The McKay correspondence, the Coxeter element and representation theory.

*(English)*Zbl 0605.22010
Élie Cartan et les mathématiques d’aujourd’hui. The mathematical heritage of Élie Cartan, Semin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 209-255 (1985).

[For the entire collection see Zbl 0573.00010.]

Let \(\Gamma\) be a nontrivial finite subgroup of \(SU(2)\) and let \({\hat \Gamma}=\{\gamma_ 0,\gamma_ 1,...,\gamma_{\ell})\) be the set of equivalence classes of irreducible finite-dimensional complex representations of \(\Gamma\). Let \(\gamma: \Gamma\to SU(2)\) be the given 2-dimensional representation, and let \(A(\Gamma)\) denote the \((\ell+1)\times (\ell+1)\) matrix whose (0,j) entry \(A(\Gamma)_{ij}\) is the multiplicity of \(\gamma_ i\) in \(\gamma_ j\otimes \gamma\). It was observed by McKay that corresponding to \(\Gamma\) there exists a complex simple Lie algebra \({\mathfrak g}=\mu(\Gamma)\) of rank \(\ell\) such that \(2- A(\Gamma)\) is the Cartan matrix \(C(\tilde{\mathfrak g})\) of the affine Kac- Moody Lie algebra \(\tilde{\mathfrak g}\) associates with \({\mathfrak g}\), and that the correspondence \(\Gamma \mapsto \mu (\Gamma)={\mathfrak g}\) sets up a bijection between the set of isomorphism classes of finite subgroups \(\Gamma\) of \(SU(2)\) and the set of all isomorphism classes of complex simple Lie algebras of types A, D or E.

(Explicitly, the correspondence \(\mu\) is as follows. Let \(C_ n,\Delta_ n,A_ n,S_ n\) denote respectively the cyclic group of order n, the dihedral group of order 2n, the alternating group on n letters and the symmetric group on n letters. Then every finite subgroup F of \(SO(3)\) is isomorphic to one of \(C_ n,\Delta_ n,A_ 4,S_ 4\) or \(A_ 5\). Now let \(SU(2)\to SO(3)\) be the usual double covering, and for any finite subgroup F of \(SO(3)\) let \(F^*\) denote its inverse image in \(SU(2)\), so that \(| F^*| =2| F|\). Then every finite subgroup \(\Gamma\) of \(SU(2)\) that is not cyclic of odd order is of the form \(F^*\) for some finite subgroup F of \(SO(3)\), and the groups \(C^*_ n,\Delta^*_ n,A^*_ 4,S^*_ 4,A^*_ 5\) correspond respectively (in the correspondence \(\mu)\) to the Lie algebras of types \(A_{2n-1},D_{n+2},E_ 6,E_ 7,E_ 8.)\)

The Cartan matrix \(C(\tilde{\mathfrak g})\) is relative to an ordered set of simple roots \(\alpha_ i\in \tilde{\mathfrak h}'\) \((0\leq i\leq \ell)\), where \({\mathfrak h}\subset \tilde{\mathfrak h}\) are respectively Cartan subalgebras of \({\mathfrak g}\subset \tilde{\mathfrak g}\). The indexing may be chosen so that \(\gamma_ 0\) is the trivial representation of \(\Gamma\) and \(\alpha_ 0\in \tilde{\mathfrak h}'\) is the added simple root corresponding to the negative of the highest root \(\psi\in {\mathfrak h}'\) of \({\mathfrak g}.\)

Let \(\pi_ n\) be the representation of \(SU(2)\) on the nth symmetric power \(S^ n({\mathbb{C}}^ 2)\) \((n\geq 0)\); these are a complete set of irreducible unitary representations of the compact group SU(2). Consider the restriction of \(\pi_ n\) to the finite subgroup \(\Gamma\): if \(\pi_ n| \Gamma =\sum^{\ell}_{i=0}m_ i \gamma_ i\), we associate with \(\pi_ n\) the element \(v_ n\in \tilde {\mathfrak h}'\) in the root lattice of \((\tilde{\mathfrak g},\tilde{\mathfrak h})\) defined by \(v_ n=\sum^{\ell}_{i=0}m_ i \alpha_ i\). We may then form the generating function \(P_{\Gamma}(t)=\sum^{\infty}_{n=0}v_ n t^ n\), with coefficients in \(\tilde{\mathfrak h}'\). The problem to which this paper is devoted is the determination of \(P_{\Gamma}(t)\) in terms of data derived from the root structure of \({\mathfrak g}=\mu(\Gamma)\). The author assumes throughout that the Coxeter number h of \({\mathfrak g}\) is even, or equivalently that \(\Gamma =F^*\) for some finite subgroup F of \(SO(3)\). Thus g is not of type \(A_{\ell}\) with \(\ell\) even. The Dynkin diagram of \({\mathfrak g}\) then has a special node \(i_*\), which is the branch point if \({\mathfrak g}\) is of type D or E, and the middle node if \({\mathfrak g}\) is of type \(A_{2n-1}.\)

