Let $ f:\ (\mathbb{C}^N,0) \to (\mathbb{C},0)$ be the germ of a holomorphic function with an isolated critical point at $ 0$, with $ f(0) = 0$. We will be interested only in polynomial functions (from the list below, cf. Sect. 2.4), so $ f\in \mathbb{C}[x_1,\ldots, x_N]$. The Milnor ring of $ f$ is defined by \begin{eqnarray*} \Miln(f,0) = \mathbb{C}[[x_1,\ldots, x_N]]/(\dpar_1f,\ldots, \dpar_Nf) \end{eqnarray*} where $ \dpar_i := \dpar/\dpar x_i$; it is a finite-dimensional commutative $ \mathbb{C}$-algebra. (In fact, it is a Frobenius, or, equivalently, a Gorenstein algebra.) The number \begin{eqnarray*} \mu := \dim_\mathbb{C} \Miln(f,0) \end{eqnarray*} is called the multiplicity or Milnor number of $ (f,0)$.

A Milnor fiber is \begin{eqnarray*} V_z = f^{-1}(z)\cap \bar B_\rho \end{eqnarray*} where \begin{eqnarray*} \bar B_\rho = \{(x_1,\ldots, x_N)|\ \sum |x_i|^2 \leq \rho\} \end{eqnarray*} for $ 1\gg \rho\gg |z|> 0$.

For $ z$ belonging to a small disc $ D_\epsilon = \{z\in \mathbb{C}|\ |z| < \epsilon\}$, the space $ V_z$ is a complex manifold with boundary, homotopically equivalent to a bouquet $ \vee S^{N-1}$ of $ \mu$ spheres, [Milnor2016].

The family of free abelian groups

Take $ t\in\mathbb{R}_{> 0}\cap \overset{\bullet}D_\epsilon$; the lattice $ Q(f;t)$ does not depend, up to a canonical isomorphism, on the choice of $ t$. Let us call this lattice $ Q(f)$. The linear operator

In all the examples below $ T(f)$ has finite order $ h$. The eigenvalues of $ T(f)$ have the form $ e^{2\pi i k/h},\ k\in \mathbb{Z}$. The set of suitably chosen $ k$'s for each eigenvalue are called the spectrum of our singularity.

The $ \mathbb{C}$-vector space $ \Miln(f,0)$ may be identified with the tangent space to the base $ B$ of the miniversal deformation of $ f$. For \begin{eqnarray*} \lambda \in B^0 = B{\setminus} \Delta \end{eqnarray*} where $ \Delta\subset B$ is an analytic subset of codimension $ 1$, the corresponding function $ f_\lambda: \mathbb{C}^N\to \mathbb{C}$ has $ \mu$ nondegenerate Morse critical points with distinct critical values, and the algebra $ \Miln(f_\lambda)$ is semisimple, isomorphic to $ \mathbb{C}^\mu$.

Let $ 0\in B$ denote the point corresponding to $ f$ itself, so that $ f = f_0$, and pick $ t\in \mathbb{R}_{> 0}\cap \overset{\bullet}D_\epsilon$ as in Sect. 2.1.

Afterwards pick $ \lambda\in B^0$ close to $ 0$ in such a way that the critical values $ z_1, \ldots z_\mu$ of $ f_\lambda$ have absolute values $ \ll t$.

As in Sect. 2.1, for each \begin{eqnarray*} z\in \tD_\epsilon := D_\epsilon{\setminus}\{z_1, \ldots z_\mu\} \end{eqnarray*} the Milnor fiber $ V_z$ has the homotopy type of a bouquet $ \vee S^{N-1}$ of $ \mu$ spheres, and we will be interested in the middle homology \begin{eqnarray*} Q(f_\lambda;z) = \tH_{N-1}(V_z;\mathbb{Z})\isom \mathbb{Z}^\mu \end{eqnarray*} The lattices $ Q(f_\lambda;z)$ carry a natural bilinear product induced by the cup product in the homology which is symmetric (resp. skew-symmetric) when $ N$ is odd (resp. even).

The collection of these lattices, when $ z\in \tD_\epsilon$ varies, carries a flat Gauss–Manin connection.

Consider an "octopus" \begin{eqnarray*} Oct(t)\subset\mathbb{C} \end{eqnarray*} with the head at $ t$: a collection of non-intersecting paths $ p_i$ ("tentacles") connecting $ t$ with $ z_i$ and not meeting the critical values $ z_j$ otherwise. It gives rise to a base \begin{eqnarray*} \{b_1,\ldots, b_\mu\}\subset Q(f_\lambda) := Q(f_\lambda;t) \end{eqnarray*} (called "distinguished") where $ b_i$ is the cycle vanishing when being transferred from $ t$ to $ z_i$ along the tentacle $ p_i$, cf. [Gabrielov1973], [Arnold et al.1988].

The Picard–Lefschetz formula describes the action of the fundamental group $ \pi_1(\tD_\epsilon;t)$ on $ Q(f_\lambda)$ with respect to this basis. Namely, consider a loop $ \gamma_i$ which turns around $ z_i$ along the tentacle $ p_i$, then the corresponding transformation of $ Q(f_\lambda)$ is the reflection (or transvection) $ s_i := s_{b_i}$, cf. [Lefschetz1950], Théorème fondamental, Ch. II, p. 23.

The loops $ \gamma_i$ generate the fundamental group $ \pi_1(\tD_\epsilon)$. Let \begin{eqnarray*} \rho:\ \pi_1(\tD_\epsilon;t)\to GL(Q(f_\lambda)) \end{eqnarray*} denote the monodromy representation. The image of $ \rho$, denoted by $ G(f_\lambda)$ and called the monodromy group of $ f_\lambda$, lies inside the subgroup $ O(Q(f_\lambda))\subset GL(Q(f_\lambda))$ of linear transformations respecting the above mentioned bilinear form on $ Q(f_\lambda)$.

The subgroup $ G(f_\lambda)$ is generated by $ s_i, 1\leq i \leq \mu$.

As in Sect. 2.1, we have the monodromy operator \begin{eqnarray*} T(f_\lambda)\in G(f_\lambda), \end{eqnarray*} the image by $ \rho$ of the path $ p\subset \tD_\epsilon$ starting at $ t$ and going around all points $ z_1, \ldots, z_\mu$.

