Reductive algebraic groups and root data

Reductive algebraic groups and root data

Let us fix an algebraically closed field K and let 𝐆 be a connected reductive algebraic group over K. Let 𝐓 be a maximal torus of 𝐆, let X(𝐓) be the character group of 𝐓 (resp. Y(𝐓) the dual lattice of one-parameter subgroups of 𝐓) and Φ (resp Φᵛ) the roots (resp. coroots) of 𝐆 with respect to 𝐓.

Then 𝐆 is determined up to isomorphism by the root datum (X(𝐓),Φ, Y(𝐓),Φᵛ). In algebraic terms, this consists in giving a free ℤ-lattice X=X(𝐓) of dimension the rank of 𝐓 (which is also called the rank of 𝐆), and a root system Φ ⊂ X, and similarly giving the dual roots Φ^⊂ Y=Y(𝐓).

This can be constructed by giving integral matrices as arguments to our function rootdatum. We assume that the canonical basis of the vector space V on which W acts is a ℤ-basis of X, and the rows of the first argument simpleroots(W) of rootdatum are the simple roots Π expressed in this basis of X. Similarly the rows of the second argument simplecoroots(W) describes the simple coroots in the basis of Y dual to the chosen basis of X. The duality pairing between X and Y is the canonical one, that is the pairing between vectors x∈ X and y∈ Y is given by transpose(x)*y. Thus, we must have the relation simplecoroots(W)*permutedims(simpleroots(W))=cartan(W).

The roots need not generate V, so the matrices need not be square. For example, the root datum of the linear group of rank 3 can be obtained as:

julia> W=rootdatum([-1 1 0;0 -1 1],[-1 1 0;0 -1 1])
A₂Φ₁

julia> reflrep(W,W(1))
3×3 Matrix{Int64}:
 0  1  0
 1  0  0
 0  0  1

here the Weyl group is the symmetric group on 3 letters acting by permutation of the basis of X. The dimension of X, size(simpleroots(W),2), is the rank(W)] and the dimension of the subspace generated by the roots, size(simpleroots(W),1), is the semisimplerank(W). In the example the rank is 3 and the semisimple rank is 2.

The default representation obtained as W=coxgroup(:A,2) corresponds to the adjoint algebraic group (the group in its isogeny class with a trivial centre). In this case Φ is a basis of X, so simpleroots(W) is the identity matrix and simplecoroots(W) is the Cartan matrix cartan(W) of the root system.

The form coxgroup(:A,2,sc=true) constructs the semisimple simply connected algebraic group, where simpleroots(W) is the transposed of cartan(W) and simplecoroots(W) is the identity matrix.

An extreme form of root data can be specified more conveniently: when W is the trivial coxgroup() (in this case 𝐆 is a torus), the root datum has no roots, thus is entirely determined by the rank r. The function torus(r) constructs such a root datum (it could be also obtained by giving two 0×r matrices to rootdatum).

Finally, the rootdatum function also understands some familiar names for the algebraic groups for which it gives the results that could be obtained by giving the appropriate simpleroots(W) and simplecoroots(W) matrices:

julia> rootdatum(:gl,3)   # same as the previous example
gl₃

The types of root data which are understood are :gl, :pgl, :sl, :slmod, :tgl :sp, :csp, :psp, :so, :pso, :cso, :halfspin, :gpin, :spin, :E6, :E6sc, :CE6, :E7, :E7sc, :CE7, :E8, :F4, :G2.

The group 𝐆 is semisimple if the rank is equal to the semisimple rank. In this case, things are constrained: the lattice X, having an integral pairing with the coroots, is in the dual lattice of the coroot lattice. This dual lattice is called the weight lattice. The dual basis to the simple coroots is called the fundamental weights. Similarly we have the coweight lattice and the fundamental coweights. In general the lattice X is an intermediate lattice between the root and the weight lattices.

