Transitions between root subsets associated with Carter diagrams

For any two root subsets associated with two Carter diagrams that have the same $ADE$ type and the same size, we construct the transition matrix that maps one subset to the other. The transition between these two subsets is carried out in some canonical way affecting exactly one root, so that this root is mapped to the minimal element in some root subsystem. The constructed transitions are involutions. It is shown that all root subsets associated with the given Carter diagram are conjugate under the action of the Weyl group. A numerical relationship is observed between enhanced Dynkin diagrams $\Delta(E_6)$, $\Delta(E_7)$ and $\Delta(E_8)$ (introduced by Dynkin-Minchenko) and Carter diagrams. This relationship echoes the $2-4-8$ assertions obtained by Ringel, Rosenfeld and Baez in completely different contexts regarding the Dynkin diagrams $E_6$, $E_7$, $E_8$.


Diagrams with cycles
In 1972, R. Carter introduced the so-called admissible diagrams representing semi-Coxeter elements of conjugacy classes in the Weil group. The admissible diagrams also represent root subsets 1 of the root systems associated with the Weyl group. These root subsets sometimes form strange cycles, strange because the extended Dynkin diagram A l cannot be part of any admissible diagram. The explanation for this fact was that in the case of extended Dynkin diagrams, the inner products of roots of cycle A l are negative, while in the case of admissible diagrams, there are necessarily both positive and negative inner products, see [18,Lemma A.1]. Thus was born the concept of the Carter diagram, see [18]. They differ from admissible diagrams in that they take into account the sign of the inner product on the pair of roots; for this, the language of solid and dotted edges is used, see Section 2.2.
Carter diagrams distinguish between negative inner products (solid edges) and positive inner products (dotted edges). The number of solid edges must necessarily be odd, as well as the number of dotted edges. This agrees well with the fact that, by definition, the length of an admissible diagram is even. In the theorem on exclusion of long cycles [18,Theorem 3.1] it was shown that any Carter diagram with cycles of arbitrary even length can be reduced to another Carter diagram containing only 4-cycles. In proving this theorem, for each particular case, it is checked that a semi-Coxeter element associated with a Carter diagram with long cycles is conjugate to a semi-Coxeter element associated with another Carter diagram containing only cycles of length 4.
The Carter diagrams of the same type and the same index constitute a homogeneous class of Carter diagrams. Denote by C(Γ) the homogeneous class containing the Carter diagram Γ, see (1) and Fig. 1.
Transitions between root subsets associated with Carter diagrams 261 Figure 1: Carter diagrams of D and E types adjacency list is not a complete list, but a minimal list that chains all Carter diagrams of to the same homogeneous class using the transition matrices constructed in Theorem 4.1.

A group of transitions M
Let C(Γ) be a homogeneous class of Carter diagrams out of (1), and S be the set of all root subsets associated with diagrams of C(Γ). Let M be some subgroup of the group generated by transitions of type M I operating on S. The Weyl group W operates simply transitively on the set of bases of the corresponding root system, all bases are associated with a single Dynkin diagram (which is also the Carter diagram). This is not true for the group M : any element M t ∈ M (of type M I ) can map some root subset S 1 ∈ S to another root subset S 2 ∈ S. The root subsets S 1 and S 2 are associated, generally speaking, with different Carter diagrams Γ 1 ∈ C(Γ) and Γ 2 ∈ C(Γ). We will say that S 1 (resp. S 2 ) is a Γ 1 -set (resp. Γ 2 -set).