This operator $ T(f_\lambda)$ is now a product of $ \mu$ simple reflections \begin{eqnarray*} T(f_\lambda) = s_1s_2\ldots s_\mu, \end{eqnarray*} this is because the only critical value $ 0$ of $ f$ became $ \mu$ critical values $ z_1, \ldots, z_\mu$ of $ f_\lambda$.

One can identify the relative (reduced) homology $ \tH_{N-1}(V_t, \dpar V_t;\mathbb{Z})$ with the dual group $ \tH_{N-1}(V_t;\mathbb{Z})^*$, and one defines a map \begin{eqnarray*} \text{var}: \tH_{N-1}(V_t, \dpar V_t;\mathbb{Z}) \to \tH_{N-1}(V_t;\mathbb{Z}), \end{eqnarray*} called a variation operator , which translates to a map \begin{eqnarray*} L: Q(f_\lambda)^* \iso Q(f_\lambda) \end{eqnarray*} ("Seifert form") such that the matrix $ A(f_\lambda)$ of the bilinear form in the distinguished basis is \begin{eqnarray*} A(f_\lambda) = L + (-1)^{N-1}L^t, \end{eqnarray*} and \begin{eqnarray*} T(f_\lambda) = (-1)^{N-1}LL^{-t}. \end{eqnarray*} Cf. [Lamotke1975]. A choice of a path $ q$ in $ B$ connecting $ 0$ with $ \lambda$, enables one to identify $ Q(f)$ with $ Q(f_\lambda)$, and $ T(f)$ will be identified with $ T(f_\lambda)$.

The image $ G(f)$ of the monodromy group $ G(f_\lambda)$ in $ GL(Q(f)) \isom GL(Q(f_\lambda))$ is called the monodromy group of $ f$; it does not depend on a choice of a path $ q$.

If $ g\in \mathbb{C}[y_1,\ldots, y_M]$ is another function, the sum, or join of two singularities $ f\oplus g:\ \mathbb{C}^{N+M}\to \mathbb{C}$ is defined by \begin{eqnarray*} (f\oplus g)(x,y) = f(x) + g(y) \end{eqnarray*} Obviously we can identify \begin{eqnarray*} \Miln(f\oplus g)\isom \Miln(f)\otimes \Miln(g) \end{eqnarray*} Note that the function $ g(y) = y^2$ is a unit for this operation.

It follows that the singularities $ f(x_1,\ldots, x_N)$ and \begin{eqnarray*} f(x_1,\ldots, x_N) + x_{M+1}^2 + \cdots + x^2_{N+M} \end{eqnarray*} are "almost the same". In order to have good signs (and for other purposes) it is convenient to add some squares to a given $ f$ to get $ N\equiv 3\mod(4)$.

The fundamental Sebastiani–Thom theorem, [Sebastiani1971], says that there exists a natural isomorphism of lattices \begin{eqnarray*} Q(f\oplus g) \isom Q(f)\otimes_\mathbb{Z} Q(g), \end{eqnarray*} and under this identification the full monodromy decomposes as \begin{eqnarray*} T_{f\oplus g} = T_f\otimes T_g \end{eqnarray*} Thus, if \begin{eqnarray*} \Spec(T_f) = \{e^{\mu_p\cdot 2\pi i/h_1}\},\quad \Spec(T_f) = \{e^{\nu_q\cdot 2\pi i/h_2}\} \end{eqnarray*} then \begin{eqnarray*} \Spec(T_{f\oplus g}) = \{e^{(\mu_p h_2 + \nu_qh_1)\cdot 2\pi i/h_1h_2}\} \end{eqnarray*}

Cf. [Arnold et al.1988] (a), 15.1. They are: \begin{eqnarray*} &\displaystyle x^{n+1},\quad n\geq 1, \qquad {\rm(A_n)}& \\ &\displaystyle x^2y + y^{n-1},\quad n\geq 4\qquad {\rm(D_n)}& \\ &\displaystyle x^4 + y^3 \qquad {\rm(E_6)}& \\ &\displaystyle xy^3 + x^3\qquad {\rm(E_7)}& \\ &\displaystyle x^5 + y^3 \qquad{\rm(E_8)}& \end{eqnarray*}

Their names come from the following facts:

• their lattices of vanishing cycles may be identified with the corresponding root lattices;
• the monodromy group is identified with the corresponding Weyl group;
• the classical monodromy $ T_f$ is a Coxeter element, therefore its order $ h$ is equal to the Coxeter number, and \begin{eqnarray*} \Spec(T_f) = \{e^{2\pi i k_1/h},\ldots, e^{2\pi i k_r/h}\} \end{eqnarray*} where the integers \begin{eqnarray*} 1 = k_1 < k_2 < \cdots < k_r = h - 1, \end{eqnarray*} are the exponents of our root system.

Let us call a lattice a pair $ (Q, A)$ where $ Q$ is a free abelian group, and \begin{eqnarray*} A: Q\times Q\to \mathbb{Z} \end{eqnarray*} a symmetric bilinear map ("Cartan matrix"). We shall identify $ A$ with a map \begin{eqnarray*} A: Q \to Q^\vee := Hom(Q,\mathbb{Z}). \end{eqnarray*} A polarized lattice is a triple $ (Q, A, L)$ where $ (Q, A)$ is a lattice, and \begin{eqnarray*} L:\ Q\iso Q^\vee \end{eqnarray*} ("variation", or "Seifert matrix") is an isomorphism such that

where \begin{eqnarray*} L^\vee: Q = Q^{\vee\vee}\iso Q^\vee \end{eqnarray*} is the conjugate to $ L$.

