4. Types C and D
4.2. The classification theorem
The discussion above leads to the following theorem. Slightly abusing the terminology, by a mirror of type D, we mean the top mirror for Ce or an either one for De (i.e., when the corresponding end of the affine Dynkin diagram is of type D).
Theorem 4.2.1. (i) The group cW0 of type Ce can be naturally identified with the
group fW of type Be, where s0 ∈ Wf is interpreted as the generator of Π0 ⊂ Wc0 = fW.
Accordingly, the configurations for the elements from Wf become configurations for cW0 and an arbitrary element of Wc0 can be obtained from a configuration of type Be.
(ii) The resulting elements belong toWf0 of type Ce if and only if the number of the top reflections of the corresponding configuration is even. Furthermore, they belong to Wf0 of
type De if the number of bottom reflections is even too. Given we0 ∈fW0, it can be obtained
from a Be-type configuration where all s0 and sn are included in the events s00 = s0s1s0 and s0
n =snsn−1sn.
(iii) All minimal NGT from Wf0 of type Ce or De come from minimal NGT of type Be
satisfying (ii). However, the latter may become non-minimal in fW0; it occurs only if the horizontal line involved in the NGT is a unique one in its (horizontal) bunch and if the corresponding mirror is of type D.
Proof. The statements from (i, ii) have been already discussed. Concerning Ce, we move all s0 to the beginning (or the end) of a given reduced decomposition of we ∈ Wf, replacing s0s1s0 by s00 when necessary. It will give a word from Wf0 possibly multiplied bys0 on the right (or on the left).
As for De, we can use that the group fW for Be can be naturally identified with the extension of Wf0 of type Be by Z2 ×Z2 generated by the elements s0, sn (pairwise com-
mutative) treated as outer automorphisms of the corresponding affine Dynkin diagram. The element s0 is from cW0, the element s
n is not; both are of zero length by definition.
One can move all such elements in a given reduced Be-decomposition to its beginning or to its end. The elements s0
0 and s0n may be produced during this process. The top (or
bottom) parity corrections are needed if the elementss0 (or sn) do not cancel each other.
Algebraically, (ii) means that for any given we0 ∈ fW0, its reduced decompositions with the minimal possible numbers of s0
0 and s0n remain reduced in Wf, where s00 and s0n
are expressed in terms of s0 and sn. The geometric approach guarantees that at least
one such reduced decomposition exists. The construction we7→we0 (subject to the parity conditions) consists of moving alls0, snto the beginning (or to the end) of a given reduced
Concerning (iii), let us use the right graph in Figure 4.2 to demonstrate what happens in the beginning of the configuration if there is only one line parallel to the corresponding mirror. If (the lowest) line 4 is removed, then the corresponding reduced decomposition reads s2s1s03s1s2. Using the Coxeter D-relation, it can be transformed to s2s03s1s03s2 = s0
3s2s1s2s03. Thus, the beginning of the resulting reduce decomposition is movable (not
unique) and such B-minimal NGT will not remainD-minimal.
This example is actually a general one; it is sufficient to manage arbitrary Cen and
e
Dn if the horizontal line involved in the NGT is near the mirror of type D and is the
only one in its horizontal bunch. Then the right end of the resultingλ-sequence becomes movable upon the switch to Cen or Den in this case (and only in such case).
For instance, let us consider the configuration from Figure 4.1 including the dashed line to 1? and excluding line 6. It represents a minimal NGT for B5e , which will not
remain minimal upon its recalculation to D5e (which is possible because the numbers of top and bottom reflections are both even). It is analogous to the constraint “at least two horizontal bottom lines” from [CS] in the nonaffine D-case.
The following theorem is an explicit form of claim (iii), reformulated in terms of the parity corrections.
Theorem 4.2.2. (i) An arbitrary minimal NGT from fW of type Cen(n ≥ 3) can be obtained as we0, we00 or we∗ as follows.
Let we be a B-positive minimal NGT of type Ben(n ≥ 3). If no top parity correction is needed for we (i.e., the total number of top reflections in we0 is even), then we0 := we, (considered as Ce-words) and, moreover, we00 = ı
C(we) = s0wse 0 are minimal NGT of type
e
Cn. The element we∗ is minimal NGT if it has even number of the top reflections and
the initial we has at least two bottom horizontal lines; we∗ is involutive and coincides with ıC(we∗).
Otherwise, if the parity corrections are needed, we0 :=ws0e (the results of the top-right
parity correction) is a minimal NGT ofCen-type. For the top-left correction, we00:=s0we= 30
ıC(we0) = (we0)−1, so such element can be represented as u˜0 for certain B-positive u˜. The element we∗ defined now as ı
B(we)s0 =s0ıB(we), is a minimal NGT subject to the same “2 line constraint” as above.
(ii) Minimal NGT of typeDare given in terms of theC-wordswe0,we00,we∗ from part (i)
as follows. If no bottom parity correction is needed, i.e., the total number of the bottom reflections in the initial we (the cases of we0 and we00) or ıB(we) (the case of we∗) is even, then each of these elements is a minimal NGT of type Den. We require here the existence of at least two horizontal lines near the bottom for we in the cases of we0 and we00.
Otherwise, if the total number of the bottom reflections for we0, we00 or for we∗ is odd,
then the elements snwe0 = we0sn and snwe00 = we00sn, as well as the elements we∗sn and snwe∗ = (we∗sn)−1 are minimal NGT of D-type. We impose the same constraint as above in the cases of we0,we00, namely, the configuration forwe is supposed to contain at least two
horizontal lines near the bottom.
We note that the operation we 7→ s0ws0e for B-positive we is trivial if we contains at least one top horizontalline. It is obvious geometrically that the configuration fors0ws0e
can be transformed to ensure the cancellation of such two s0 if they are “performed” on the same top horizontal line. Similarly, the transformation we 7→ snwse n is not needed
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