The classic Mckay correspondence gives a connection between finite subgroups of $\operatorname{SU}(2)$ and the simply-laced Dynkin diagrams. In this article, a direct proof is presented. The bipartite structure of the Mckay diagrams is introduced. After that, the similar method can be used on finite subgroups of $\operatorname{SO}(4)$, we get a edges-coloured graph. We finally get some applications.
In this paper characters of the normaliser of $d$-split Levi subgroups in $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer.These results are applied to verify the Alperin-McKay conjecture for primes $\ell$ with $\ell\nmid 6(q^2-1)$ and the Alperin weight conjecture for $\ell$-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of $\mathrm {SL}_n(q)$ and $\mat…
Given $Γ$ a finite subgroup of $\mathbf{SL}_3\mathbb{C}$, we determine how an arbitrary finite dimensional irreducible representation of $\mathbf{SL}_3\mathbb{C}$ decomposes under the action of $Γ$. To the subgroup $Γ$ we attach a generalized Cartan matrix $C_Γ$. Then, inspired by B. Kostant, we decompose the Coxeter element of the Kac-Moody algebra attached to $C_Γ$ as a product of reflections of a special form, thereby suggesting an algebraic form for the McKay correspondence in dimension 3.
According to McKay (1980) the irreducible characters of finite subgroups of SU(2) are in a natural 1-1 correspondence with the extended Coxeter-Dynkin graphs of type ADE. We show that the character values themselves can be given by an uniform formula, as special values of polynomials which arise naturally as numerators of Poincare series associated to finite subgroups of SU(2) acting on polynomials in two variables. These polynomials have been the subject of a number of investigations, but their interpretation as characters has apparently not been noticed.
A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…
Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classica…
Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq 0}V^{\otimes k}$ are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of $\mathrm{SU}_2$ give rise to the exponents of all untwisted and twisted affine Lie algebras except ${\rm A}_{2n}^{(1)}$.
In this paper, we study the relationship between the McKay quivers of a finite subgroups $G$ of special linear groups general linear groups, via some natural extension and embedding. We show that the McKay quiver of certain extension of a finite subgroup $G$ of $\mathrm{SL}(m,\mathbb C)$ in $\mathrm{GL}(m,\mathbb C)$ is a regular covering of the McKay quiver of $G$, and when embedding $G$ in a canonical way into $\mathrm{GL}(m-1,\mathbb C)$, the new McKay quiver is obtained by adding an arrow from the Nakayama translation of $i$ back to $i$ for each $i$. We also show that certain interesting e…
For a finite group G and a finite-dimensional G-module V, we prove a general result on the Poincaré series for the G-invariants in the tensor algebra T(V). We apply this result to the finite subgroups G of the 2-by-2 special unitary matrices and their natural module V of 2-by-1 column vectors. Because these subgroups are in one-to-one correspondence with the simply laced affine Dynkin diagrams by the McKay correspondence, the Poincaré series obtained are the generating functions for the number of walks on the simply laced affine Dynkin diagrams.
The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to $p$ ($p$ a prime) of a finite group $G$ and those of the subgroup $N$, the normalizer of Sylow $p$-subgroup. In this paper we observe that MC implies the existence of analogous bijections involving various pairs of algebras, including certain crossed products, and that MC is \emph{equivalent} to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is \emph{equivalent} to parallel st…
For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of $SL(2)$. Moreover, for a normal subgroup scheme $\mathcal{N}$ of a finite group scheme $\mathcal{G}$, we show that there is a connection between the ramification indices of the restriction morphism $\mathbb{P}(\mathcal{V}_{\mathcal{N}})\rightarrow\mathbb{P}(\mathcal{V}_{\mathcal{G}})$ between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quiver…
For a finite subgroup $G$ of the special unitary group $SU_2$, we study the centralizer algebra $Z_k(G) = End_G(V^{\otimes k})$ of $G$ acting on the $k$-fold tensor product of its defining representation $V= \mathbb{C}^2$. These subgroups are in bijection with the simply-laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of $Z_k(G)$ as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram.
In this paper we show that for any affine complete rational surface singularity there is a correspondence between the dual graph of the minimal resolution and the quiver of the endomorphism ring of the special CM modules. We thus call such an algebra the reconstruction algebra. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,C), it means that the endomorphism ring of the special CM C[[x,y]]^G modules can be used to build the dual graph of the…
We prove the Alperin-McKay Conjecture for all $p$-blocks of finite groups with metacyclic, minimal non-abelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order $p$. In the special case $p=3$, we also verify Alperin's Weight Conjecture for these defect groups. Moreover, in case $p=5$ we do the same for the non-abelian defect groups $C_{25}\rtimes C_{5^n}$. The proofs do not rely on the classification of the finite simple groups.
Let $G$ be a finite subgroup of $\mbox{GL}(2)$ acting on $\mathbf{A}^2\setminus\{0\}$ freely. The $G$-orbit Hilbert scheme $G\mbox{-Hilb}(\mathbf{A}^2)$ is a minimal resolution of the quotient $\mathbf{A}^2/G$. We determine the generator sheaf of the ideal defining the universal $G$-cluster over $G\mbox{-Hilb}(\mathbf{A}^2)$, which somewhat strengthens the well-known McKay correspondence for a finite subgroup of $\mbox{SL}(2)$. We also study the quiver structure of $G\mbox{-Hilb}(\mathbf{A}^2)$ at every $G$-cluster $O_{Z_y}=O_{\mathbf{A}^2}/I_y$ in terms of a collection of sort of minimal $G$-…
We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of $SL_2(\mathbb C)$ via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of $ADE$ types.