Theorems on transition matrix
In this paper, we show that each homogeneous class of Carter diagrams essentially depicts the same subset of the root system given in different bases. There are several chains of diagrams containing homogeneous classes of Carter diagrams. The transition between neighboring bases in any chain can be performed in some canonical way, affecting exactly one root, which is mapped to the minimum element in some root subsystem. The transition matrix connecting two adjacent diagrams is an involution.
The main result of this paper can be formulated as follows: Let { Γ, Γ} be a pair out of the adjacency list (2), and let S (resp. S) be a Γ-set (resp. Γ-set). We construct the matrix M I having the following properties ( and M I acts also as involution on S: The values of t i in (3) and (4) are given in Tables 2-5 of Section 4.6.
(d) In most cases of Section 4.6, the mapping M I given in (3) eliminates one circle (or one endpoint), the mapping M I given in (4) builds one circle (or one endpoint). In case (16) of Section 4.6, M I eliminates 3 cycles.
Only ADE root systems are considered. In [19], [20] Vavilov and Migrin combined both types of considered diagrams: Carter diagrams and enhanced Dynkin diagrams, they applied the language of solid and dotted edges to enhanced Dynkin diagrams. The obtained diagrams are called signed enhanced Dynkin diagrams. They showed that any Carter diagram of the homogeneous class containing the Dynkin diagrams E 6 , E 7 , E 8 can be embedded into the signed enhanced Dynkin diagram ∆(Γ) associated with Γ such that the "solid and dotted" correspondence is preserved. For a further discussion of the relationship between enhanced Dynkin diagrams and Carter diagrams, see Section C.1.

McKee-Smyth diagrams: Eigenvalues in (−2, 2)
Much to my surprise, I found a complete list of 8-vertex Carter diagrams with circles in the paper of McKee and Smyth [11,. The {0, 1}-matrices with zeros on the diagonal can be regarded as adjacency matrices of graphs. Assume that the off-diagonal elements of such a matrix to be chosen from the set {−1, 0, 1}. Then, we get so-called a signed graph, a non-zero (α, β)th entry denotes a sign of −1 or 1 on the edge connecting vertices α and β. The signed graphs exactly correspond to our diagrams with solid and dotted edges. The matrix with zeros on the diagonal is called an uncharged matrix. By [11,Theorem 4], the signed graphs maximal with respect to having all their eigenvalues in (−2, 2) are exactly Carter diagrams E 8 (a i ), 1 ≤ i ≤ 8 and D l (a i ), i < l/2 − 1, see Fig. 1. If the diagonal matrix 2I is added to such an uncharged matrix, then exactly partial Cartan matrix will be obtained, see Section 3.2. Then, the eigenvalues of the partial Cartan matrices should lie in the interval (0, 4). Using eigenvalues one can get an invariant description of Carter diagrams, see [16,Section 4.4]. For some details on the relationship between Carter diagrams and eigenvalues of partial Cartan matrices, see Section C.2.
Similar results to [11] were also obtained by Mulas and Stanic in [12].
2 Diagrams containing cycles 2.1 Admissible diagrams: Conjugacy classes of W Let Φ be the root system corresponding to the Weyl group W . Each element w ∈ W can be expressed in the form Carter proved that k in the decomposition (5) is the smallest if and only if the subset of roots {α 1 , α 2 , . . . , α k } is linearly independent; such a decomposition is said to be reduced. The admissible diagram corresponding to the given element w is not unique, since the reduced decomposition of the element w is not unique.
Denote by l C (w) the smallest value k corresponding to any reduced decomposition (5). The corresponding set of roots {α 1 , α 2 , . . . , α k } consists of linearly independent and not necessarily simple roots, see Lemma 2.1. If l(w) is the smallest value k in any expression like (5) such that all roots α i are simple, then l C (w) ≤ l(w). The Cartan matrix (resp. quadratic form) associated with Φ is denoted by B (resp. B). The inner product induced by B is denoted by (·, ·).
Let us take the subset of linearly independent, but not necessarily simple roots S ⊂ Φ. To the subset S we associate some diagram Γ that provides one-to-one correspondence between roots of S and nodes of Γ. The diagram Γ is said to be admissible if the following two conditions hold: (a) The nodes of Γ correspond to a set of linearly independent roots in Φ.
(b) If a subdiagram of Γ is a cycle, then it contains an even number of nodes.
Note that the admissible diagram may contain cycles, since the roots of S are non necessarily simple, see [17, Section 1.2.1]. Let us fix some basis of roots corresponding to the given admissible diagram Γ: The admissible diagram is bicolored, i.e., the set of nodes can be partitioned into two disjoint subsets S α = {α 1 , . . . , α k } and S β = {β 1 , . . . , β h }, where roots of S α (resp. S β ) are mutually orthogonal. The element is called the semi-Coxeter element; it represents the conjugacy class associated with the admissible diagram Γ and root subset S (not necessarily root system).