The Coxeter automorphism of a polarized lattice is defined by

We shall say that the operators $ A$ and $ C$ are in a Cartan–Coxeter correspondence . Example Let $ (Q, A)$ be a lattice, and $ \{e_1, \ldots, e_n\}$ an ordered $ \mathbb{Z}$- base of $ Q$. With respect to this base $ A$ is expressed as a symmetric matrix $ A = (a_{ij}) = A(e_i, e_j)\in \fgl_n(\mathbb{Z})$. Let us suppose that all $ a_{ii}$ are even. We define the matrix of $ L$ to be the unique upper triangular matrix $ (\ell_{ij})$ such that $ A = L + L^t$ (in particular $ \ell_{ii} = a_{ii}/2$; in our examples we will have $ a_{ii} = 2$.) We will call $ L$ the standard polarization associated to an ordered base. $\square$

Polarized lattices form a groupoid:

an isomorphism of polarized lattices $ f:\ (Q_1, A_1, L_1) \iso (Q_2, A_2, L_2)$ is by definition an isomorphism of abelian groups $ f: Q_1\iso Q_2$ such that \begin{eqnarray*} L_1(x, y) = L_2(f(x), f(y)) \end{eqnarray*} (and whence $ A_1(x, y) = A_2(f(x), f(y))$).

Cf. [Steinberg1985], [Casselman2017]. Let $ \alpha_1, \ldots, \alpha_r$ be a base of simple roots of a finite reduced irreducible root system $ R$ (not necessarily simply laced). Let \begin{eqnarray*} A = (a_{ij}) = (\langle \alpha_i, \alpha_j^\vee \rangle) \end{eqnarray*} be the Cartan matrix.

Choose a black/white coloring of the set of vertices of the corresponding Dynkin graph $ \Gamma(R)$ in such a way that any two neighbouring vertices have different colours; this is possible since $ \Gamma(R)$ is a tree (cf. Sect. 5.2).

Let us choose an ordering of simple roots in such a way that the first $ p$ roots are black, and the last $ r - p$ roots are white. In this base $ A$ has a block form \begin{eqnarray*} A = \left( \begin{matrix} 2I_p &\quad X\\ Y &\quad 2I_{r-p} \end{matrix}\right) \end{eqnarray*} Consider a Coxeter element

where \begin{eqnarray*} C_B = \prod_{i=1}^p s_i,\quad C_W = \prod_{i=p+1}^r s_i. \end{eqnarray*} Here $ s_i$ denotes the simple reflection corresponding to the root $ \alpha_i$.

The matrices of $ C_B, C_W$ with respect to the base $ \{\alpha_i\}$ are \begin{eqnarray*} C_B = \left( \begin{matrix} -I & \quad-X\\ 0 &\quad I \end{matrix}\right),\quad C_W = \left( \begin{matrix} I &\quad 0\\ -Y &\quad -I \end{matrix}\right), \end{eqnarray*} so that

This is an observation due to R.Steinberg, cf. [Steinberg1985], p. 591.

We can also rewrite this as follows. Set \begin{eqnarray*} L = \left( \begin{matrix} I &\quad 0\\ Y &\quad I \end{matrix}\right),\quad U = \left( \begin{matrix} I &\quad X\\ 0 &\quad I\end{matrix}\right). \end{eqnarray*} Then $ A = L + U$, and one checks easily that

so we are in the situation Sect. 3.1. This explains the name "Cartan–Coxeter correspondence".

Proof.

We consider the Dynkin graph of $ A_n$ with the obvious numbering of the vertices.

The Coxeter number $ h = n + 1$, the set of exponents: \begin{eqnarray*} \Exp(A_n) = \{1, 2, \ldots, n\} \end{eqnarray*}

The eigenvalues of any Coxeter element are $ e^{i\theta_k}$, and the eigenvalues of the Cartan matrix $ A(A_n)$ are $ 2 - 2\cos\theta_k$, $ \theta_k = 2\pi k/h$, $ k\in \Exp(A_n)$.

An eigenvector of $ A(A_n)$ with the eigenvalue $ 2 - 2\cos\theta$ has the form

Denote by $ C(A_n)$ the Coxeter element \begin{eqnarray*} C(A_n) = s_1s_2\cdots s_n \end{eqnarray*} Its eigenvector with the eigenvalue $ e^{2i\theta}$ is: \begin{eqnarray*} X_{C(A_n)} = (\sum_{k=0}^{n-j} e^{2ik\theta})_{1\leq j\leq n} \end{eqnarray*} For example, for $ n = 4$: \begin{eqnarray*} C_{A_4} = \begin{pmatrix} 0 &\quad 0 &\quad 0 &\quad -1 \\ 1 &\quad 0 &\quad 0 &\quad -1 \\ 0 &\quad 1 &\quad 0 &\quad -1 \\ 0 &\quad 0 &\quad 1 &\quad -1 \end{pmatrix} \quad\text{ and }\quad X_{C(A_4 )} = \begin{pmatrix} 1 + e^{2i\theta} + e^{4i\theta} + e^{6i\theta} \\ 1 + e^{2i\theta} + e^{4i\theta} \\ 1 + e^{2i\theta} \\ 1 \end{pmatrix}\end{eqnarray*}

is an eigenvector with eigenvalue $ e^{2i\theta}$.

Similarly, for $ n = 2$: s \begin{eqnarray*} C_{A_2} = \begin{pmatrix} 0 &\quad -1 \\ 1 &\quad -1 \end{pmatrix} ,\quad X_{C(A_2 )} = \begin{pmatrix} 1+ e^{2i\gamma} \\ 1 \end{pmatrix} \end{eqnarray*} $\square$

Suppose we are given two polarized lattices $ (Q_i, A_i, L_i)$, $ i = 1, 2$.

Set $ Q = Q_1\otimes Q_2$, whence \begin{eqnarray*} L:= L_1\otimes L_2: Q\iso Q^\vee, \end{eqnarray*} and define \begin{eqnarray*} A: = A_1*A_2 := L + L^\vee: Q\iso Q^\vee \end{eqnarray*}

The triple $ (Q, A, L)$ will be called the join , or Sebastiani–Thom , product of the polarized lattices $ Q_1$ and $ Q_2$, and denoted by $ Q_1*Q_2$.

Obviously \begin{eqnarray*} C(L) = - C(L_1)\otimes C(L_2)\in GL(Q_1\otimes Q_2). \end{eqnarray*}

It follows that if $ \Spec(C(L_i)) = \{e^{2\pi i k_{i}/h_i},\ k_i\in K_i\}$ then

The ranks: \begin{eqnarray*} r(E_8) = 8 = r(A_4)r(A_2)r(A_1); \end{eqnarray*} the Coxeter numbers: \begin{eqnarray*} h(E_8) = h(A_4)h(A_2)h(A_1) = 5\cdot 3\cdot 2 = 30. \end{eqnarray*} It follows that \begin{eqnarray*} |R(E_8)| = 240 = |R(A_4)||R(A_2)||R(A_1)|. \end{eqnarray*} The exponents of $ E_8$ are: \begin{eqnarray*} 1, 7, 13, 19, 11, 17, 23, 29. \end{eqnarray*} All these numbers, except $ 1$, are primes, and these are all primes $ \leq$30, not dividing $ 30$.

