Let R be a root system in the real vector space V as in Chapter~refRoot systems and finite Coxeter groups. Let F_0 be an automorphism of V. We say that F_0 is an automorphism of R if it is of finite order, if F_0 preserves the set of roots Rsubset V and if the adjoint automorphism F_0^ast of Vdual preserves the set of coroots Rdualsubset Vdual.
An automorphism F_0 of R normalizes the reflection group W:=W(R). More precisely it induces an automorphism F:;W rightarrow W, defined by wmapsto F_0wF_0^{-1}. A Coxeter coset is the coset WF_0 in the group of automorphisms of V, generated by W and F_0.
Let Deltasubset R be a set of simple roots of R. Then the set Delta F_0 is again a set of simple roots. So there is a unique element w_1 in W such that phi = w_1F_0 stabilizes Delta.
A subset C of a Coxeter coset WF_0 is called a conjugacy class if one of the following three equivalent conditions is fulfilled: item C is the orbit of an element in WF_0 under the conjugation action of W. item C is a conjugacy class of < W,F_0 > contained in WF_0. item The set {w in Wmid;wF_0 in C} is an F-conjugacy class of W (two elements v,w in W are called F-conjugate, if and only if there exists x in W with v = xwF(x^{-1})).
Let us consider the map from the set of irreducible characters of < W,F_0 > to the set of characters of W given by restriction. We get the following facts from Clifford theory: The characters having a nonzero value on a class in the coset WF_0 are exactly those characters which restrict to irreducible ones on W. The irreducible characters of W in the image of this map are exactly those which are fixed under the canonical action of F. Let chi in hat{W} be such a character and chi_1,ldots,chi_r the characters of < W,F_0 > which restrict to chi. Then r is the smallest positive integer such that F_0^r in W and the restrictions of the chi_i, i=1,ldots,r to the conjugacy classes of WF_0 are equal up to scalar multiplication with r-th roots of unity.
In CHEVIE we choose (following Lusztig) for each F-stable character
of W a single (not canonical) extension to a character of <
W,F_0 > , which we will call a preferred extension. The table of
the restrictions of the preferred extensions to the coset WF_0 is
called the character table of the coset WF_0. (See also the
subsequent section on CharTable for Coxeter cosets.)
We define a scalar product on the class functions of a Coxeter coset WF_0 by [langle chi,psi rangle := frac1mid Wmidsum_win W chi(wF_0)barpsi(wF_0).] Then the character table of WF_0 contains an orthonormal set of class functions on WF_0.
A subcoset W_1w_0F_0 of WF_0 is given by a reflection subgroup W_1 of W and an element w_0 of W such that w_0F_0 is an automorphism of the root system R_1 of W_1.
We then have a natural notion of restriction of class functions on WF_0 to class functions on W_1w_0F_0 as well as of induction in the other direction. These maps are adjoint with respect to the scalar product defined above (see BMM93, p.15). The question of finding the conjugacy classes and character table of a Coxeter coset can be reduced to the case of irreducible root systems R: First we assume that F_0=phi, i.e., F_0 fixes a chosen set Delta of simple roots, and consider the canonical monomorphism < W,F_0 > rightarrow Wrtimes < F> , where F is the automorphism of W defined above. It is clear that this map induces a bijection from WF_0 to WFsubset Wrtimes < F> which preserves conjugacy classes in the cosets. The preferred extensions are defined such that they factorize over this map. Let W_1,W_2leq W be F-stable reflection subgroups of W such that W=W_1times W_2 and let F_i the restriction of F on W_i, i=1,2. Here the preferred extension is defined via restriction to the image of Wrtimes < F> hookrightarrow (W_1rtimes < F_1> ) times (W_2rtimes < F_2> ). Thus we can reduce the determination of conjugacy classes and the character table of WF_0 to the case were W is decomposed in irreducible components W=W_1times cdots times W_k which are cyclically permuted by F. In this case there are natural bijections from the F-conjugacy classes of W to the F^k-conjugacy classes of W_1 as well as from the F-stable characters of W to the F^k-stable characters of W_1. The definition of preferred extensions on WF can be reduced to the definition of preferred extensions for W_1F^k. So, we are reduced to the case that W is the Coxeter group of an irreducible root system and F permutes the simple roots, hence induces a graph automorphism on the corresponding Dynkin diagram. If F=1 then conjugacy classes and characters coincide with those of the Coxeter group W. The nontrivial cases to consider are (the order of F is written as left exponent to the type): ^2A_n, ^2D_n, ^3D_4, ^2E_6 and ^2I_2(2k+1). (Note that the exceptional automorphisms of order 2 which permute the Coxeter generators of the Coxeter groups of type B_2, G_2, F_4 or I_2(2k) do not come from automorphisms of the underlying root systems.)
In case ^3D_4 the group Wrtimes < F> can be embedded into the Coxeter group of type F_4, which induces a labeling for the conjugacy classes of the coset. The preferred extension is chosen as the (single) extension with rational values. In case ^2D_n the group Wrtimes < F> is isomorphic to a Coxeter group of type B_n. This induces a canonical labeling for the conjugacy classes of the coset and allows to define the preferred extension in a combinatorial way using the labels (pairs of partitions) for the characters of the Coxeter group of type B_n. In the remaining cases the group Wrtimes < F> is in fact isomorphic to a direct product Wtimes < w_0F> where w_0 is the longest element of W. So, there is a canonical labeling of the conjugacy classes and characters of the coset by those of W. The preferred extensions are defined by describing the signs of the character values on w_0F.