The author’s main results can now be stated. Theorem 1: The series \(P_{\Gamma}(t)\) is of the form \(z(t)/(1-t^ a)(1-t^ b)\), where a and b are even integers satisfying \(2\leq a\leq b\leq h\), and \(z(t)=\sum^{h}_{i=0}z_ i t^ i\) with \(z_ i\in \tilde {\mathfrak h}'\). Theorem 2: We have \(a=2d\), where d is the coefficient of \(\alpha_{i_*}\) in the highest root \(\psi\) ; moreover \(a+b=h+2\) and \(ab=2| \Gamma |.\)

Next let W be the (finite) Weyl group of (\({\mathfrak g},{\mathfrak h})\). There is a Coxeter element \(\sigma\in W\) corresponding to \(\Pi =\{\alpha_ 1,...,\alpha_{\ell}\}\) such that \(\sigma =\tau_ 2\tau_ 1\), where \(\tau_ 2,\tau_ 1\in W\) have order \(\leq 2\) and correspond to a decomposition \(\Pi =\Pi_ 1\cup \Pi_ 2\) into orthogonal subsets. The order may be chosen so that \(\tau_ 2\psi =\psi\). For \(n\geq 1\) let \(\tau_ n\) denote \(\tau_ 1\) or \(\tau_ 2\) according as n is odd or even, and let \(\tau^{(n)}=\tau_ n\tau_{n-1}...\tau_ 1\). Theorem 3: We have \(z_ 0=z_ h=\alpha_ 0\). For \(1\leq n\leq h-1\), we have \(z_ n\in {\mathfrak h}'\) (not just \(\tilde{\mathfrak h}')\) and indeed \(z_ n=(\tau^{(n-1)}-\tau^{(n)})\psi\). Moreover \(z_ g=2\alpha_{i_*}\), where \(g=h/2\), and \(z_{g+k}=z_{g-k}\) for \(1\leq k\leq g.\)

The Poincaré series \(P_{\Gamma}(t)_ i\) for each individual representation \(\gamma_ i\) is obtained by considering only the ith coefficient of the vectors \(v_ n\). By Th. 1, we have \(P_{\Gamma}(t)_ i=z(t)_ i/(1-t^ a)(1-t^ b)\), where \(z(t)_ i\) is the coefficient of \(\alpha_ i\) in z(t). When \(i=0\) we have \(z(t)_ 0=1+t^ h\), and hence Theorem 4: The Poincaré series \(P_{\Gamma}(t)_ 0\) of the algebra of invariants \(S({\mathbb{C}}^ 2)^{\Gamma}\) is \((1+t^ h)/(1-t^ a)(1-t^ b)\). Next, when \(i=i_*\), the representation \(\gamma_{i_*}\) is an irreducible representation of \(\Gamma\) of maximum dimension, and we have Theorem \(5: z(t)_{i_*}=\sum^{d-1}_{j=0}(t^{g-2j}+t^{g+2j}).\)

To determine \(z(t)_ i\) for the remaining nodes i, let \(\Phi\) be the set of positive roots of (\({\mathfrak g},{\mathfrak h})\) not orthogonal to the highest root \(\psi\). Let \(\Phi_ i\) be the intersection of \(\Phi\) with the orbit of \(\alpha_ i\) under the subgroup of W generated by the Coxeter element \(\sigma\). Then \(\Phi\) is the disjoint union of the \(\Phi_ i\), \(1\leq i\leq \ell\), and \(card(\Phi_ i)\) is equal to \(2d_ i\) if \(i\neq i_*\), and to \(2d_ i-1\) if \(i=i_*\), where \(d_ i\) is the coefficient of \(\alpha_ i\) in \(\psi\) (and is the degree of the representation \(\gamma_ i)\). For each \(n\geq 1\), let \(\Pi_ n\) denote \(\Pi_ 1\) or \(\Pi_ 2\) according as n is odd or even. Then for each positive root \(\phi\) of (\({\mathfrak g},{\mathfrak h})\) there is a unique integer \(n=n(\phi)\) such that \(1\leq n\leq h\) and \(\phi \in (\tau^{(n-1)})^{-1} \Pi_ n\), and we have Theorem 6: if \(i\neq 0\) or \(i_*\) then \(z(t)_ i=\sum_{\phi \in \Phi_ i}t^{n(\phi)}\), and all the coefficients of \(z(t)_ i\) are 1 or 0. Furthermore the coefficients of \(t^{g-k}\) and \(t^{g+k}\) (where \(g=h/2)\) are equal for \(k=1,...,g\) and vanish for \(k=0\).