They may be determined from the formula \begin{eqnarray*} \frac{i}{5} + \frac{j}{3} + \frac{1}{2} = \frac{30 + k(i,j)}{30},\quad 1\leq i\leq 4,\quad 1\leq j\leq 2, \end{eqnarray*} so \begin{eqnarray*} &&k(i, 1)= 1 + 6(i-1) = 1, 7, 13, 19; \\ &&k(i,2) = 1 + 10 + 6(i-1) = 11, 17, 23, 29. \end{eqnarray*} This shows that the exponents of $ E_8$ are the same as the exponents of $ A_4*A_2*A_1$.

The following theorem is more delicate.

In the left hand side $ Q(A_n)$ means the root lattice of $ A_n$ with the standard Cartan matrix and the standard polarization \begin{eqnarray*} A(A_n) = L(A_n) + L(A_n)^t \end{eqnarray*} where the Seifert matrix $ L(A_n)$ is upper triangular.

In the process of the proof, given in Sects. 4.4–4.6 below, the isomorphism $ \Gamma$ will be written down explicitly. Cf. [Arnold et al.1988], Chapter I, Sect. 4 (especially Fig. 39), and references to the articles by A'Campo and Gusein-Zade therein.

For $ n = 4, 2, 1$, we consider the bases of simple roots $ e_1,\ldots, e_n$ in $ Q(A_n)$, with scalar products given by the Cartan matrices $ A(A_n)$.

The tensor product of three lattices \begin{eqnarray*} Q_* = Q(A_4)\otimes Q(A_2)\otimes Q(A_1) \end{eqnarray*} will be equipped with the "factorizable" basis in the lexicographic order: \begin{eqnarray*} (f_1,\ldots, f_8) &:=& (e_1\otimes e_1\otimes e_1, e_1\otimes e_2\otimes e_1, e_2\otimes e_1\otimes e_1, e_2\otimes e_2\otimes e_1,\nonumber\\ &&e_3\otimes e_1\otimes e_1, e_3\otimes e_2\otimes e_1, e_4\otimes e_1\otimes e_1, e_4\otimes e_2\otimes e_1). \end{eqnarray*} Introduce a scalar product $ (x, y)$ on $ Q_*$ given, in the basis $ \{f_i\}$, by the matrix \begin{eqnarray*} A_* = A_4*A_2*A_1. \end{eqnarray*}

Let $ (Q, ( , ))$ be a lattice of rank $ r$. We introduce the following two sets of transformations $ \{\alpha_m\}, \{\beta_m\}$ on the set $ Bases-cycl(Q)$ of cyclically ordered bases of $ Q$.

If $ x = (x_i)_{i\in \mathbb{Z}/r\mathbb{Z}}$ is a base, and $ m\in \mathbb{Z}/r\mathbb{Z}$, we set \begin{eqnarray*} (\alpha_m(x))_i = \left\lbrace \begin{array}{ll} x_{m+1} + (x_{m+1}, x_{m})x_m\ &\quad \text{if}\ i = m\\ x_m &\quad \text{if}\ i = m + 1 \\ x_i &\quad \text{otherwise} \end{array}\right. \end{eqnarray*} and \begin{eqnarray*} (\beta_m(x))_i = \left\lbrace \begin{array}{ll} x_m &\text{if}\ i = m - 1 \\ x_{m-1} + (x_{m-1}, x_{m})x_m\ &\text{if}\ i = m\\ x_i &\text{otherwise} \end{array}\right. \end{eqnarray*} We define also a transformation $ \gamma_m$ by \begin{eqnarray*} (\gamma_m(x))_i = \left\lbrace \begin{matrix} - x_m &\quad \text{if}\ i = m \\ x_i &\quad \text{otherwise} \end{matrix}\right. \end{eqnarray*}

Consider the base $ f = \{f_1, \ldots f_8\}$ of the lattice $ Q_* := Q(A_4)\otimes Q(A_2) \otimes Q(A_1)$ described in Sect. 4.4, and apply to it the following transformation

Then the base $ G'(f)$ has the intersection matrix given by the Dynkin graph of $ E_8$, with the ordering indicated in Fig. 1 below.

Figure 1 Gabrielov's ordering of $ E_8$.

This concludes the proof of Theorem 2. $\square$

By definition, the isomorphism of lattices $ \Gamma$, (22), induces a bijection between the bases \begin{eqnarray*} g:\ \{f_1,\ldots, f_8\} \iso \{\alpha_1,\ldots, \alpha_8\}\subset R(E_8). \end{eqnarray*} where in the right hand side we have the base of simple roots, and a map \begin{eqnarray*} G:\ R(A_4)\times R(A_2)\times R(A_1)\to R(E_8),\ G(x,y,z) = \Gamma(x\otimes y \otimes z) \end{eqnarray*} of sets of the same cardinality $ 240$ which is not a bijection however: its image consists of $ 60$ elements.

Note that the set of vectors $ \alpha\in Q(E_8)$ with $ (\alpha, \alpha) = 2$ coincides with the root system $ R(E_8)$, cf. [Serre1970], Première Partie, Ch. 5, 1.4.3.

The isomorphism $ G'$ (19) is given by a matrix $ G'\in GL_8(\mathbb{Z})$ such that \begin{eqnarray*} A_G(E_8) = G^{\prime t}A_*G' \end{eqnarray*} where we denoted \begin{eqnarray*} A_* = A(A_4)*A(A_2)*A(A_1), \end{eqnarray*} the factorized Cartan matrix, and $ A_G$ denotes the Cartan matrix of $ E_8$ with respect to the numbering of roots indicated on Fig. 1.

Now let us pass to the numbering of vertices of the Dynkin graph of type $ E_8$ indicated in [Bourbaki2007] (the difference with Gabrielov's numeration is in three vertices $ 2, 3$, and $ 4$).

