The Coxeter elements of the classical types A_n, B_n, D_n and I_2(m): characteristic polynomials, orders and spectral exponents from their reflection models

(Model and identification.) Let $E$ be the model space of [F1] with its standard inner product, with simple roots ${\alpha }_1,\dots ,{\alpha }_n$: for $A_n$, $E=\{(y_1,\dots ,y_{n+1}):\sum y_i=0\}\subseteq {\mathbb{R}}^{n+1}$ and ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$; for $B_n$, $E={\mathbb{R}}^n$, ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ $(i<n)$ and ${\alpha }_n={\epsilon }_n$; for $D_n$, $E={\mathbb{R}}^n$, ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ $(i\le n-1)$ and ${\alpha }_n={\epsilon }_{n-1}+{\epsilon }_n$. These are the standard simple roots and diagram numbering of [F1] and of the classification [F3]: the branch node of the $D_n$ star is ${\alpha }_{n-2}$ with arms ${\alpha }_{n-1},{\alpha }_n$ of one vertex each and ${\alpha }_1,\dots ,{\alpha }_{n-3}$ of $n-3$ vertices. Define the linear map $\phi :V\to E$ by $\phi (e_i):={\alpha }_i\mathrm{/}\mathrm{\mid }{\alpha }_i\mathrm{\mid }$ on the basis $(e_i)$ of $V$. For every pair: $({\alpha }_i,{\alpha }_i)=2$ except $({\alpha }_n,{\alpha }_n)=1$ in type $B_n$, so $\phi$ preserves norms of basis vectors; adjacent simple roots with all labels $3$ satisfy $({\alpha }_i,{\alpha }_j)\mathrm{/}(\mathrm{\mid }{\alpha }_i\mathrm{\mid }\mathrm{\mid }{\alpha }_j\mathrm{\mid })=-1\mathrm{/}2=B(e_i,e_j)$, the $B_n$ edge of label $4$ satisfies $({\alpha }_{n-1},{\alpha }_n)\mathrm{/}(\mathrm{\mid }{\alpha }_{n-1}\mathrm{\mid }\mathrm{\mid }{\alpha }_n\mathrm{\mid })=-1\mathrm{/}\sqrt{2}=B(e_{n-1},e_n)$, and non-adjacent pairs are orthogonal, so $B(e_i,e_j)=(\phi e_i,\phi e_j)$ for all $i,j$ the normalized simple roots form a basis of $E$, so $\phi$ is invertible and is an isometry of bilinear forms [F2, F1]. Consequently for $a\in V$ with $B(a,a)=1$ and $y=\phi v$ one has $\phi r_a{\phi }^{-1}(y)=\phi (v-2B(v,a)a)=y-2(y,\phi a)\phi a=s_{\phi a}(y)$, so conjugation by $\phi$ carries the reflection with normal $a$ to the reflection with normal $\phi a$; applied to $a=e_i$ this gives $\phi \rho (s_i){\phi }^{-1}=s_{{\alpha }_i}$. Hence $\psi :=\phi \rho {\phi }^{-1}:W\to \mathrm{G}\mathrm{L}(E)$ is a homomorphism (a conjugate of the homomorphism $\rho$) with $\psi (s_i)=s_{{\alpha }_i}$, and $\psi (c)=\psi (a)\psi (b)$ is the product of the two colour-class reflection products in the model. Since listing the two classes in the other order replaces $c$ by a conjugate element [F3], characteristic polynomials and orders computed for one listing apply to $c$ in either convention.

($I_2(m)$.) Here the model is $V$ itself, $c=s_1{s}_2$ is the product $A$ of the two simple reflections, and [F4] gives trace $2\cos (2\pi \mathrm{/}m)$, determinant $1$ and exact order $m$. For a $2\times 2$ matrix the characteristic polynomial is $X^2-\mathrm{tr}(A)X+\det (A)$, as one sees by expanding $\det (XI-A)$ [F5], so $\det (X-\rho (c))=X^2-2\cos (2\pi \mathrm{/}m)X+1$. Since $\exp (2\pi i\mathrm{/}m)=\cos (2\pi \mathrm{/}m)+i\sin (2\pi \mathrm{/}m)$ and $\exp (-2\pi i\mathrm{/}m)=\cos (2\pi \mathrm{/}m)-i\sin (2\pi \mathrm{/}m)$ by [F7], the two numbers $\exp (\pm 2\pi i\mathrm{/}m)$ have sum $2\cos (2\pi \mathrm{/}m)$ and modulus one, so $(X-\exp (2\pi i\mathrm{/}m))(X-\exp (-2\pi i\mathrm{/}m))=X^2-2\cos (2\pi \mathrm{/}m)X+1$ and the eigenvalues are $\exp (2\pi i\mathrm{/}m)$ and $\exp (-2\pi i\mathrm{/}m)$. Moreover $\exp (-2\pi i\mathrm{/}m)=\exp (2\pi i(m-1)\mathrm{/}m)$: indeed $\exp (2\pi i(m-1)\mathrm{/}m)=\exp (2\pi i)\exp (-2\pi i\mathrm{/}m)$ and $\exp (2\pi i)=\exp (i\pi )\exp (i\pi )=(-1{)}^2=1$ by multiplicativity and $e^{i\pi }+1=0$ [F7]. Hence the spectral exponents are $1$ and $m-1$, with $h=m$.