Let \(\Gamma\) be a nontrivial finite subgroup of \(SU(2)\) and let \({\hat \Gamma}=\{\gamma_ 0,\gamma_ 1,...,\gamma_{\ell})\) be the set of equivalence classes of irreducible finite-dimensional complex representations of \(\Gamma\). Let \(\gamma: \Gamma\to SU(2)\) be the given 2-dimensional representation, and let \(A(\Gamma)\) denote the \((\ell+1)\times (\ell+1)\) matrix whose (0,j) entry \(A(\Gamma)_{ij}\) is the multiplicity of \(\gamma_ i\) in \(\gamma_ j\otimes \gamma\). It was observed by McKay that corresponding to \(\Gamma\) there exists a complex simple Lie algebra \({\mathfrak g}=\mu(\Gamma)\) of rank \(\ell\) such that \(2- A(\Gamma)\) is the Cartan matrix \(C(\tilde{\mathfrak g})\) of the affine Kac- Moody Lie algebra \(\tilde{\mathfrak g}\) associates with \({\mathfrak g}\), and that the correspondence \(\Gamma \mapsto \mu (\Gamma)={\mathfrak g}\) sets up a bijection between the set of isomorphism classes of finite subgroups \(\Gamma\) of \(SU(2)\) and the set of all isomorphism classes of complex simple Lie algebras of types A, D or E.

(Explicitly, the correspondence \(\mu\) is as follows. Let \(C_ n,\Delta_ n,A_ n,S_ n\) denote respectively the cyclic group of order n, the dihedral group of order 2n, the alternating group on n letters and the symmetric group on n letters. Then every finite subgroup F of \(SO(3)\) is isomorphic to one of \(C_ n,\Delta_ n,A_ 4,S_ 4\) or \(A_ 5\). Now let \(SU(2)\to SO(3)\) be the usual double covering, and for any finite subgroup F of \(SO(3)\) let \(F^*\) denote its inverse image in \(SU(2)\), so that \(| F^*| =2| F|\). Then every finite subgroup \(\Gamma\) of \(SU(2)\) that is not cyclic of odd order is of the form \(F^*\) for some finite subgroup F of \(SO(3)\), and the groups \(C^*_ n,\Delta^*_ n,A^*_ 4,S^*_ 4,A^*_ 5\) correspond respectively (in the correspondence \(\mu)\) to the Lie algebras of types \(A_{2n-1},D_{n+2},E_ 6,E_ 7,E_ 8.)\)

The Cartan matrix \(C(\tilde{\mathfrak g})\) is relative to an ordered set of simple roots \(\alpha_ i\in \tilde{\mathfrak h}'\) \((0\leq i\leq \ell)\), where \({\mathfrak h}\subset \tilde{\mathfrak h}\) are respectively Cartan subalgebras of \({\mathfrak g}\subset \tilde{\mathfrak g}\). The indexing may be chosen so that \(\gamma_ 0\) is the trivial representation of \(\Gamma\) and \(\alpha_ 0\in \tilde{\mathfrak h}'\) is the added simple root corresponding to the negative of the highest root \(\psi\in {\mathfrak h}'\) of \({\mathfrak g}.\)

Let \(\pi_ n\) be the representation of \(SU(2)\) on the nth symmetric power \(S^ n({\mathbb{C}}^ 2)\) \((n\geq 0)\); these are a complete set of irreducible unitary representations of the compact group SU(2). Consider the restriction of \(\pi_ n\) to the finite subgroup \(\Gamma\): if \(\pi_ n| \Gamma =\sum^{\ell}_{i=0}m_ i \gamma_ i\), we associate with \(\pi_ n\) the element \(v_ n\in \tilde {\mathfrak h}'\) in the root lattice of \((\tilde{\mathfrak g},\tilde{\mathfrak h})\) defined by \(v_ n=\sum^{\ell}_{i=0}m_ i \alpha_ i\). We may then form the generating function \(P_{\Gamma}(t)=\sum^{\infty}_{n=0}v_ n t^ n\), with coefficients in \(\tilde{\mathfrak h}'\). The problem to which this paper is devoted is the determination of \(P_{\Gamma}(t)\) in terms of data derived from the root structure of \({\mathfrak g}=\mu(\Gamma)\). The author assumes throughout that the Coxeter number h of \({\mathfrak g}\) is even, or equivalently that \(\Gamma =F^*\) for some finite subgroup F of \(SO(3)\). Thus g is not of type \(A_{\ell}\) with \(\ell\) even. The Dynkin diagram of \({\mathfrak g}\) then has a special node \(i_*\), which is the branch point if \({\mathfrak g}\) is of type D or E, and the middle node if \({\mathfrak g}\) is of type \(A_{2n-1}.\)