Carter diagrams: Language of "solid and dotted" edges
In [18], it was observed that the cycles in the admissible diagrams with necessity contains at least one pair of roots {α 1 , β 1 } with (α 1 , β 1 ) > 0 and at least one pair of roots {α 2 , β 2 } with (α 2 , β 2 ) < 0, where (·, ·) is the Tits bilinear form associated with the root system Φ. This observation motivated me to distinguish such pairs of roots: Let us draw the dotted (resp. solid ) edge {α, β} if (α, β) > 0 (resp. (α, β) < 0). The admissible diagram with dotted and solid edges is said to be the Carter diagram. Up to dotted edges, the classification of Carter diagrams coincides with the classification of admissible diagrams.
In the theorem on exclusion of long cycles [18], it was shown that any Carter diagram with cycles of arbitrary even length can be reduced to diagrams with cycles of length 4 only. This explains why the admissible diagrams D l (b1 Table 2] do not appear in the lists of conjugacy classes. The Carter diagrams with conjugate semi-Coxeter elements are said to be equivalent. The Carter diagrams (with cycles) representing non-Coxeter conjugacy classes are given in Fig. 1. For another view of these diagrams, see [18, Table 1].

Carter diagrams: Eliminating the cycle.
The semi-Coxeter elements generated by reflections {s α 1 , s α 2 , s β 1 , s β 2 } constitute exactly two conjugacy classes with representatives w t and w o , see  The element w o is conjugate to the w o = s α 1 s α 2 s −(α 1 +β 1 +β 2 ) s β 2 since The elements w o and w o are conjugate, the corresponding sets of roots are as follows: There is a map M : S 1 −→ S 2 acting as follows: Note that M is an involution, M : S 1 → S 2 and M : S 2 → S 1 , since Thus, M transforms the root β 1 into the minimal element of the root subsystem {α 1 , β 1 , β 2 }. In this paper, we will encounter a number of involution mappings M that map a certain element to the minimal element of some root subsystem of Φ. So, we observe that there are two different orders of reflections: • The cyclic order of reflections o. Then, we get a 4-cycle leading to the Coxeter class • The bicolored order of reflections t. Then, we get an indestructible 4-cycle leading to the semi-Coxeter class D 4 (a 1 ).

Connection diagrams
In [18], in addition to Carter diagrams, the so-called connection diagrams were introduced. Let S be a set of linearly independent and not necessarily simple roots, o be the order of reflections in the decomposition (5) of element w associated with the set of roots In [18], three equivalence transformations operating on the connection diagrams and Carter diagrams were introduced: similarities, conjugations and s-permutations. The Carter diagrams are studied there up to equivalence. In what follows, we only need similarity. Let α be a root in the Γ-set S. The similarity transformation L α reflects the root α: Two connection diagrams obtained from each other by a sequence of reflections (6), are called similar connection diagrams, see Fig. 4.

Bicolored partition
Let Γ be a Carter diagram and be any Γ-set (of not necessarily simple roots), where roots of the set are mutually orthogonal. According to condition 2.1(a), there exists a certain set (7) of linearly independent roots, and thanks to condition 2.1(b), there exists a partition S = S α S β which is said to be the bicolored partition. Let w = w 1 w 2 be the decomposition of w into the product of two involutions. By [4, Lemma 5] each of w 1 and w 2 can be expressed as a product of reflections as follows: be the linearly independent root subset. Then, the decomposition (8) is reduced, see Lemma 2.1, and k + h = l C (w). The decomposition (8) is said to be a bicolored decomposition.

The generalized Cartan matrix
The is called a generalized Cartan matrix, [10], [16, Section 2.1]. For the Carter diagram Γ, which is not a Dynkin diagram, the condition (C2) fails: The elements k ij associated with dotted edges are positive. If the Carter diagram does not contain any cycle, then the Carter diagram is the Dynkin diagram, the corresponding conjugacy class is the conjugacy class of the Coxeter element, and the partial Cartan matrix is the classical Cartan matrix, which is the particular case of a generalized Cartan matrix.