We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
We give a new moduli construction of the minimal resolution of the singularity of type 1/r(1,a) by introducing the Special McKay quiver. To demonstrate that our construction trumps that of the G-Hilbert scheme, we show that the induced tautological line bundles freely generate the bounded derived category of coherent sheaves on X by establishing a suitable derived equivalence. This gives a moduli construction of the Special McKay correspondence for abelian subgroups of GL(2).
Given a finite group $G$ and its representation $ρ$, the corresponding McKay graph is a graph $Γ(G,ρ)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $π,τ$ of $Γ(G,ρ)$ is $dim Hom_G(π\otimes ρ, τ) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (u…
We establish a $q$-analog of our recent work on vertex representations and the McKay correspondence. For each finite group $Γ$ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ\times \mathbb C^{\times}$ and the symmetric groups. An important special case is obtained when $Γ$ is a finite subgroup of $SU_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.
The classical McKay correspondence for finite subgroups $G$ of $\SL(2,\C)$ gives a bijection between isomorphism classes of nontrivial irreducible representations of $G$ and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity $\A^2_\C/G$. Over non algebraically closed fields $K$ there may exist representations irreducible over $K$ which split over $\bar{K}$. The same is true for irreducible components of the exceptional divisor. In this paper we show that these two phenomena are related and that there is a bijection between nontrivial irreduc…
The $n$-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $n$-slice algebras via their $(n+1)$-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame $n$-slice algebras to the McKay quiver of a finite subgroup of $\mathrm{GL}(n+1, \mathbb C)$. In the case of $n=2$, we describe the relations for the $2$-slice algebras related to the McKay quiver of finite Abelian subgroups of $\mathrm{SL}(3, \mathbb C)$ and of the finite subgroups obtained from em…
We show that for every prime $r$ all $r$-subgroups in the normalized units of the integral group ring of $\operatorname{PSL}(2,p^3)$ are isomorphic to subgroups of $\operatorname{PSL}(2,p^3)$. This answers a question of M. Hertweck, C.R. Höfert and W. Kimmerle for this series of groups.
Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent b…
We are interested in the McKay quiver $Γ(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group…
Let $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $Π$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $Π$ occurs inside a principal series representation of $G$, induced from the $\operatorname{O}(n)$-representation $\bigwedge\nolimits^p(\mathbb{C}^n)$ and characters of a minimal parabolic subgroup of $G$ at the limit of the complementary series. Considering the subgroup $G'=\operatorname{O}(1,n)$ of $G$ with maximal compact subgroup $K'$, we prove branching laws and explicit Plancherel formulas for the restrictions to…
We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and $\operatorname{G}_2$. These subgroups are images of Hitchin representations. In particular, we show that every uniform lattice of $\operatorname{Sp}(2n,\mathbb{R})$, of $\operatorname{SO}(k+1,k)$ with $k\equiv1,2[4]$ and of $\operatorname{G}_2$ contains infinitely many mapping class group orbits of Zariski-dense Hitchin representations of fixed genus. Toget…
We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case $G$ is a finite group satisfying some mild conditions. Crucial in the proof is the construction of amalgamated decompositions of the elementary group $\operatorname{E}_2(\mathcal{O})$, where $\mathcal{O}$ is an order in a rational division algebra. A major step is to introduce subgroups $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ inside the so-called higher modular groups $\operatorname{SL}…
Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property $(\operatorname{T})$, both in terms of the finite group $G$ and in terms of the simple components of the semisimple algebra $\mathbb{Q}G$. Furthermore, it is shown that for $\mathcal{U}( \mathbb{Z} G)$ this property is equivalent to the weaker property $\operatorname{FAb}$ (i.e. every subgroup of finite index has finite abelianization), and in particular also to a h…
Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2 to nodes of the affine E8 Dynkin diagram. The purpose of this article is to give an explanation for this latter correspondence using elementary properties of the group PSL2. We also obtain a super analogue of McKay's observation, in which conjugacy classes of the Monster are replaced by conjugacy classes of Conway's group -- the …
This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the F-coinvariants of the component group of the center of G has prime order and if p is a good prime for G, then the relative McKay conjecture holds for G at the prime p. In particular, this conjecture is true for G^F in defining characteristic for G a simple and simply-connected group of type B_n, C_n, E_6 and E_7. Our main tools are the theory of Gelfand-Grae…
In this paper we study the scalar generalized Verma module $M$ associated to a character of a parabolic subgroup of $\operatorname{SL}(E)$. Here $E$ is a finite dimensional vector space over an algebraically closed field $K$ of characteristic zero. The Verma module $M$ has a canonical simple quotient $L$ with a canonical filtration $F$. In the case when the quotient $L$ is finite dimensional we use left annihilator ideals in $U(\mathfrak{sl}(E))$ and geometric results on jet bundles to generalize to an algebraically closed field of characteristic zero a classical formula of W. Smoke on the str…
