subgroup of Wn). Let XC = {x ∈ WC | l(xw) ≥ l(x) ∀ w ∈ WC}. Note that XC is the set of minimal length left coset representatives of Wn/WC. Now, let xC=∑w ∈ XC w. We now define Σ'(Wn) = ⊕C ℤ xC, where C runs over the signed compositions of n. By BH05, this is a subalgebra of ZWn. Now, let YC be the set of elements of XC which are not in any other XD and let yC=∑w ∈ YC w. Then Σ'(Wn) = ⊕C ℤ yC. Moreover, the linear map θ' : Σ'(Wn) → K(Wn) defined by θ'(xC) = IndWCWn 1 is a surjective homomorphism of algebras (see BH05). We still call it the Solomon homomorphism.

115.40 GeneralizedSolomonAlgebra

GeneralizedSolomonAlgebra(n,K)

returns the generalized Solomon algebra Σ'(Wn) defined above. If C is a signed composition of n, the element xC is printed as X(C) Note that A:=GeneralizedSolomonAlgebra(n,K) is endowed with the following fields:

A.group:
the group CoxeterGroup("B",n)

A.xbasis:
the function sending the signed composition C to xC.

A.ybasis:
the function sending the signed composition C to yC.

A.injection:
the injection of A in the group algebra of W.

Note that GeneralizedSolomonAlgebra(W,K) endows W with the field W.generalizedsolomon which is a record containing the following fields:

W.generalizedsolomon.signedcompositions:
the set of signed compositions of n

W.generalizedsolomon.conjugacy:
conjugacy classes of reflection subgroups WC of W (presented as sublists of [1..2*3^(n-1)] as in the classical Solomon algebra case).

W.generalizedsolomon.mackey:
essentially the structure constants of the generalized Solomon algebra over the rationals. 

    gap> A:=GeneralizedSolomonAlgebra(3);
    GeneralizedSolomonAlgebra(CoxeterGroup("B",3),Rationals)
    gap> W:=A.group;
    CoxeterGroup("B",3)
    gap> W.generalizedsolomon.signedcompositions;
    [ [ 3 ], [ -3 ], [ 1, 2 ], [ 2, 1 ], [ 2, -1 ], [ -1, 2 ], [ 1, -2 ],
      [ -2, 1 ], [ -1, -2 ], [ -2, -1 ], [ 1, 1, 1 ], [ 1, -1, 1 ], [ 1, 1, -1 ],
      [ -1, 1, 1 ], [ 1, -1, -1 ], [ -1, 1, -1 ], [ -1, -1, 1 ], [ -1, -1, -1 ] ]
    gap> W.generalizedsolomon.conjugacy;
    [ [ 1 ], [ 2 ], [ 3, 4 ], [ 5, 6 ], [ 7, 8 ], [ 9, 10 ], [ 11 ],
      [ 12, 13, 14 ], [ 15, 16, 17 ], [ 18 ] ]
    gap> X:=A.xbasis;
    function ( arg ) ... end
    gap> X(2,1)*X(1,-2);
    X(1,-2)+X(1,-1,1)+X(1,1,-1)+X(1,-1,-1)

115.41 SolomonHomomorphism

SolomonHomomorphism(x)

returns the image of the element x of A=SolomonAlgebra(W,K) or A=GeneralizedSolomonAlgebra(n,K) in GrothendieckRing(W,K) under Solomon homomorphism. 

    gap> A:=GeneralizedSolomonAlgebra(2);
    GeneralizedSolomonAlgebra(CoxeterGroup("B",2),Rationals)
    gap> Display(CharTable(A.group));
    B2
         2   3   2   3   2   2

           11. 1.1 .11  2.  .2
        2P 11. 11. 11. 11. .11

    11.      1   1   1  -1  -1
    1.1      2   .  -2   .   .
    .11      1  -1   1  -1   1
    2.       1   1   1   1   1
    .2       1  -1   1   1  -1

    gap> A.basis[3]*A.basis[2];
    -X(1,-1)+X(-1,1)+X(-1,-1)
    gap> SolomonHomomorphism(last);
    X(1)+2*X(2)+X(3)+X(4)+X(5)

115.42 ZeroHeckeAlgebra

ZeroHeckeAlgebra(W)

This constructs the 0-Hecke algebra of the finite Coxeter group W. 

    gap> W:=CoxeterGroup("B",2);
    CoxeterGroup("B",2)
    gap> A:=ZeroHeckeAlgebra(W);
    ZeroHeckeAlgebra(CoxeterGroup("B",2))
    gap> Radical(A);
    TwoSidedIdeal(ZeroHeckeAlgebra(CoxeterGroup("B",2)),
    [ T(21)-T(12), T(21)-T(212), T(21)-T(121), T(21)-T(1212) ])

115.43 Performance

We just present here some examples of computations with the above programs (on a usual PC: 2 GHz, 256 Mo).

Constructing the group algebra of a Weyl group of type F4 (1124 elements): 4 seconds 

    gap> W:=CoxeterGroup("F",4);
    CoxeterGroup("F",4)
    gap> A:=GroupAlgebra(W);
    GroupAlgebra(CoxeterGroup("F",4),Rationals)
    gap> time;
    4080

Constructing the Grothendieck ring of the Weyl group of type E8 (696 729 600 elements, 112 irreducible characters): 5 seconds 

    gap> W:=CoxeterGroup("E",8);
    CoxeterGroup("E",8)
    gap> A:=GrothendieckRing(W);
    GrothendieckRing(CoxeterGroup("E",8),Rationals)
    gap> time;
    5950

Computing with the Solomon algebra of the Weyl group of type E6

(51 840 elements):

• Constructing the algebra:

less than 5 seconds

• Computing the Loewy length: 1 second

• Computing the Cartan Matrix: around 12 seconds 

    gap> W:=CoxeterGroup("E",6);
    CoxeterGroup("E",6)
    gap> A:=SolomonAlgebra(W);
    SolomonAlgebra(CoxeterGroup("E",6),Rationals)
    gap> time;
    4610
    gap> LoewyLength(A);
    5
    gap> time;
    1060
    gap> CartanMatrix(A);

             1
             2  1  1  1  1
             3  2  2  2  3  1  1  1  1                    2
             4  3  3  3  4  2  2  2  3  1  1  1           3
             5  4  4  5  5  3  3  4  5  2  2  3  1  1     4
             6  5  6  6  6  4  5  5  6  3  5  4  2  3  1  5  0

    123456   1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
     12345   1  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
     12346   1  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .
     12356   .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .
     13456   .  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .  .
      1234   1  1  .  .  1  1  .  .  .  .  .  .  .  .  .  .  .
      1235   2  .  1  1  .  .  1  .  .  .  .  .  .  .  .  .  .
      1245   1  .  1  .  1  .  .  1  .  .  .  .  .  .  .  .  .
      1356   .  .  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .
       123   2  1  1  .  2  1  .  1  1  1  .  .  .  .  .  .  .
       125   1  1  1  1  .  .  .  .  .  .  1  .  .  .  .  .  .
       134   1  1  .  .  1  1  .  .  .  .  .  1  .  .  .  .  .
        12   2  1  1  .  1  1  .  1  1  1  .  .  1  .  .  .  .
        13   1  1  .  .  1  1  .  .  .  .  .  .  .  1  .  .  .
         1   1  1  .  .  1  1  .  .  .  .  .  1  .  .  1  .  .
      2345   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1  .
         0   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  1

    gap> time;
    12640
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gap3-jm
27 Nov 2023