($A_n$.) The model reflection $s_{{\alpha }_i}$ acts on $E$ by the transposition $(ii+1)$ of coordinates: for $x\in E$, $s_{{\alpha }_i}(x)=x-(x_i-x_{i+1})({\epsilon }_i-{\epsilon }_{i+1})$ swaps the $i$-th and $(i+1)$-st coordinates and fixes the others. For $n=1$, $c=s_1$ acts on the line $E=\mathbb{R}({\epsilon }_1-{\epsilon }_2)$ as $-\mathrm{i}\mathrm{d}$, so $\det (X-\psi (c))=X+1=X^1+\cdots +1$, the order is $2$, and the spectral exponent is $1$ [F1, F2, step 1.1]. For $n\ge 2$ write $O={\prod }_{i\mathrm{o}\mathrm{d}\mathrm{d}}(ii+1)$ and $E_0={\prod }_{i\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}(ii+1)$ for the two colour-class products on the coordinates $1,\dots ,n+1$; they are products of pairwise disjoint transpositions, so each product has commuting factors, and $\psi (c)=O{E}_0$ (the reverse product is conjugate because $O^2=1$). On basis vectors, using the four cases: $E_0{\epsilon }_1={\epsilon }_1$ and $O{\epsilon }_1={\epsilon }_2$, so $1\to 2$; for even $2\le j\le n-1$ one has $E_0{\epsilon }_j={\epsilon }_{j+1}$ and $O{\epsilon }_{j+1}={\epsilon }_{j+2}$, so $j\to j+2$; for even $j=n$ (only when $n$ is even) $E_0{\epsilon }_n={\epsilon }_{n+1}$ and $O{\epsilon }_{n+1}={\epsilon }_{n+1}$, so $n\to n+1$; for even $j=n+1$ (only when $n$ is odd), $E_0{\epsilon }_{n+1}={\epsilon }_{n+1}$ and $O{\epsilon }_{n+1}={\epsilon }_n$, so $n+1\to n$; for odd $j\ge 3$ one has $E_0{\epsilon }_j={\epsilon }_{j-1}$ and $O{\epsilon }_{j-1}={\epsilon }_{j-2}$, so $j\to j-2$. Tracing these rules gives the single cycle $1\to 2\to 4\to 6\to \cdots \to [\text{largest even}\le n+1]\to [\text{largest odd}\le n+1]\to \cdots \to 5\to 3\to 1$ on the coordinates [F9]. In the basis $({\epsilon }_1,{\epsilon }_2,{\epsilon }_4,\dots )$ ordered along this cycle, the matrix of the cycle has $X$ on the diagonal, $-1$ on the subdiagonal and $-1$ in the top-right corner; in the Leibniz sum for $\det (XI-M)$ the only nonzero terms are the identity permutation, contributing $X^N$, and the $N$-cycle $(1NN-1\cdots 2)$, of sign $(-1{)}^{N-1}$ and entry product $(-1{)}^N$, contributing $-1$; hence the characteristic polynomial is $X^N-1$ with $N=n+1$ [F5]. The full coordinate space decomposes as the invariant direct sum $E_{\mathbb{C}}\oplus \mathbb{C}(1,\dots ,1)$, and the second summand is fixed, so on the $n$-dimensional space $E$ the operator $\psi (c)$ has characteristic polynomial $(X^{n+1}-1)\mathrm{/}(X-1)=X^n+X^{n-1}+\cdots +1$; by [F8] its eigenvalues are the $n$ numbers $\exp (2\pi ik\mathrm{/}(n+1))$, $k=1,\dots ,n$, each once. The restricted spectrum includes the primitive $(n+1)$-st root $\exp (2\pi i\mathrm{/}(n+1))$, so the order of $\psi (c){\mathrm{\mid }}_E$ is divisible by $n+1$; the full cycle has order $n+1$, so the restricted order is exactly $n+1$. Thus the spectral exponents are $1,2,\dots ,n$ with $h=n+1$.

