Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group is largely determined by a linearly reductive subgroup scheme of SL(2), with the McKay quiver of relative to its standard module being the Gabriel quiver of the principal block . The graphs underlying these quivers are extended Dynkin diagrams of type or , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.     

p -Sparse BEM for weakly singular integral equation with random data

We consider the weakly singular boundary integral equation 𝒱u = g on a randomly perturbed smooth closed surface Γ(ω) with deterministic g or on a deterministic closed surface Γ with stochastic g (ω). The aim is the computation of the centered moments ℳ︁^k u, k ⩾ 1, if the corresponding moments of the perturbation are known. The problem on the stochastic surface is reduced to a problem on the nominal deterministic surface Γ with the random perturbation parameter κ (ω). Resulting formulation for the k th moment is posed in the tensor product Sobolev spaces and involve the k -fold tensor product operators. The standard full tensor product Galerkin BEM requires 𝒪(N^k) unknowns for the k th moment problem, where N is the number of unknowns needed to discretize the nominal surface Γ. Based on [1], we develop the p -sparse grid Galerkin BEM to reduce the number of unknowns to 𝒪(N (log N)^{k –1}) (cf. [2], [3] for the wavelet approach). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Yoneda Algebras of Serial QF-Algebras

The Yoneda algebras of serial QF-algebras over an algebraically closed field k are described in terms of quivers with relations. We also describe the Ext-algebras of simple modules over the k-algebras in question. Bibliography: 7 titles.     

Various Structures Associated to the Representation Categories of Eight-Dimensional Nonsemisimple Hopf Algebras

We determine various additional structures on all nonsemisimple Hopf algebras of dimension 8 over an algebraically closed field k of characteristic 0, including their representation rings and quasitriangular structures. As a consequence, it is shown that for two such Hopf algebras, the tensor categories of their representations are monoidally equivalent if and only if the representation rings of them are isomorphic as rings. An erratum to this article is available at .     

Universal Deformation Rings for the Symmetric Group <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S ₄ denote the symmetric group on 4 letters. We determine the universal deformation ring R(S ₄,V) for every kS ₄-module V which has stable endomorphism ring k and show that R(S ₄,V) is isomorphic to either k, or W[t]/(t ²,2t), or the group ring W[ℤ/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.     

Commuting Nilpotent Operators and Maximal Rank

Let X, [(X)\tilde]{\widetilde X} be commuting nilpotent matrices over k with nilpotency p ^t , where k is an algebraically closed field of positive characteristic p. We show that if X- [(X)\tilde]{X- \widetilde X} is a certain linear combination of products of pairwise commuting nilpotent matrices, then X is of maximal rank if and only if [(X)\tilde]{\widetilde X} is of maximal rank.     

Basic Hopf algebras and quantum groups

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers given in terms of a finite group G and sequence of elements of G in the following way can occur. The quiver has vertices and arrows , where the set is closed under conjugation with elements in G and for each g in G, the sequences W and are the same up to a permutation. We show how is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R]. Received August 15, 1994; in final form May 16, 1997     

Decomposing thick subcategories of the stable module category

Let be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of . It is shown that every thick tensor-ideal of (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module into indecomposable modules. If is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring , then the decomposition of reflects the decomposition of W into connected components. Received: 27 April 1998 / In revised form: 16 July 1998     

Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

In this paper, we investigate the tensor structure of the category of finite- dimensional weight modules over the Hopf–Ore extensions kG(χ^?1,a,0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same.     

Gaussian fluctuations of characters of symmetric groups and of Young diagrams

We study asymptotics of reducible representations of the symmetric groups S _q for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, the left-regular representation (Plancherel measure), irreducible representations and tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.     

When is a ring of torus invariants a polynomial ring?

Let ρ:T→GL(V) be a finite dimensional rational representation of a torus over an algebraically closed fieldk. We give necessary and sufficient conditions on the arrangement of the weights ofV within the character lattice ofT for the ring of invariants,k[V] ^T , to have a homogeneous system of parameters consisting of monomials (Theorem 4.1). Using this we give two simple constructive criteria each of which gives necessary and sufficient conditions fork[V] ^T to be a polynomial ring (Theorem 5.8 and Theorem 5.10). Research supported in part by NSERC Grant OGP 137522     

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k‐Einstein (in the sense that their 2k‐Ricci tensor is constant) or have constant 2k‐Gauss‐Bonnet curvature. The results hold for a family of manifolds containing all non‐flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.     

A polynomial path-following interior point algorithm for general linear complementarity problems

Linear Complementarity Problems (LCPs) belong to the class of \mathbbNP{\mathbb{NP}} -complete problems. Therefore we cannot expect a polynomial time solution method for LCPs without requiring some special property of the coefficient matrix. Our aim is to construct interior point algorithms which, according to the duality theorem in EP (Existentially Polynomial-time) form, in polynomial time either give a solution of the original problem or detects the lack of property P_*([(k)\tilde]){\mathcal{P}_*(\tilde\kappa)} , with arbitrary large, but apriori fixed [(k)\tilde]{\tilde\kappa}). In the latter case, the algorithms give a polynomial size certificate depending on parameter [(k)\tilde]{\tilde{\kappa}} , the initial interior point and the input size of the LCP). We give the general idea of an EP-modification of interior point algorithms and adapt this modification to long-step path-following interior point algorithms.     

Tensor Product Representations of the Lie Superalgebra 𝔭(n) and Their Centralizers

Abstract

We construct an associative algebra A _k and show that there is a representation of A _k on V ^?k , where V is the natural 2n-dimensional representation of the Lie superalgebra 𝔭(n). We prove that A _k is the full centralizer of 𝔭(n) on V ^?k , thereby obtaining a “Schur-Weyl duality” for the Lie superalgebra 𝔭(n). This result is used to understand the representation theory of the Lie superalgebra 𝔭(n). In particular, using A _k we decompose the tensor space V ^?k , for k = 2 or 3, and show that V ^?k is not completely reducible for any k ≥ 2.     

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.