The partial Cartan matrix
Similarly to the Cartan matrix associated with Dynkin diagrams, we determine the Cartan matrix for each pair {Γ, S} consisting of the connection or Carter diagram Γ and Γ-set S: where S = {τ 1 , . . . , τ n }. We call the matrix B Γ a partial Cartan matrix corresponding to the diagram Γ. The partial Cartan matrix B Γ is well-defined since products (τ i , τ j ) in (10) do not depend on the choice of the Γ-set S. The elements of the partial Cartan matrix are uniquely determined by the diagram Γ as follows: Let L be the subspace spanned by the vectors {τ 1 , . . . , τ n }. We write this fact as follows: The subspace L is said to be the S-associated subspace. Let B be the Cartan matrix corresponding to the primary root system Φ.
(ii) For every Carter diagram, the matrix B Γ is positive definite.
If Γ is a Dynkin diagram, the partial Cartan matrix B Γ is the Cartan matrix associated with Γ. By (13) the matrix B Γ is positive definite. The symmetric bilinear form associated with B Γ is denoted by (·, ·) Γ and the corresponding quadratic form is denoted by B Γ . For example, this is so in the case of Lorentzian algebras, see [8], [5]. However, in these cases the Cartan matrices are of hyperbolic type, whereas the partial Cartan matrices are positive definite.
(ii) I would like to quote S. Brenner's article: ". . . it is amusing to note that there is a surprisingly large intersection between the finite type quivers with commutativity conditions and the diagrams by Carter in describing conjugacy classes of the classical Weyl groups . . . ", [3, p.43]. On various other cases arising in the representation theory of quivers, algebras and posets with Cartan matrices containing positive offdiagonal elements, see [2], [7, 10.7], [15].

First transition theorem
Let { Γ, Γ} be a homogeneous pair of Carter diagrams, S be a Γ-set, and S be a Γ-set. In this section, we construct a mapping connecting S and S. This mapping represents the transition matrix connecting S and S as bases in the linear spaces. The transition matrix has some good properties that are presented in Theorems 4.1 and 4.3. Let Γ ′ be the subdiagram of Γ, and subset S ′ ⊂ S be a Γ ′ -set. If Γ ′ is the Dynkin diagram, we call S ′ the Dynkin subset.
The image S = M I S is the set { S\ α}⊔{α} that satisfies to the orthogonality relations of the Carter diagram Γ.
(ii) The transformation M I : S −→ S is an involution 1 on the set S ⊔ {α}: For each pair of diagrams { Γ, Γ} from list (2), the matrix M I is defined in Tables 2-5 of Section 4.6. The matrix M I is the transition matrix transforming each basis S into some basis S.
The proof of Theorem 4.1 is given in Section 4.6. It is carried out separately for each pair { Γ, Γ} in the adjacency list (2).

The chain of homogeneous pairs
Let Γ be a Carter diagram. Denote by C( Γ) the homogeneous class containing Γ. For any Carter diagram Γ and the Dynkin diagram Γ from C( Γ), there exists the chain of homogeneous pairs connecting Γ and Γ as follows: This fact follows easily from consideration of the adjacency list (2).
There are 16 cases in Section 4.6. Denote by M I (n) the transition matrix of the n-th case. The similarity transformation L τ i from (6) is the diagonal matrix of the form with −1 in the {i, i}th entry. The homogeneous pairs are bound by matrices M I and similarity matrices L τ i . Consider, for example, the chain diagrams (15) and Section 1.2.
In eq. (15), we mean that instead of each diagram Γ there is some Γ-set. The matrices M I (n) are given in Appendix B. Consider the product of matrices of (15): The matrix F maps the E 8 (a 8 )-basis S to a certain E 8 -basis S = F S: The chain (15) is parallel to the ascending chain of maximal eigenvalues of the corresponding partial Cartan matrices, see Section C.2.

Alternative transitions
The transition matrices from the adjacency list (2) followed by actions of L τ , with τ = α 4 .