($B_n$.) Here ${\alpha }_n={\epsilon }_n$ and the model reflections are $s_i=(ii+1)$ for $i<n$ and $s_n$ negating the $n$-th coordinate. Let $O={\prod }_{i\mathrm{o}\mathrm{d}\mathrm{d}\le n}{s}_i$ and $E_0={\prod }_{i\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\le n}{s}_i$ be the two colour-class products of the path $1-2-\cdots -n$ and let $\psi (c)=O{E}_0$. Direct composition on basis vectors gives, uniformly in the parity of $n$: $c({\epsilon }_1)={\epsilon }_2$; $c({\epsilon }_j)={\epsilon }_{j+2}$ for even $j\le n-2$; $c({\epsilon }_{n-1})=-{\epsilon }_n$ and $c({\epsilon }_n)={\epsilon }_{n-2}$ when $n$ is odd; $c({\epsilon }_n)=-{\epsilon }_{n-1}$ when $n$ is even; and $c({\epsilon }_j)={\epsilon }_{j-2}$ for odd $j\ge 3$. For example, once the chain of even indices reaches the last coordinates the sign from $s_n$ appears and the negative coordinates are traversed in reverse: for $n=4$, ${\epsilon }_1\mapsto {\epsilon }_2\mapsto {\epsilon }_4\mapsto -{\epsilon }_3\mapsto -{\epsilon }_1$; for $n=5$, ${\epsilon }_1\mapsto {\epsilon }_2\mapsto {\epsilon }_4\mapsto -{\epsilon }_5\mapsto -{\epsilon }_3\mapsto -{\epsilon }_1$. Consequently the orbit of ${\epsilon }_1$ under $c$ consists of the $2n$ vectors $\pm {\epsilon }_1,\dots ,\pm {\epsilon }_n$, each exactly once: its first $n$ elements $f_0:={\epsilon }_1,f_1:=c{\epsilon }_1,\dots ,f_{n-1}:=c^{n-1}{\epsilon }_1$ are $\pm {\epsilon }_j$ with pairwise distinct $j$ (for $n=2$: ${\epsilon }_1,{\epsilon }_2$; $n=3$: ${\epsilon }_1,{\epsilon }_2,-{\epsilon }_3$; $n=4$: ${\epsilon }_1,{\epsilon }_2,{\epsilon }_4,-{\epsilon }_3$; $n=5$: ${\epsilon }_1,{\epsilon }_2,{\epsilon }_4,-{\epsilon }_5,-{\epsilon }_3$), hence a basis, and $c^n{\epsilon }_1=-{\epsilon }_1$. In this basis $c{f}_i=f_{i+1}$ $(i<n-1)$ and $c{f}_{n-1}=-f_0$, so the matrix of $c$ has $X$ on the diagonal, $-1$ on the subdiagonal and $+1$ in the top-right corner of $XI-M$; as in 2.1 the only nonzero Leibniz terms give $\det (XI-M)=X^n+1$ [F5]. Since $c^n$ commutes with $c$ and $c^n{\epsilon }_1=-{\epsilon }_1$, it sends each basis vector $f_j=c^j{\epsilon }_1$ to $-f_j$, so $c^n=-\mathrm{i}\mathrm{d}$ on $E$ and $c^{2n}=\mathrm{i}\mathrm{d}$. The orbit of ${\epsilon }_1$ already contains the $2n$ distinct signed coordinate vectors, so the order of $c$ is exactly $2n$. By [F8] applied to $z=-1$ the eigenvalues are the numbers $\exp (i(\pi +2\pi k)\mathrm{/}n)=\exp (2\pi i(2k+1)\mathrm{/}(2n))$, $k=0,\dots ,n-1$: the spectral exponents are the odd residues $1,3,\dots ,2n-1$ with $h=2n$.