It follows that the finite abelian group obtained by quotienting the coweight lattice by the coroot lattice describes all possibilities. This group is called the fundamental group of the root system; the fundamental group of 𝐆 is the quotient of Y by the coroot lattice. The fundamental group of the root system is also called the adjoint fundamental group since for an adjoint group Y is the coweight lattice. The fundamental group of 𝐆 is a subgroup of the adjoint fundamental group. It turns out that the non-trivial elements of the adjoint fundamental group are in bijection with the minuscule coweights, that is the coweights whose pairing with every root is in -1,0,1 (this bijection is compatible with the group structure, taking coweights modulo the coroots). The function intermediate_group can construct any semisimple group using this.

The fundamental group has another incarnation which can be more convenient.

• The extended simple roots for an irreducible root system is Π̃=Π∪{-α₀} where α₀ is the highest root. In general it is the union for each irreducible component of the extended simple roots.

Via the theory of the affine Weyl group (see affine), the fundamental group is isomorphic to the subgroup of W which stabilizes Π̃. We represent Π̃ by the indices of the roots it contains and the function fundamental_group returns it as a group of permutations of these indices.

torus(rank::Integer)

This function returns the object corresponding to the notion of a torus of dimension rank, a Coxeter group of semisimple rank 0 and given rank. This corresponds to a split torus; the extension to Coxeter cosets is more useful.

julia> torus(3)
Φ₁³

istorus(W) whether W is a torus

radical(W)

A torus of dimension rank(W)-semisimplerank(W)).

The radical datum of a root datum (X,Φ,Y,Φ^∨) is (X/(X∩ ℚ Φ),∅ ,Φ^⟂,∅), a toral datum.

derived_datum(G)

The derived datum of (X,Φ,Y,Φ^∨) is (X/Φ^{∨⟂}, Φ, Y∩ ℚ Φ^∨, Φ^∨) where ⟂ denotes the orthogonal. This function starts with a root datum object G and returns the root datum object corresponding to the derived datum.

SubTorus(W,Y::Matrix{<:Integer})

The function returns the subtorus 𝐒 of the maximal torus 𝐓 of the reductive group represented by the Weyl group W such that Y(𝐒) is the (pure) sublattice of Y(𝐓) generated by the (integral) vectors Y. A basis of Y(𝐒) in Hermite normal form (for easy memebership testing) is computed and stored in the field S.gens of the returned SubTorus struct. A basis of Y(𝐓) adapted to Y(𝐒) is also computed. This means a set of integral vectors, stored in S.complement, is computed such that M=vcat(S.gens,S.complement) is a basis of Y(𝐓) (equivalently M∈GL(Z^{rank(W)}). An error is raised if Y does not define a pure sublattice.

julia> W=coxgroup(:A,4)
A₄

julia> SubTorus(W,[1 2 3 4;2 3 4 1;3 4 1 1])
SubTorus(A₄,[1 0 3 -13; 0 1 2 7; 0 0 4 -3])

julia> SubTorus(W,[1 2 3 4;2 3 4 1;3 4 1 2])
ERROR: not a pure sublattice
Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] Chevie.Weyl.SubTorus(::FiniteCoxeterGroup{Perm{Int16},Int64}, ::Matrix{Int64}) at /home/jmichel/julia/Chevie.jl/src/Weyl.jl:1082
 [3] top-level scope at REPL[25]:1

fundamental_group(W;full=false)

This function returns the fundamental group of the algebraic group defined by the root datum W; for an adjoint root datum this is the subgroup of W which stabilises the extended root basis (the set of simple roots enriched by the lowest root of each irreducible component). By default, this is returned as a group of permutations of the indices of the roots of the extended root basis. If full=true is given, then the result is given as a subgroup of W (thus permuting other roots besides those in the extended root basis).

For a general root datum the result is a subgroup of the group returned for an adjoint root datum. The definition we take of the fundamental group of a (not necessarily semisimple) reductive group is (P∩ Y(𝐓))/Q where P is the coweight lattice (the dual lattice in Y(𝐓)⊗ ℚ of the root lattice) and Q is the coroot latice. The bijection between elements of P/Q and permutations of the extended root basis is explained in the context of non-irreducible groups for example in (Bonnafé, 2005; §3.B).

julia> W=coxgroup(:A,3) # the adjoint group
A₃

julia> fundamental_group(W) # the 12th root is the lowest one
Group((1,2,3,12))

julia> fundamental_group(W;full=true)
Group((1,2,3,12)(4,5,10,11)(6,7,8,9))

julia> W=rootdatum(:sl,4) # the semisimple simply connected group
sl₄

julia> fundamental_group(W)
Group(Perm{Int16}[])

intermediate_group(W, indices...)

