Twisted Tori

The functionality presented here deals with the computation of the twisted tori of a finite group of Lie type.

Note that for a given group G(k), the twisted tori are returned as subgroups of the standard torus of G(K) for the smallest field extension K of k, where this is possible.

For finite fields and an untwisted group of Lie type G(k), a twisted torus Tw(k) of G(k) has the form

Tw(k) = { t∈T(K) | tσ w = t },

where T(K) is the standard K-split torus of G(K), σ is the generator of the Galois group Gal(K:k) and w is an element of the Weyl group of G(k).

TwistedTorusOrder(R, w) : RootDtm, GrpPermElt -> SeqEnum
Given the root datum R and a Weyl group element w, computes the orders of the cyclic components of the twisted torus Tw(k)⊂G(R, k) as sequence of polynomials in q, the order of the field k.
TwistedToriOrders(G) : GrpLie -> SeqEnum
TwistedToriOrders(R) : RootDtm -> SeqEnum
Given a group of Lie type G or a root datum R, takes for every conjugacy class of the Weyl group of G a representative w, and computes TwistedTorusOrder(R, w). Returns the sequence of the lists consisting of TwistedTorusOrder(R, w) and w for every conjugacy class.
TwistedTorus(G, w) : GrpLie, GrpPermElt -> List
Computes the twisted torus Tw(k) of the group of Lie type G for the given element w of the Weyl group of G. Returned is the list consisting of three elements, first of them being the sequence of orders of cyclic parts of the torus, the second being the sequence of generators of the respective orders and the third being w. See [Hal05] for the algorithm used.
TwistedTori(G) : GrpLie -> SeqEnum
Computes one twisted torus Tw(k) of the group of Lie type G for each conjugacy class wW of the Weyl group W of G. A sequence of them is returned. See [Hal05] for the algorithm used.

Example GrpLie_GrpLieTori (H110E21)

We compute all twisted tori of A1(5): 
> G := GroupOfLieType("A1", 5);
> TwistedToriOrders(G);
[ [*
    [
        q - 1
    ],
    Id($)
*], [*
    [
        q + 1
    ],
    (1, 2)
*] ]
> TwistedTori(G);
[ [*
    [ 4 ],
    [ (2)  ],
    Id($)
*], [*
    [ 6 ],
    [ ( k.1^4)  ],
    (1, 2)
*] ]
As we may notice, the second one is contained in the group over the quadratic field extension: 
> Universe($1[2][2]);
$: Group of Lie type A1 over Finite field of size 5^2

Example GrpLie_GrpLieTori2 (H110E22)

These are the orders of the decompositions of all (up to conjugacy) maximal tori of the group G2(q) as polynomials in q: 
> R := RootDatum("G2");
> [ t[1] : t in TwistedToriOrders(R) ];
[
    [  q - 1,  q - 1  ],
    [  q + 1,  q + 1  ],
    [  q^2 - 1        ],
    [  q^2 - 1        ],
    [  q^2 + q + 1    ],
    [  q^2 - q + 1    ]
]
V2.28, 13 July 2023