($D_n$.) Here ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i\le n-1$ and ${\alpha }_n={\epsilon }_{n-1}+{\epsilon }_n$; the simple reflections $s_i$ for $i<n$ swap coordinates $i,i+1$, while $s_n$ sends ${\epsilon }_{n-1}\mapsto -{\epsilon }_n$ and ${\epsilon }_n\mapsto -{\epsilon }_{n-1}$. The branch node is $n-2$. Choose the color class containing the branch node as $J$ and use $c=ab$ with $J$ first; reversing the classes conjugates the product by [F3]. If $n$ is even, $J=\{2,4,\dots ,n-2\}$ and $K=\{1,3,\dots ,n-3,n-1,n\}$; applying the commuting reflections in $b$ and then $a$ gives $c({\epsilon }_1)={\epsilon }_3$, $c({\epsilon }_j)={\epsilon }_{j+2}$ for odd $3\le j\le n-3$, $c({\epsilon }_{n-1})=-{\epsilon }_{n-2}$, $c({\epsilon }_j)={\epsilon }_{j-2}$ for even $4\le j\le n-2$, $c({\epsilon }_2)={\epsilon }_1$, and $c({\epsilon }_n)=-{\epsilon }_n$. If $n$ is odd, $J=\{1,3,\dots ,n-2\}$ and $K=\{2,4,\dots ,n-3,n-1,n\}$; the same reflection formula gives $c({\epsilon }_1)={\epsilon }_2$, $c({\epsilon }_j)={\epsilon }_{j+2}$ for even $2\le j\le n-3$, $c({\epsilon }_{n-1})=-{\epsilon }_{n-2}$, $c({\epsilon }_j)={\epsilon }_{j-2}$ for odd $3\le j\le n-2$, and $c({\epsilon }_n)=-{\epsilon }_n$. The formulas show that, on $E^{\mathrm{\prime }}:=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}({\epsilon }_1,\dots ,{\epsilon }_{n-1})$, the first $m:=n-1$ iterates of ${\epsilon }_1$ are a signed coordinate basis: for even $n$ they are ${\epsilon }_1,{\epsilon }_3,\dots ,{\epsilon }_{n-1},-{\epsilon }_{n-2},-{\epsilon }_{n-4},\dots ,-{\epsilon }_2$, and for odd $n$ they are ${\epsilon }_1,{\epsilon }_2,{\epsilon }_4,\dots ,{\epsilon }_{n-1},-{\epsilon }_{n-2},-{\epsilon }_{n-4},\dots ,-{\epsilon }_3$. In this basis $c$ advances each vector to the next and sends the last to $-{\epsilon }_1$, so the companion determinant computation of 2.2 gives $\det (XI-c{\mathrm{\mid }}_{E^{\mathrm{\prime }}})=X^m+1$ and $c^m{\epsilon }_1=-{\epsilon }_1$. Since $c^m$ commutes with $c$, it is $-\mathrm{i}\mathrm{d}$ on the orbit basis of $E^{\mathrm{\prime }}$, so the orbit of ${\epsilon }_1$ and its negatives contains the $2m$ distinct signed basis vectors and the order on $E^{\mathrm{\prime }}$ is exactly $2m$; the remaining coordinate line has eigenvalue $-1$, so the full order is $\mathrm{lcm}(2m,2)=2m=2n-2$. Thus $\det (XI-\psi (c))=(X^{n-1}+1)(X+1)$. By [F8], the roots of $X^m+1$ are $\exp (2\pi i(2k+1)\mathrm{/}(2m))$ for $k=0,\dots ,m-1$, and the extra root $-1$ is $\exp (2\pi im\mathrm{/}(2m))$; the spectral exponents are $1,3,\dots ,2n-3$ and one additional $n-1$ (counted again when $n$ is even), with $h=2n-2$.

The map $\psi =\phi \rho {\phi }^{-1}$ is conjugate to $\rho$, and complexification preserves characteristic polynomials and operator orders, so steps 1.2-2.3 compute $\det (X-{\rho }_{\mathbb{C}}(c))$ and its order for the four types. Since ${\rho }_{\mathbb{C}}$ is faithful by [F2], this operator order equals the order of $c$ in $W$, which is the Coxeter number $h$ used in the statement. The eigenvalues listed there are the spectral exponents in the sense fixed in the statement, since conjugate linear maps have the same eigenvalues with the same multiplicities. Finally ${\rho }_{\mathbb{C}}(c)$ has finite order because $W$ is finite, so it is diagonalisable by [F6] over $\mathbb{C}$; this holds in each type and shows both the diagonalisability and, together with the characteristic polynomials, the displayed spectra. Nothing above uses invariant degrees, Hilbert series or any degree table: only the reflection models, the rank-two computation and roots of unity are used, and the statement makes no assertion about $h={\max }_i(d_i)$, which is proved only by the final determination theorem of this page.