Frank Lübeck

Centralizers and numbers of semisimple classes in exceptional groups of Lie type

The text files which can be accessed below provide information about the conjugacy classes of semisimple elements in exceptional groups of Lie type.

For a connected reductive group G with Frobenius morphism F and group of fixed points G(q) = G^F we collect the semisimple classes of these groups into semisimple class types, where two classes are in the same class type iff their elements have G(q)-conjugate centralizers in G.

Let s in G(q) be semisimple, C be the centralizer of s in G and C⁰ the connected component of C. The group C⁰ is again a connected reductive group and F induces a Frobenius morphism on C and C⁰.

In the tables given here, we parameterize the G(q)-conjugacy classes of centralizers (and so the semisimple class types) by triple indices [i,j,k]. Let [i,j,k] and [i',j',k'] be two such triples. Then i=i' if and only if the corresponding connected centralizers C⁰ are conjugate in the algebraic group G. The first two indices are equal (i=i' and j=j') if and only if the corresponding full centralizers C are conjugate in the algebraic group G. The third index k distinguishes the G(q)-conjugacy classes of the F-stable elements in the G-conjugacy class of C.

We consider an example to explain the information given in these files. This example is from the case ²E₆(q)_ad, the algebraic group G is of adjoint type with root system of type E₆. The file contains for each triple index [i,j,k] a block like:

i = 16: Pi = [ 1, 2, 5 ]
  j = 2: Omega of order 3, action on Pi: <(1,5,2)>
    k = 5: F-action on Pi is (1,2,5)               [16,2,5]
      Dynkin type is (A_1(q^3) + T(phi2 phi6)).3
      Order of center |Z^F|: phi2 times
        1, q congruent 0 modulo 3
        1, q congruent 1 modulo 3
        3, q congruent 2 modulo 3
      Numbers of classes in class type:
        q congruent 1 modulo 6:   0
        q congruent 2 modulo 6:   ( q-2 )
        q congruent 3 modulo 6:   0
        q congruent 4 modulo 6:   0
        q congruent 5 modulo 6:   ( q-3 )
      Fusion of maximal tori of C^F in those of G^F:
        [ 22, 6 ]
      elements of other class types in center:
[ [ 1, 1, 1, 1 ], [ 2, 1, 1, 1 ], [ 3, 2, 3, 2 ], [ 8, 1, 2, 2 ], 
  [ 13, 2, 2, 1 ], [ 14, 2, 2, 1 ] ]

Here the line starting 'i = ' indicates the root system of the corresponding connected centralizer C⁰ as subsystem of the root system of G; in the example it is generated by the simple roots number 1,2 and 5 of the root system of type E₆ (in standard Bourbaki labelling, a bigger number corresponds to the highest negative root). The line starting 'j = ' tells that the component group Ω = C/C⁰ is of order 3 (Ω is isomorphic to a subgroup of the stabilizer of the simple roots Pi of C⁰ and its action on Pi is also given). The line starting 'k = ' indicates the action induced by the Frobenius morphism on the set Pi of simple roots of C⁰ and the following line describes the Dynkin type of the finite group C(q): this contains a description of the Dynkin diagram of the semisimple part of the connected centralizer and the action of the Frobenius on that diagram, in a T(...) part the order coming from the central torus of the connected component (that is |Z(C⁰)⁰(q)|, if not trivial), and if the F-stable subgroup of the component group is not trivial, its size is given as well.

In the given example the connected centralizer C⁰ has semisimple part with Dynkin diagram three components of type A₁ where the Frobenius permutes these components cyclically. The torus Z(C⁰)⁰(q) has order phi2 phi6 (see below), and the component group C/C⁰ is fixed pointwise by the Frobenius.

In the following line Z is the center of the full centralizer C, and the order of the subgroup of F-stable points is given. This order is given as a polynomial evaluated at q. All given polynomials are printed in factorized form, where phii denotes the i-th cyclotomic polynomial evaluated at q. So, in the example Z^F is of order 3 phi2 = 3 (q+1) if q is congruent to 2 mod 3 and the order is phi2 otherwise.

Then the number of semisimple conjugacy classes whose elements have centralizer in G which is G(q)-conjugate to C is given. In the example this number depends on the congruence class of q mod 6 (in particular there are no elements in this semisimple class type if q is not 2 mod 3).

The information about 'Fusion of maximal tori' shows how many C(q)-conjugacy classes of F-stable maximal tori in C^0 exist and how these classes fuse into the classes of maximal tori of G(q) (the numbering of the classes of tori is the same as towards the end of the table where the index i correspond to the empty root system (Pi = [ ]) and j = 1).

Finally, the information on 'elements of other class types in center' indicates which types of semisimple elements are contained in Z(q) = Z^F. This is a list of 4-tuples, like [ 13, 2, 2, 1 ] in the example. Here, the first three entries refer to the conjugacy classes of semisimple elements which have centralizer indexed by [13,2,2], and the fourth entry 1 means that each of these classes have 1 representative in Z(q).

Other groups: D₄(q)_sc = Spin₈(q), D₄(q)_ad, D₆(q)_ad

Last updated: Wed Jan 30 23:22:45 2019 (CET)