The Gabrielov's Coxeter element (the full monodromy) in the Bourbaki numbering looks as follows: \begin{eqnarray*} C_{G}(E_8 ) = s_1 \circ s_3 \circ s_4 \circ s_2 \circ s_5 \circ s_6 \circ s_7 \circ s_8 \end{eqnarray*}

Figure 2 Bourbaki ordering of $ E_8$.

To obtain the Cartan eigenvectors of $ E_8$, one should pass from $ C_G(E_8)$ to the "black/white" Coxeter element (as in Sect. 3.3) \begin{eqnarray*} C_{BW}(E_8) = s_1 \circ s_4 \circ s_6 \circ s_8 \circ s_2 \circ s_3 \circ s_5 \circ s_7 \end{eqnarray*}

Any two Coxeter elements are conjugate in the Weyl group $ W(E_8)$.

The elements $ C_G(E_8)$ and $ C_{BW}(E_8)$ are conjugate by the following element of $ W(E_8)$: \begin{eqnarray*} C_G(E_8) = w^{-1}C_{BW}(E_8)w \end{eqnarray*}

where \begin{eqnarray*} w = s_7 \circ s_5 \circ s_3 \circ s_2 \circ s_6 \circ s_4 \circ s_5 \circ s_1 \circ s_3 \circ s_2 \circ s_4 \circ s_1 \circ s_3 \circ s_2 \circ s_1 \circ s_2 \end{eqnarray*}

This expression for $ w$ can be obtained using an algorithm described in [Casselman2017], cf. also [Brieskorn1988].

Thus, if $ x_*$ is an eigenvector of $ C_*(E_8)$ then \begin{eqnarray*} x_{BW} = wG^{-1}x_* \end{eqnarray*} is an eigenvector of $ C_{BW}(E_8)$. But we know the eigenvectors of $ C_*(E_8)$, they are all factorizable.

This provides the eigenvectors of $ C_{BW}(E_8)$, which in turn have very simple relation to the eigenvectors of $ A(E_8)$, due to Theorem 1. Conclusion: an expression for the eigenvectors of $ {A}({E}_{8})$.

Let $ \theta = \frac{a\pi}{5},\ 1\leq a\leq 4$, $ \gamma = \frac{b\pi}{3},\ 1\leq b\leq 2$, $ \delta = \frac{\pi}{2}$, \begin{eqnarray*} &&\alpha = \theta + \gamma + \delta = \pi + \frac{k\pi}{30}, \\ &&k\in \{1, 7, 11, 13, 17, 19, 23, 29\}. \end{eqnarray*} The $ 8$ eigenvalues of $ A(E_8)$ have the form \begin{eqnarray*} \lambda(\alpha) = \lambda(\theta,\gamma) = 2 - 2\cos \alpha \end{eqnarray*} An eigenvector of $ A(E_8)$ with the eigenvalue $ \lambda(\theta,\gamma)$ is

\begin{eqnarray*} &&X_{E_8}(\theta,\gamma)\nonumber\\ &&\quad = \begin{pmatrix} \cos(\gamma + \theta - \delta) + \cos ( \gamma - 3 \theta - \delta) + \cos (\gamma - \theta - \delta) \\ \cos( 2 \gamma + 2 \theta)\\ \cos(2\gamma) + \cos (2\gamma + 2 \theta) + \cos ( 2\gamma - 2 \theta) + \cos (4\theta) + \cos(2 \theta) \\ \cos(\gamma + 3 \theta - \delta) + \cos ( \gamma + \theta - \delta ) + \cos ( -\gamma + 3 \theta - \delta) \\ 2 \cos(2\gamma) + 2 \cos(2 \gamma + 2 \theta) + \cos(2\gamma - 2 \theta) + \cos(2 \gamma + 4 \theta) + \cos(4\theta) + 2 \cos (2 \theta) + 1 \\ \cos(\gamma + 3 \theta - \delta) + \cos( \gamma + \theta - \delta) \\ \cos (2 \gamma) + \cos ( 2 \theta - 2 \delta) \\ \cos (\gamma - \theta - \delta ) \end{pmatrix} \end{eqnarray*}

One can simplify it as follows:

The Perron–Frobenius eigenvector corresponds to the eigenvalue \begin{eqnarray*} 2 - 2 \cos \frac{\pi}{30}, \end{eqnarray*} and may be chosen as \begin{eqnarray*} v_{PF} = \begin{pmatrix} 2 \cos \frac{\pi}{5} \cos \frac{11\pi}{30} {\\[1.2mm]}\\ \cos \frac{\pi}{15} {\\[1.2mm]}\\ 2\cos^{2}\frac{\pi}{5} {\\[1.2mm]}\\ 2\cos \frac{2\pi}{30} \cos \frac{\pi}{30} {\\[1.2mm]}\\ 2 \cos \frac{4\pi}{15} \cos \frac{\pi}{5} + \frac{1}{2} {\\[1.2mm]}\\ 2 \cos \frac{\pi}{5} \cos \frac{7\pi}{30} {\\[1.2mm]}\\ 2 \cos \frac{\pi}{30} \cos \frac{11\pi}{30} {\\[1.2mm]}\\ \cos \frac{11\pi}{30} \end{pmatrix} \end{eqnarray*}

Ordering its coordinates in the increasing order, we obtain \begin{eqnarray*} v_{PF < } = \begin{pmatrix} \cos \frac{11\pi}{30} {\\[1.2mm]}\\ 2 \cos \frac{\pi}{5} \cos \frac{11\pi}{30} {\\[1.2mm]}\\ 2 \cos \frac{\pi}{30} \cos \frac{11\pi}{30} {\\[1.2mm]}\\ \cos \frac{\pi}{15} {\\[1.2mm]}\\ 2 \cos \frac{\pi}{5} \cos \frac{7\pi}{30} {\\[1.2mm]}\\ 2\cos^{2}\frac{\pi}{5} {\\[1.2mm]}\\ 2 \cos \frac{4\pi}{15} \cos \frac{\pi}{5} + \frac{1}{2} {\\[1.2mm]}\\ 2\cos \frac{2\pi}{30} \cos \frac{\pi}{30} \end{pmatrix} \end{eqnarray*}