The product of transition matrices
As in Section 4.2.1, for any chain (14), one can construct the matrices F and F −1 , where F is the product of corresponding transition matrices M I (n) and similarity matrices L τ i . The matrix F is invertible since all M I (n) and L τ i are invertible.
This means that for any Γ-set S there exists Γ-set S such that 1 The property of the edge to be solid or dotted is called the edge type.
Transitions between root subsets associated with Carter diagrams

273
The matrix F does not depend on S and S. If S = {τ 1 , . . . , τ n }, F transforms τ i so that where f ji are some coefficients that depend only on diagrams Γ and Γ.

Action of the Weyl group on a Carter diagram
We suppose that { Γ, Γ} is the homogeneous pair of Carter diagrams, where Γ is the simply-laced Dynkin diagram, and W is the Weyl group associated with Γ.
Lemma 4.2. Let F , F −1 be the matrices described in (16), (17). Then, (i) The matrix F commutes with the Weyl group W on any Γ-set S as follows: (ii) Let Γ-sets S and S ′ be conjugate by some w ∈ W : wS = S ′ . Then, F S and F S ′ are conjugate by the same element w ∈ W , i.e, Proof. (i) It suffices to prove eq. (19) for each element τ i ∈ S. Each element w ∈ W transforms basis S to another basis S ′ = wS, where wτ i = τ ′ i , and S ′ = {τ ′ 1 , . . . , τ ′ n }. In our case, Therefore, F wτ i = wF τ i for any τ i ∈ S.
(ii) If wS = S ′ then by (i), we have wF S = F wS = F S ′ , i.e., F S and F S ′ conjugate by the same element w ∈ W .  Let S ′ and S ′′ be any Γ-sets. We will prove that S ′ and S ′′ are conjugate under the Weyl group W , i.e., w S ′ = S ′′ , for some w ∈ W.

Second transition theorem
There exist Γ-sets S ′ and S ′′ such that (ii) First, by Theorem 4.1, we transform S to some Γ-set S ′ by the mapping F as in (i), see Fig. 6. Further, as in (i), there exists w ∈ W such that wS ′ = S. Thus, wF S = wS ′ = S.

Conjugacy of all Γ ′ -sets
As above, { Γ, Γ} is a homogeneous pair of Carter diagrams, Γ is the simply-laced Dynkin diagram, W is the Weyl group associated with Γ.
Similarly to the fact that all Γ-bases (where Γ is the Dynkin diagram) are conjugate under the Weyl group [9, Theorem 1.4], the same fact holds for Carter diagrams. In other words, F w S = F S ′ , and w S = S ′ .
• Minimal root: α 3 is the minimal root in Φ.
• Checking relations: (2) Pair {D l (a k ), D l }: M I maps D l (a k )-set to D l -set.
• Mapping: • Root systems: Φ(S 1 ) is a root system of type D k+2 , for k ≥ 2, Φ(S 2 ) is a root system of type A 3 , for k = 1.
• Minimal root: β 3 is the minimal root in Φ.
• Minimal root: α 3 is the minimal root in Φ.
• Minimal root: α 1 is the minimal root in Φ.
• Minimal root: α 3 is the minimal root in Φ.
• Checking relations: (ii) Let us prove that M I is an involution. There exists a certain root α ∈ S such that The image M I α = α is the root in S. Then, In other words, M I : α −→ α and M I : α −→ α.

Relation of partial Cartan matrices
Consider pairs { Γ, Γ} out of the adjacency list (2). The transition matrix M I maps the Γ-basis S to the Γ-basis S: If the matrix M does not change a certain root τ i , the designation of this root and the corresponding node is the same for Γ-basis and Γ-basis, namely: M τ i = τ i . The transition matrix M relates the partial Cartan matrices B Γ and B Γ as follows: Eq. (25) is the relation of partial Cartan matrices B Γ given in different bases. Then, The matrix t M B E 6 (a 1 ) M is the Cartan matrix B E 6 for Dynkin diagram E 6 .
Then, m 2 is also the maximal element in the E 6 -set {α 1 , α 2 , α 3 , α 4 , α 4 , α 6 }. A.2 Enhanced Dynkin diagram ∆(E 7 ) As in the case E 6 , the extra nodes m 1 and m 2 are as follows: Further, the extra node m 3 is as follows: Here, m 3 is the maximal element in the D 6 -set {α 2 , α 3 , α 4 , α 4 α 6 }. The extra node m 4 : The node m 4 is the maximal element in the E 7 -set (see Fig. 22): The numerical relation in Remark 1.1 motivates to the following assumption: There is a correspondence between Carter diagrams (with cycles) and extra nodes in the Dynkin-Minchenko completion procedure.
One more conjecture on relationship between enhanced Dynkin diagrams and Carter diagrams is as follows: Conjecture C.2. If { Γ, Γ} is a homogeneous pair of Carter diagrams, then the signed enhanced Dynkin diagrams associated with Γ and Γ coincide up to similarities.