The author’s main results can now be stated. Theorem 1: The series \(P_{\Gamma}(t)\) is of the form \(z(t)/(1-t^ a)(1-t^ b)\), where a and b are even integers satisfying \(2\leq a\leq b\leq h\), and \(z(t)=\sum^{h}_{i=0}z_ i t^ i\) with \(z_ i\in \tilde {\mathfrak h}'\). Theorem 2: We have \(a=2d\), where d is the coefficient of \(\alpha_{i_*}\) in the highest root \(\psi\) ; moreover \(a+b=h+2\) and \(ab=2| \Gamma |.\)

Next let W be the (finite) Weyl group of (\({\mathfrak g},{\mathfrak h})\). There is a Coxeter element \(\sigma\in W\) corresponding to \(\Pi =\{\alpha_ 1,...,\alpha_{\ell}\}\) such that \(\sigma =\tau_ 2\tau_ 1\), where \(\tau_ 2,\tau_ 1\in W\) have order \(\leq 2\) and correspond to a decomposition \(\Pi =\Pi_ 1\cup \Pi_ 2\) into orthogonal subsets. The order may be chosen so that \(\tau_ 2\psi =\psi\). For \(n\geq 1\) let \(\tau_ n\) denote \(\tau_ 1\) or \(\tau_ 2\) according as n is odd or even, and let \(\tau^{(n)}=\tau_ n\tau_{n-1}...\tau_ 1\). Theorem 3: We have \(z_ 0=z_ h=\alpha_ 0\). For \(1\leq n\leq h-1\), we have \(z_ n\in {\mathfrak h}'\) (not just \(\tilde{\mathfrak h}')\) and indeed \(z_ n=(\tau^{(n-1)}-\tau^{(n)})\psi\). Moreover \(z_ g=2\alpha_{i_*}\), where \(g=h/2\), and \(z_{g+k}=z_{g-k}\) for \(1\leq k\leq g.\)

The Poincaré series \(P_{\Gamma}(t)_ i\) for each individual representation \(\gamma_ i\) is obtained by considering only the ith coefficient of the vectors \(v_ n\). By Th. 1, we have \(P_{\Gamma}(t)_ i=z(t)_ i/(1-t^ a)(1-t^ b)\), where \(z(t)_ i\) is the coefficient of \(\alpha_ i\) in z(t). When \(i=0\) we have \(z(t)_ 0=1+t^ h\), and hence Theorem 4: The Poincaré series \(P_{\Gamma}(t)_ 0\) of the algebra of invariants \(S({\mathbb{C}}^ 2)^{\Gamma}\) is \((1+t^ h)/(1-t^ a)(1-t^ b)\). Next, when \(i=i_*\), the representation \(\gamma_{i_*}\) is an irreducible representation of \(\Gamma\) of maximum dimension, and we have Theorem \(5: z(t)_{i_*}=\sum^{d-1}_{j=0}(t^{g-2j}+t^{g+2j}).\)

To determine \(z(t)_ i\) for the remaining nodes i, let \(\Phi\) be the set of positive roots of (\({\mathfrak g},{\mathfrak h})\) not orthogonal to the highest root \(\psi\). Let \(\Phi_ i\) be the intersection of \(\Phi\) with the orbit of \(\alpha_ i\) under the subgroup of W generated by the Coxeter element \(\sigma\). Then \(\Phi\) is the disjoint union of the \(\Phi_ i\), \(1\leq i\leq \ell\), and \(card(\Phi_ i)\) is equal to \(2d_ i\) if \(i\neq i_*\), and to \(2d_ i-1\) if \(i=i_*\), where \(d_ i\) is the coefficient of \(\alpha_ i\) in \(\psi\) (and is the degree of the representation \(\gamma_ i)\). For each \(n\geq 1\), let \(\Pi_ n\) denote \(\Pi_ 1\) or \(\Pi_ 2\) according as n is odd or even. Then for each positive root \(\phi\) of (\({\mathfrak g},{\mathfrak h})\) there is a unique integer \(n=n(\phi)\) such that \(1\leq n\leq h\) and \(\phi \in (\tau^{(n-1)})^{-1} \Pi_ n\), and we have Theorem 6: if \(i\neq 0\) or \(i_*\) then \(z(t)_ i=\sum_{\phi \in \Phi_ i}t^{n(\phi)}\), and all the coefficients of \(z(t)_ i\) are 1 or 0. Furthermore the coefficients of \(t^{g-k}\) and \(t^{g+k}\) (where \(g=h/2)\) are equal for \(k=1,...,g\) and vanish for \(k=0\).

Reviewer: I.G.Macdonald

##### MSC:

22E46 | Semisimple Lie groups and their representations |

17B20 | Simple, semisimple, reductive (super)algebras |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

20E07 | Subgroup theorems; subgroup growth |