Classtypes

closed_subsystems(W)

W should be a Weyl group. The function returns the Poset of closed subsystems of the root system of W. Each closed subsystem is represented by the list of indices of its simple roots. If W is the Weyl group of a reductive group 𝐆, then closed subsystem correspond to reductive subgroups of maximal rank. And all such groups are obtained this way, apart from some exceptions in characteristics 2 and 3 (see Malle-Testerman 2011 Proposition 13.4).

julia> W=coxgroup(:G,2)
G₂

julia> closed_subsystems(W)
1 2<1 4<4<∅
1 2<1 5<1<∅
1 2<2 6<6<∅
1 2<3 5<5<∅
1 4<1
1 5<6
1 5<5
2 6<2<∅
3 5<3<∅

ClassTypes(G[,p])

G should be a root datum or a twisted root datum representing a finite reductive group $𝐆 ^F$ and p should be a prime. The function returns the class types of G in characteristic p (in good characteristic if p is omitted). Two elements of $𝐆 ^F$ have the same class type if their centralizers are conjugate. If su is the Jordan decomposition of an element x, the class type of x is determined by the class type of its semisimple part s and the unipotent class of u in $C_𝐆 (s)$.

The function ClassTypes is presently only implemented for simply connected groups, where $C_𝐆 (s)$ is connected. This section is a bit experimental and may change in the future.

ClassTypes returns a struct which contains a list of classtypes for semisimple elements, which are represented by subspets and contain additionnaly information on the unipotent classes of $C_𝐆 (s)$.

Let us give some examples:

julia> t=ClassTypes(rootdatum(:sl,3))
ClassTypes(A₂,good characteristic)
┌──────────┬─────────┐
│C_G(s)    │ |C_G(s)|│
├──────────┼─────────┤
│A₂₍₎=Φ₁²  │      Φ₁²│
│A₂₍₎=Φ₁Φ₂ │     Φ₁Φ₂│
│A₂₍₎=Φ₃   │       Φ₃│
│A₂₍₁₎=A₁Φ₁│   qΦ₁²Φ₂│
│A₂        │q³Φ₁²Φ₂Φ₃│
└──────────┴─────────┘

By default, only information about semisimple centralizer types is returned: the type, and its generic order.

julia> xdisplay(t;unip=true)
ClassTypes(A₂,good characteristic)
┌──────────┬───────────────┐
│C_G(s)    │    u |C_G(su)|│
├──────────┼───────────────┤
│A₂₍₎=Φ₁²  │    1       Φ₁²│
│A₂₍₎=Φ₁Φ₂ │    1      Φ₁Φ₂│
│A₂₍₎=Φ₃   │    1        Φ₃│
│A₂₍₁₎=A₁Φ₁│   11    qΦ₁²Φ₂│
│          │    2       qΦ₁│
│A₂        │  111 q³Φ₁²Φ₂Φ₃│
│          │   21      q³Φ₁│
│          │    3       3q²│
│          │ 3_ζ₃       3q²│
│          │3_ζ₃²       3q²│
└──────────┴───────────────┘

Here we have displayed information on unipotent classes, with their centralizer.

julia> xdisplay(t;nClasses=true)
ClassTypes(A₂,good characteristic)
┌──────────┬─────────────────────────┐
│C_G(s)    │       nClasses  |C_G(s)|│
├──────────┼─────────────────────────┤
│A₂₍₎=Φ₁²  │(q²-5q+2q₃+4)/6       Φ₁²│
│A₂₍₎=Φ₁Φ₂ │       (q²-q)/2      Φ₁Φ₂│
│A₂₍₎=Φ₃   │  (q²+q-q₃+1)/3        Φ₃│
│A₂₍₁₎=A₁Φ₁│       (q-q₃-1)    qΦ₁²Φ₂│
│A₂        │             q₃ q³Φ₁²Φ₂Φ₃│
└──────────┴─────────────────────────┘

Here we have added information on how many semisimple conjugacy classes of 𝐆 ^F have a given type. The answer in general involves variables of the form qₐ which represent gcd(q-1,a).

Finally an example in bad characteristic:

julia> t=ClassTypes(coxgroup(:G,2),2);xdisplay(t;nClasses=true)
ClassTypes(G₂,char. 2)
ClassTypes(G₂,char. 2)
┌──────────┬──────────────────────────────┐
│C_G(s)    │         nClasses     |C_G(s)|│
├──────────┼──────────────────────────────┤
│G₂₍₎=Φ₁²  │(q²-8q+2q₃+10)/12          Φ₁²│
│G₂₍₎=Φ₁Φ₂ │        (q²-2q)/4         Φ₁Φ₂│
│G₂₍₎=Φ₁Φ₂ │        (q²-2q)/4         Φ₁Φ₂│
│G₂₍₎=Φ₆   │    (q²-q-q₃+1)/6           Φ₆│
│G₂₍₎=Φ₃   │    (q²+q-q₃+1)/6           Φ₃│
│G₂₍₎=Φ₂²  │ (q²-4q+2q₃-2)/12          Φ₂²│
│G₂₍₁₎=A₁Φ₁│       (q-q₃-1)/2       qΦ₁²Φ₂│
│G₂₍₁₎=A₁Φ₂│       (q-q₃+1)/2       qΦ₁Φ₂²│
│G₂₍₂₎=Ã₁Φ₁│          (q-2)/2       qΦ₁²Φ₂│
│G₂₍₂₎=Ã₁Φ₂│              q/2       qΦ₁Φ₂²│
│G₂        │                1 q⁶Φ₁²Φ₂²Φ₃Φ₆│
│G₂₍₁₅₎=A₂ │         (q₃-1)/2    q³Φ₁²Φ₂Φ₃│
│G₂₍₁₅₎=²A₂│         (q₃-1)/2    q³Φ₁Φ₂²Φ₆│
└──────────┴──────────────────────────────┘

We notice that if q is a power of 2 such that q≡2 (mod 3), so that q₃=1, some class types do not exist. We can see what happens by giving a specific value to q₃:

julia> xdisplay(t(;q_3=1);nClasses=true)
ClassTypes(G₂,char. 2) q₃=1
┌──────────┬──────────────────────────┐
│C_G(s)    │     nClasses     |C_G(s)|│
├──────────┼──────────────────────────┤
│G₂₍₎=Φ₁²  │(q²-8q+12)/12          Φ₁²│
│G₂₍₎=Φ₁Φ₂ │    (q²-2q)/4         Φ₁Φ₂│
│G₂₍₎=Φ₁Φ₂ │    (q²-2q)/4         Φ₁Φ₂│
│G₂₍₎=Φ₆   │     (q²-q)/6           Φ₆│
│G₂₍₎=Φ₃   │     (q²+q)/6           Φ₃│
│G₂₍₎=Φ₂²  │   (q²-4q)/12          Φ₂²│
│G₂₍₁₎=A₁Φ₁│      (q-2)/2       qΦ₁²Φ₂│
│G₂₍₁₎=A₁Φ₂│          q/2       qΦ₁Φ₂²│
│G₂₍₂₎=Ã₁Φ₁│      (q-2)/2       qΦ₁²Φ₂│
│G₂₍₂₎=Ã₁Φ₂│          q/2       qΦ₁Φ₂²│
│G₂        │            1 q⁶Φ₁²Φ₂²Φ₃Φ₆│
└──────────┴──────────────────────────┘