In GAP we construct the Coxeter coset by starting from a Coxeter datum
specified by the matrices of simpleRoots and simpleCoroots, and
giving in addition the matrix F0Mat of the map F_0:V rightarrow V
(see the commands CoxeterCoset and CoxeterSubCoset). As it is true
for the Coxeter groups the elements of WF_0 are uniquely determined by
the permutation they induce on the set of roots R. We consider these
permutations as Elements of the Coxeter coset.
Coxeter cosets are implemented in GAP by a record which points to a
Coxeter datum record and has additional fields holding F0Mat, F0Perm
and the corresponding reduced element phi. Functions on the coset work
with elements of the group coset W F_0 (for example,
ChevieClassInfo); however, most definitions for elements of untwisted
Coxeter groups apply without change to elements in W F_0: e.g., if we
define the length of an element wphi in W F_0 as the number of
positive roots it sends to negative ones, it is the same as the length of
w, i.e., phi is of length 0, since phi has been chosen to
preserve the set of positive roots. Similarly, the CoxeterWord
describing wphi is the same as the one for w, etcldots
We associate to a Coxeter coset WF_0 a twisted Dynkin diagram,
consisting of the Dynkin diagram of W and the graph automorphism
induced by phi on this diagram (this specifies the group
Wrtimes < F> , mentioned above, up to isomorphism). See the
commands CartanType, CartanName and PrintDynkinDiagram for Coxeter
cosets.
The motivation for introducing this notion comes from the theory of Chevalley groups, or more generally reductive algebraic groups over finite fields. Let bG be a reductive algebraic group over the algebraic closure overlineFF_q of a finite field FF_q, which is defined over FF_q, with corresponding Frobenius endomorphism F, so the finite group of rational points bG(FF_q) identifies to the subgroup bG^F of fixed points under F. Let bT be an F-stable maximal torus of bG. The Weyl group of bG with respect to bT is the quotient W=N_bG(bT)/bT. Let X(bT) be the character group of bT, that is the group of rational homomorphisms bTtooverline FF_q. The group W acts naturally on X(bT), and thus also on the vector space V=RR otimes X(bT). It is a fundamental fact in the theory of reductive groups that there is a canonical root system Phi in V defined by (bG,bT) such that W is the Weyl group of that root system.
The Frobenius endomorphism F acts also naturally on X(T) and defines thus an endomorphism of V, which is of the form q F_0, where F_0 is an automorphism of finite order of V. We get thus a Coxeter datum (V,W, F_0).
To completely specify bG^F up to isomorphism, we need a little more information. Let Y(bT) be the group of cocharacters of bT, that is of rational homomorphisms overlineFF_qtobT. The ZZ-modules X(bT) and Y(bT) are naturally dual to each other, and there is a canonical root system Phidual in Y(bT) dual to Phi. Let Vdual= RR otimes Y(bT). The classification theorems on reductive groups show that the isomorphism type of bG^F is completely determined by the datum (V,Phi,Vdual,Phidual, F_0), and the integer q. Thus we can think of this datum as a way of representing in GAP the essential information which determines a Chevalley group. Indeed, all properties of Chevalley groups can be computed from its Weyl datum: symbols representing unipotent characters, conjugacy classes, and finally the whole character table of bG^F.
It turns out that an interesting part of the objects attached to this datum depends only on (V,W, F_0): the order of the maximal tori, the ``fake degrees", the order of bG^F, Deligne-Lusztig induction in terms of ``almost characters", symbols representing unipotent characters and Deligne-Lusztig induction, etcldots (see, e.g., BMM93). It is thus possible to extend their construction to non-crystallographic groups (or even to more general complex reflection groups); this is why we did not include Phi in the definition of a Coxeter coset.
However, in GAP we will always have the whole datum
(V,Phi,Vdual,Phidual, F_0). We assume that we have chosen a Borel
subgroup of bG containing bT. This defines an order on the roots,
and thus a basis Pi of Phi as well as a basis Pidual of
Phidual. Any element of W F_0 induces a permutation of the roots,
and there is a unique element phi in W F_0, which we call the reduced
element in the coset, which preserves the set of positive roots. This
element is stored in the component phi of the coset, and can be used to
test isomorphism of cosets. The coset W F_0 is completely defined by
the permutation F0Perm of the roots induced by F_0 when bG is
semi-simple (in which case Phi generates V). So in this case we
just need to specify F0Perm when defining the coset.
We should mention also a special case of Chevalley groups which does not exactly fit the above description: the Ree and Suzuki groups. In these cases, the group is defined as bG^F where F is not a Frobenius endomorphism, but an isogeny such that either F^2 or F^3 is a Frobenius endomorphism. Here, F still defines an endomorphism of V which normalizes W (and induces the automorphism of order 2 of W --- W is of type G_2, B_2 or F_4), but this endomorphism is no longer q times an endomorphism of finite order; however, up to some power of q, F_0 still takes a form independent of q (but here, q is a power of a fixed prime p equal to 2 or 3 depending on the group considered). This has not yet been implemented.
GAP 3.4.4