In the Ref. [Zamolodchikov1989a, Zamolodchikov1989b], obtains the following expression for the PF vector: \begin{eqnarray*} v_{Zam}(m) = \begin{pmatrix} m \\ 2 m \cos \frac{\pi}{5} {\\[1.2mm]}\\ 2 m \cos \frac{\pi}{30} {\\[1.2mm]}\\ 4 m \cos \frac{\pi}{5} \cos \frac{7\pi}{30} {\\[1.2mm]}\\ 4 m \cos \frac{\pi}{5} \cos \frac{2\pi}{15} {\\[1.2mm]}\\ 4 m \cos \frac{\pi}{5} \cos \frac{\pi}{30} {\\[1.2mm]}\\ 8 m \cos^{2} \frac{\pi}{5} \cos \frac{7\pi}{30} {\\[1.2mm]}\\ 8 m \cos^{2} \frac{\pi}{5} \cos \frac{2\pi}{15} \end{pmatrix} \end{eqnarray*}

Setting $ m = \cos \frac{11\pi}{30}$, we find indeed : \begin{eqnarray*} v_{PF < } = v_{Zam} \left(\cos \frac{11\pi}{30}\right) \end{eqnarray*}

The proof is exactly the same as for $ Q(E_8)$. The passage from $ A_3 * A_2 * A_1$ to $ E_6$ is obtained by the following transformation \begin{eqnarray*} G'_{E_6} = \gamma_4 \gamma_1 \alpha_1 \alpha_2 \alpha_3 \alpha_4 \beta_6 \beta_3 \alpha_1 \end{eqnarray*}

cf. [Gabrielov1973], Example 2.

After a passage from Gabrielov's ordering to Bourbaki's, we obtain a transformation \begin{eqnarray*} G_{E_6} = \begin{pmatrix} 0&\quad -1&\quad 1&\quad 0&\quad 0&\quad 0\\ -1&\quad 0&\quad 1&\quad 0&\quad 0&\quad 0\\ 0&\quad -1&\quad 0&\quad 1&\quad 0&\quad 0\\ -1&\quad 0&\quad 0&\quad 0&\quad 1&\quad 0\\ 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 1\\ -1&\quad 0&\quad 0&\quad 0&\quad 0&\quad 1 \end{pmatrix} \in GL_{6} ( \mathbb{Z} ) \end{eqnarray*}

such that \begin{eqnarray*} A(E_6) = G_{E_6}^{t} A_{*} G_{E_6} \quad\text{ and }\quad C_{G} (E_6) = G_{E_6}^{-1} C_{*} G_{E_6} \end{eqnarray*} where $ A_{*} = A(A_3) * A(A_2) * A(A_1)$ and $ C_{*} = C(A_3) \otimes C(A_2) \otimes C(A_1)$ and \begin{eqnarray*} C_{G}(E_6) = s_1 \circ s_3 \circ s_4 \circ s_2 \circ s_5 \circ s_6 \end{eqnarray*} $ C_{G}(E_6)$ is the Gabrielov's Coxeter element in the Bourbaki numbering, cf. [Bourbaki2007].

Let $ C_{BW} (E_6) = s_1 \circ s_4 \circ s_6 \circ s_2 \circ s_3 \circ s_5$ be the "black/white" Coxeter element. $ C_{G}(E_6)$ and $ C_{BW}(E_6)$ are conjugated by the following element of the Weyl group $ W(E_6)$ : \begin{eqnarray*} v = s_5 \circ s_3 \circ s_2 \circ s_4 \circ s_1 \circ s_3 \circ s_3 \circ s_1 \circ s_2 \end{eqnarray*} Thus, if $ x_{*}$ is an eigenvector of $ C_{*} ( E_6)$ then $ x_{BW} = v G^{-1}_{E_6} x_{*}$ is an eigenvector of $ C_{BW}(E_6)$.

Finally, let $ \theta = \frac{a\pi}{4} , 1 \leq a \leq 3$, $ \gamma = \frac{b\pi}{3}, 1 \leq b \leq 2$, $ \delta = \frac{\pi}{2}$ and \begin{eqnarray*} \alpha = \theta + \gamma + \delta \end{eqnarray*} The 6 eigenvalues of $ A(E_6)$ have the form $ \lambda (\alpha) = \lambda(\theta, \gamma) = 2 - 2\cos \alpha$. An eigenvector of $ A(E_6)$ with the eigenvalue $ \lambda (\alpha)$ is \begin{eqnarray*} X_{E_6} ( \theta, \lambda) = \begin{pmatrix} \cos \left(3\gamma + 3 \theta - \delta \right) \\ 2 \cos^{2} \theta \\ -2 \cos \left( 3 \gamma + 3 \theta - \delta\right) \cos \left(\gamma + \theta - \delta\right) \\ -4 \cos^{2} \theta \cos \left( \gamma + \theta - \delta\right) \\ 1 - 2 \cos \left( 2 \gamma + 3 \theta\right) \cos \theta \\ -2 \cos(\gamma) \cos \left(\theta - \delta\right) \end{pmatrix} \end{eqnarray*}

Let $ A = (a_{ij})$ be a $ n\times n$ complex matrix. We will say that $ A$ is a generalized Cartan matrix if

1. (i) for all $ i\neq j$, $ a_{ij}\neq 0$ implies $ a_{ji}\neq 0$;
2. (ii) all $ a_{ii} = 2$. If only (i) is fulfilled, we will say that $ A$ is a pseudo-Cartan matrix .

We associate to a pseudo-Cartan matrix $ A$ an unoriented graph $ \Gamma(A)$ with vertices $ 1, \ldots, n$, two vertices $ i$ and $ j$ being connected by an edge $ e = (ij)$ iff $ a_{ij}\neq 0$.

Let $ A$ be a generalized Cartan matrix. There is a unique decomposition \begin{eqnarray*} A = L + U \end{eqnarray*} where $ L = (\ell_{ij})$ (resp. $ U = (u_{ij})$) is lower (resp. upper) triangular, with $ 1$'s on the diagonal.

We define a $ q$-deformed Cartan matrix by \begin{eqnarray*} A(q) = qL + U \end{eqnarray*} This definition is inspired by the $ q$-deformed Picard–Lefschetz theory developed by [Givental1988].

The theorem will be proved after some preparations. 5.2 Let $ \Gamma$ be an unoriented tree with a finite set of vertices $ I = V(\Gamma)$.