returns a rootdatum which represents a semisimple algebraic group between the adjoint group, obtained by a call like rootdatum(:pgl,4) or coxgroup(:A,3), and the simply connected semi-simple group, obtained by a call like rootdatum(:sl,4) or coxgroup(:A,3;sc=true). This group is specified by a subset of the minuscule coweights, the coweights whose scalar product with each root is in -1,0,1 (a coweight is an element of the coweight lattice Pᵛ, the lattice dual to the root lattice). The non-trivial characters of the (algebraic) center of a semi-simple simply connected algebraic group are in bijection with the minuscule coweights, and also in bijection with Pᵛ/Qᵛ where Qᵛ is the coroot lattice. If W is irreducible, the minuscule coweights are part of the basis of the coweight lattice given by the fundamental coweights, which is the dual basis of the simple roots. They can therefore be specified by an index in the Dynkin diagram. The indices of minuscule coweights in the dynkin diagram are indices where the coefficient of the highest root on the simple roots is 1. The constructed intermediate group has lattice Y(𝐓) generated by the root lattice and the given minuscule coweights. If W is not irreducible, a minuscule coweight is a sum of minuscule coweights in different components; an element of indices is thus itself a list, interpreted as representing the sum of the corresponding coweights.

julia> W=coxgroup(:A,3)
A₃

julia> fundamental_group(intermediate_group(W)) # simply connected
Group(Perm{Int16}[])

julia> fundamental_group(intermediate_group(W,2)) # intermediate
Group((1,3)(2,12))

julia> fundamental_group(intermediate_group(W,1)) # adjoint
Group((1,2,3,12))

weightinfo(W)

W is a Coxeter group record describing an algebraic group 𝐆, or a IypeIrred. The function is independent of the isogeny type of 𝐆(so just depends on refltype(W), that is on the root system). It returns a dict with the following keys:

• :minusculeWeights the minuscule weights, described as their position in the list of fundamental weights. For non-irreducible groups, a weight is the sum of at most one weight in each irreducible component. It is represented as the list of its components. For consistency, in the case of an irreducible system, a weight is represented as a one-element list.

• :minusculeCoweights the minuscule coweights, represented in the same manner as the minuscule weights

• :decompositions for each coweight, its decomposition in terms of the generators of the adjoint fundamental group (given by the list of the exponents of the generators). Together with the next field it enables to work out the group structure of the adjoint fundamental group.

• :moduli the list of orders of the generators of the fundamental group.

• :AdjointFundamentalGroup the list of generators of the adjoint fundamental group, given as permutations of the extended root basis.

• :CenterSimplyConnected A list of semisimple elements generating the center of the universal covering of 𝐆

• :chosenAdaptedBasis A basis of the weight lattice adapted to the root lattice. In the basis of the fundamental weights, the root lattice is given by C=transpose(cartan(W)). The property specifying the :chosenAdaptedBasis is that the Hermite normal form of C*chosenAdaptedBasis is almost in Smith normal form (it is diagonal but the diagonal entries may be permuted compared to the Smith normal form; the non-trivial entries are in the positions corresponding to the generators of the fundamental group as indicated by :decompositions).

julia> weightinfo(coxgroup(:A,2)*coxgroup(:B,2))
Dict{Symbol, Array} with 8 entries:
  :moduli                  => [3, 2]
  :minusculeWeights        => [[1, 3], [1], [2, 3], [2], [3]]
  :decompositions          => [[2, 1], [2, 0], [1, 1], [1, 0], [0, 1]]
  :highestroot             => [5, 7]
  :chosenAdaptedBasis      => [1 -1 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
  :minusculeCoweights      => [[1, 4], [1], [2, 4], [2], [4]]
  :CenterSimplyConnected   => Vector{Rational{Int64}}[[1//3, 2//3, 0, 0], [0, 0…
  :AdjointFundamentalGroup => [(1,12,2), (4,14)]

weights(W) fundamental weights in the basis of X(T)

coweights(W) fundamental coweights in the basis of Y(T)