C.2 Adjacency, complexity and eigenvalues
Let us define the complexity of the Carter diagram as N c+Ke, where N is the number of cycles and K is the number of endpoints. Assume that one cycle contributes to complexity as two endpoints. One can select another proportion. In Table 6, Carter diagrams located side by side are the pairs from the adjacency list (2). The Carter diagrams from the adjacency list can be transformed to each other using the transition matrix M I constructed in Theorem 4.1, see Section 4.2.1. Denote by M ax-E the maximal eigenvalue of a partial Cartan matrix.  Table 6: Two chains arranged in ascending order of M ax-E.
In Table 6, there are two chains which are arranged in ascending order of the maximal eigenvalues of the partial Cartan matrices and in descending order of the complexity parameter, see (28).
The arrows in (28) point in the direction of increasing of maximal eigenvalue. It is not so clear the place of E 8 (a 6 ) in the homogeneous class {E, 8}: • The complexity parameter for the diagram E 8 (a 6 ) is equal to 3c.
• The maximum eigenvalue of the partial Cartan matrix for E 8 (a 6 ) is 3.902.
Based on these three factors, E 8 (a 6 ) can be placed, for example, between E 8 (a 5 ) and E 8 (a 4 ), or in the separated pair {E 8 (a 5 ), E 8 (a 6 )}. In the survey article [13], C. M. Ringel provided several notes regarding representations of Dynkin quivers. As Ringel writes: "they shed some new light on properties of Dynkin and Euclidean quivers". The following is Ringel's 2 − 4 − 8 assertion regarding Dynkin quivers E n , n = 6, 7, 8.
Let Γ be the extended Dynkin diagram (=Euclidean quiver) for the Dynkin diagram Γ. If Γ is constructed from Γ by adding the new edge to the vertex y of Γ, then y is said   to be the exceptional vertex. For the Dynkin quivers E n , n = 6, 7, 8, let x be the neighbor of y and Γ ′ = Γ\{y}, see Table 7. Let P ′ (x) (resp. I ′ (x)) be the indecomposable projective (resp. injective) representation of Γ ′ corresponding to vertex x. Let τ ′ be the Auslander-Reiten translation in the category of finite-dimensional representations rep(Γ). Then, P ′ (x) and I ′ (x) belong to the same τ ′ -orbit and there is the following 2 − 4 − 8 assertion: where n is the Ringel invariant given in the Table 7, see [13,Part 3]. For example, for the Auslander-Reiten A 5 , n = 2, see Fig. 25, and for the Auslander-Reiten D 6 , n = 4, see Fig. 26.
C.4 B. Rosenfeld: Isometry groups of the projective planes J. Baez in [1] points out another connection between the invariants 2, 4, 8 and the diagrams E 6 , E 7 and E 8 . This connection was discovered by Rosenfeld, see [14]: ". . . Boris Rosenfeld had the remarkable idea . . . that the exceptional Lie groups E 6 , E 7 and E 8 may be considered as the isometry groups of the projective planes over the following 3 algebras, respectively:" • the bioctonions C ⊗ O, • the quateroctonions H ⊗ O, • the octoctonions O ⊗ O.
Any real finite-dimensional division algebra over the reals must be only one of these: R, C, H, O 1 . The real numbers R, complex numbers C, the quaternions H, and the octonions O are division algebras of dimensions, respectively: 1, 2, 4, or 8.