Let us pick a root of $ \Gamma$, and partially order its vertices by taking the minimal vertex $ i_0$ to be the bottom of the root, and then going "upstairs". This defines an orientation on $ \Gamma$.

Proof.

Proof.

Let us consider two matrices: $ A(q) = (a(q)_{ij})$ with $ a(q)_{ii} = 1 + q$ \begin{eqnarray*} a(q)_{ij} = \left\{\begin{matrix} a_{ij} &\quad \text{if }i < j\\ qa_{ij} &\quad \text{if }i > j \end{matrix}\right. \end{eqnarray*} and \begin{eqnarray*} A'(q) = \sqrt{q}A + (1 - \sqrt{q})^2 I = (a(q)'_{ij}) \end{eqnarray*} with $ a(q)'_{ii} = 1 + q$ and $ a(q)'_{ij} = \sqrt{q}a(q)_{ij}$, $ i\neq j$.

Thus, we can apply Lemma 4 to $ A(q)$ and $ A'(q)$. So, there exists a diagonal matrix $ D$ as above such that \begin{eqnarray*} A(q) = D^{-1}A'(q)D. \end{eqnarray*} But the eigenvalues of $ A'(q)$ are obviously \begin{eqnarray*} \lambda(q) = \sqrt{q}\lambda + (1 - \sqrt{q})^2 = 1 + (\lambda - 2)\sqrt{q} + q. \end{eqnarray*} If $ v$ is an eigenvector of $ A$ for $ \lambda$ then $ v$ is an eigenvector of $ A'(q)$ for $ \lambda(q)$, and $ Dv$ will be an eigenvector of $ A(q)$ for $ \lambda(q)$. $\square$

The expression (23) resembles the number of points of an elliptic curve $ X$ over a finite field $ \mathbb{F}_q$. To appreciate better this resemblance, note that in all our examples $ \lambda$ has the form \begin{eqnarray*} \lambda = 2-2 \cos \theta, \end{eqnarray*} so if we set \begin{eqnarray*} \alpha = \sqrt{q}e^{i\theta} \end{eqnarray*} ("a Frobenius root") then $ |\alpha| = \sqrt{q}$, and \begin{eqnarray*} \lambda(q) = 1 - \alpha - \bar\alpha + q, \end{eqnarray*} cf. [Ireland and Rosen2013], Chapter 11, [U+00A7]1, [Knapp1992], Chapter 10, Theorem 10.5.

So, the Coxeter eigenvalues $ e^{2i\theta}$ may be seen as analogs of "Frobenius roots of an elliptic curve over $ \mathbb{F}_1$".

Let $ W = \BC^2$. Recall three Hermitian Pauli matrices: \begin{eqnarray*} \sigma^x = \left( \begin{matrix} 0 &\quad 1\\1 &\quad 0 \end{matrix}\right) ,\quad \sigma^y = \left( \begin{matrix} 0 &\quad -i\\i &\quad 0 \end{matrix}\right) ,\quad \sigma^z = \left( \begin{matrix} 1 &\quad 0\\0 &\quad -1 \end{matrix}\right). \end{eqnarray*} The $ \BC$-span of $ \sigma^x, \sigma^y, \sigma^z$ inside $ End(W)$ is a complex Lie algebra $ \fg = \fsl(2,\BC)$; the $ \BR$-span of the anti-Hermitian matrices $ i\sigma^x, i\sigma^y, i\sigma^z$ is a real Lie subalgebra $ \fk = \fsu(2)\subset \fg$. The resulting representation of $ \fg$ (or $ \fk$) on $ W$ is what physicists refer to as the "spin-$ \frac{1}{2}$ representation".

For a natural $ N$, consider a $ 2^N$-dimensional tensor product \begin{eqnarray*} V = \otimes_{n=1}^N W_n \end{eqnarray*} with all $ W_n = W$. We are interested in the spectrum of the following linear operator $ H$ acting on $ V$:

In keeping with the conditions of the experiment ([Coldea et al.2010]), everywhere below we assume that $ N$ is very large ($ N > > 1$), and that $ 0 < h_z < < J$.

The space $ V$ arises as the space of states of a quantum-mechanical model describing a chain of $ N$ atoms on the plane $ \BR^2$ with coordinates $ (x,z)$. The chain is parallel to the $ z$ axis, and is subject to a magnetic field with a component $ h_z$ along the chain, and a component $ h_x$ along the $ x$-axis. The $ W_n$ is the space of states of the $ n$-th atom. Only the nearest-neighbor atoms interact, and the $ J$ parameterizes the strength of this interaction.

The operator $ H$ in the Eq. (27) is called the Hamiltonian, and its eigenvalues $ \epsilon$ correspond to the energy of the system. It is a Hermitian operator (with respect to an obvious Hermitian scalar product on $ V$), thus all its eigenvalues are real.

Consider also the translation operator $ T$, acting as follows:

An eigenvector $ v_0\in V$ of $ H$ with the lowest energy eigenvalue $ \epsilon_0$ is called the ground state.

What happens as $ h_x$ varies, at fixed $ J$ and $ h_z$? When $ h_x < < J$, the ground state $ v_0$ is close to the ground state $ v_J$ of the operator $ H_J = H(J,0,0)$: \begin{eqnarray*} v_J = \otimes_{n=1}^N v_n^z , \end{eqnarray*} where $ v^z_n$ is an eigenvector of $ \sigma^z$ in $ W_i$ with eigenvalue $ 1$. Thus, the state $ v_J$ is interpreted as "all the spins pointing along the $ z$-axis".

In the opposite limit, when $ h_x > > J$, the ground state $ v_0$ is close to the ground state $ v_x$ of the operator $ H_x = H(0,0,h_x)$: \begin{eqnarray*} v_x = \otimes_{n=1}^N v_n^x , \end{eqnarray*} where $ v^x_n$ is an eigenvector of $ \sigma^x$ in $ W_n$ with eigenvalue $ 1$. Thus, the state $ v_x$ is interpreted as "all the spins pointing along the $ x$-axis".

