Algebras

This is a port of the GAP3 package Algebras by Cédric Bonnafé.

isabelian(A::FiniteDimAlgebra) whether A is commutative

GroupAlgebra(G,T=Int) group algebra of G with coefficients T

'SolomonAlgebra(W,K)'

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 Y_I={w ∈ W | R(w)=I}, X_I={w ∈ W | R(w) ⊃ I}.

Note that X_I is the set of minimal length left coset representatives of W/W_I. Now, let y_I=∑_{w ∈ Y_I} w, x_I=∑_{w ∈ X_I} w.

They are elements of the group algebra ℤ W of W over Z. Now, let $Σ(W) = ⊕_{I ⊂ S} ℤ y_I = ⊕_{I ⊂ S} ℤ x_I$. 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(x_I) = Ind_{W_I}^W 1. Solomon proved that this is an homomorphism of algebras. We call it the Solomon homomorphism.

returns the Solomon descent algebra of the finite Coxeter group (W,S) over K. If S=[s₁,…,sᵣ], the element x_I 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.W': the group W

'A.basis': the basis (x_I)_{I ⊂ S}.

'A.xbasis': the function sending the subset I (written as a number: for instance 124 for [s_1,s_2,s_4]) to x_I.

'A.ybasis': the function sending the subset I to y_I.

'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.solomonconjugacy': 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.

julia> W=coxgroup(:B,4)
B₄

julia> A=SolomonAlgebra(W)
SolomonAlgebra(B₄,Int64)

julia> X=A.xbasis; X(1,2,3)*X(2,4)
2X₂+2X₄

julia> W.solomon_subsets
16-element Vector{Vector{Int64}}:
 [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]
 []

julia> W.solomon_conjugacy
12-element Vector{Vector{Int64}}:
 [1]
 [2]
 [3]
 [4]
 [5]
 [6]
 [7, 8]
 [9, 11]
 [10]
 [12]
 [13, 14, 15]
 [16]

julia> Algebras.injection(A)(X(1,2,3))
e_+e₄+e₃₄+e₂₃₄+e₁₂₃₄+e₂₁₂₃₄+e₃₂₁₂₃₄+e₄₃₂₁₂₃₄