As a function of $ h_x$ at fixed $ J$ and $ h_z$, the system has two phases. There is a critical value $ h_x = h_c$, of the order of $ J/2$ : for $ h_x < h_c$, the ground state $ v_0$ is close to $ v_J$, and one says that the chain is in the ferromagnetic phase. By contrast, for $ h_x > h_c$, the ground state $ v_0$ is close to $ v_x$, and one says that the chain is in the paramagnetic phase. (The transition between the two phases is far less trivial than the spins simply turning to follow the field upon increasing $ h_x$: to find out more, curious reader is encouraged to consult the Ref. [Chakrabarti et al.1996].) (b) Elementary excitations at $ h_x = h_c$

Zamolodchikov's theory ([Zamolodchikov1989a, Zamolodchikov1989b]), says something spectacularly precise about the next few, after $ \epsilon_0$, eigenvalues ("energy levels") of a nearly-critical Hamiltonian $ H_c := H(J,h_z < < J, h_c)$. To see this, notice that the possible eigenvalues of the translation operator $ T$ have the form $ e^{2\pi i k/N}$, with $ -N/2\leq k \leq N/2$; let us call the number \begin{eqnarray*} p =2\pi k /N \end{eqnarray*} the momentum of an eigenstate. Since $ H$ commutes with $ T$, each eigenspace $ V_\epsilon := \{v\in V|\ H_c v = \epsilon v\}$ decomposes further as per \begin{eqnarray*} V_\epsilon = \oplus_p\ V_{p,\epsilon}, \end{eqnarray*} where \begin{eqnarray*} V_{p,\epsilon} := \{v\in V|\ H_cv = \epsilon v, Tv = e^{i p}v\} \end{eqnarray*} Let us add a constant to $ H_c$ in such a way that the ground state energy $ \epsilon_0$ becomes $ 0$ and, on the plane $ P$ with coordinates $ (p, \epsilon)$, let us mark all the points, for which $ V_{p,\epsilon} \neq 0$.

Zamolodchikov predicted ([Zamolodchikov1989a, Zamolodchikov1989b]), that there exist $ 8$ numbers $ 0 < m_1 < \cdots < m_8$ with the following property. Let us draw on $ P$ eight hyperbolae

All the marked points will be located:

• either in a vicinity of one of the hyperbolae $ \Hyp_i$ (in the limit $ N\lra \infty$ they will all lie on these hyperbolae).
• or in a shaded region separated from these hyperbolae as shown in the Fig. 3.

Figure 3 The expected joint spectrum of the operators $ T, H$.

The states $ v\in V_{p,\epsilon}$ with $ (p, \epsilon)\in \Hyp_i$ are called elementary excitations . The numbers $ m_i$ are called their masses .

The vector

These low-lying excitations (hyperbolae) are observable: one may be able to see them

1. (a) in a computer simulation, or
2. (b) in a neutron scattering experiment.

The paper ([Coldea et al.2010]) reports the results of a magnetic neutron scattering experiment on cobalt niobate CoNb$ _2$O$ _6$, a material that can be pictured as a collection of parallel non-interacting one-dimensional chains of atoms. We depict such a chain as a straight line, parallel to the $ z$-axis in our physical space $ \BR^3$ with coordinates $ x, y, z$.

The sample, at low temperature $ T < 2.95$K (Kelvin), was subject to an external magnetic field with components $ (h_x,h_z)$, with the $ h_x$ at the critical value $ h_x = h_c$, and with $ h_z < < h_c$. The system may be described as the Ising chain with a nearly-critical Hamiltonian $ H = H(J, h_z < < h_c, h_c)$ of the Eq. (27). The experiment ([Coldea et al.2010]) may be interpreted with the help of the following (oversimplified) theoretical picture.

Consider a neutron scattering off the sample. If the incident neutron has energy $ \epsilon$ and momentum $ p$, and scatters off with energy $ \epsilon'$ and momentum $ p'$, the energy and momentum conservation laws imply that the differences, called energy and momentum transfers $ \omega = \epsilon - \epsilon', q = p - p'$, are absorbed by the sample.

The energy transfer cannot be arbitrary. Suppose that, prior to scattering the neutron, the sample was in the ground state $ v_0$; upon scattering the neutron, it undergoes a transition to a state that is a linear combination of the eight elementary excitations $ v\in V_{p,\epsilon}$.

We will be interested in neutrons that scatter off with zero momentum transfer. The Zamolodchikov theory ([Zamolodchikov1989a, Zamolodchikov1989b]) predicted, that the neutron scattering intensity $ \mathcal{S}(0,\omega)$ should have peaks at $ \omega = m_a$, ($ a = 1, \ldots, 8$) of the Eq. (31). At zero momentum transfer, a neutron scattering experiment would measure the proportion of neutrons that scattered off with the energies $ m_1, \ldots, m_8$: the resulting $ \mathcal{S}(0,\omega)$ would look as in the schematic Fig. 48. Metaphorically speaking, the crystal would thus "sound" as a "chord" of eight "notes": the eigenfrequencies $ m_i$.

Figure 4 A sketch of the scattering intensity $ \mathcal{S}(0,\omega)$ at zero momentum peaks relative to $ \mathcal{S}(0, m_1)$, against the $ \omega / m_1$ ratio. The two leftmost peaks shown by thick lines correspond to the excitations with the masses $ m_1$ and $ m_2$, that were resolved in the experiment ([Coldea et al.2010]). The experimentally found mass ratio $ m_2/m_1$ is consistent with $ \frac{m_2}{m_1} = \frac{1 + \sqrt{5}}{2}$, as per the expression for the $ v_{Zam}(m)$ in the Sect. 4.10.

At the lowest temperatures, and in the immediate vicinity of $ h_x = h_c$, the experiment ([Coldea et al.2010]) succeeded to resolve the first two excitations, and to extract their masses $ m_1$ and $ m_2$. The mass ratio $ m_2/m_1$ was found to be $ \frac{m_2}{m_1} = 1.6 \pm 0.025$, consistent with $ \frac{m_2}{m_1} = \frac{1 + \sqrt{5}}{2} \approx 1.618$ of the expression for the $ v_{Zam}(m)$ in the Sect. 4.10. In other words, the experimentalists were able to hear two of the eight notes of the Zamolodchikov $ E_8$ chord.

A reader wishing to find out more about various facets of the story is invited to turn to the references ([Rajaraman1989, Delfino2004, Gosslevi2010, Borthwick and Garibaldi2011]).