Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G)={c(1)  |  c ? Irr(G)}{{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}} be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and cd(S) í cd(H){{\rm cd}(S)\subseteq {\rm cd}(H)} then S must be isomorphic to H. As a consequence, we show that if G is a finite group with X₁(G) í X₁(H){{\rm X}_1(G)\subseteq {\rm X}_1(H)} then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

Simple exceptional groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let ${{\rm cd}(G)=\{\chi(1)\;|\;\chi\in {\rm Irr}(G)\}}$ be the set of all irreducible complex character degrees of G forgetting multiplicities, and let X₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if S is a non-abelian simple group and ${{\rm cd}(S)\subseteq {\rm cd}(H)}$ then S must be isomorphic to H. As a consequence, we show that if G is a finite group with ${{\rm X}_1(G)\subseteq {\rm X}_1(H)}$ then G is isomorphic to H. In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

Quantum cohomology of <Emphasis Type="Italic">G/P</Emphasis> and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH ^*(G/P) of a flag variety is, up to localization, a quotient of the homology H _*(Gr _G ) of the affine Grassmannian Gr _G of G. As a consequence, all three-point genus-zero Gromov–Witten invariants of G/P are identified with homology Schubert structure constants of H _*(Gr _G ), establishing the equivalence of the quantum and homology affine Schubert calculi.     

Automorphisms of blocks of group algebras and character values

Let G be a finite group, χ an irreducible complex character of G and A(χ) the block ideal of the group algebra ℚG relatedℴ χ. The aim of this paper is to study the group Aut (A(χ)) of all ring (or ℚ-algebra) automorphisms of A(χ). Especially we are interested in the existence of subgroups of Aut (A(χ)), which are isomorphic to a given subgroup Γ of the Galois group of the field of character values ℚ(χ) over the rationals. In this context we prove some results related to character values.     

Analytic actions on compact surfaces and fixed points

 Consider an effective real analytic action of a connected Lie group G on a compact connected surface of Euler characteristic χ≠0. We show that if the action has no fixed point then χ≥1 and the Lie algebra 𝒢 of G is isomorphic either to a subalgebra of the affine algebra of ℝ², which is the extension of the ideal of constant vector fields by an irreducible linear subalgebra, or to sl(2,ℝ), o(3), sl(2,ℂ) and sl(3,ℝ). Received: 7 August 2001 Published online: 24 January 2003     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

Primes dividing the degrees of the real characters

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irr_rv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr_rv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained. Part of this paper was done while the second author visited the Mathematics Department of the Università di Firenze. He would like to thank the Department for its hospitality. The authors are also grateful to F. Lübeck for helping them with some computer calculations. The research of the first author was partially supported by MIUR research program “Teoria dei gruppi ed applicazioni”. This research of the second author was partially supported by the Spanish Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01. The third author gratefully acknowledges the support of the NSA and the NSF.     

A homological characterization of hyperbolic groups

A finitely presented group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0=⁽¹⁾ ₂(G, ℝ), where H ⁽¹⁾ _* (resp. ⁽¹⁾ _*) denotes the ℓ₁-homology (resp. reduced ℓ₁-homology). If Γ is a graph, then every ℓ₁ 1-cycle in Γ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0. Oblatum: 30-IV-1997 & 14-V-1998 / Published online: 14 January 1999     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)^* has the weak uniform A(G × H)^** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC ₂(G × H)^* is equal to the Fourier–Stieltjes algebra B(G × H).     

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl ^∞(G)^* the conjugate ofl ^∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l ^∞(G)^* such that γμ=0 whenever γ is an annihilator ofC ₀(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl ^∞(G)^*, and each of them generates a left ideal ofl ^∞(G)^* that contains no minimal left ideal.     

Finite groups with three composite character degrees

Let G be a finite group and let cd(G) be the set of all irreducible character degrees of G. We consider finite groups G with the property that cd(G) has at most three composite members. We derive a bound 8 for the size of character degree sets of such groups.     

On thin-complete ideals of subsets of groups

Let F ì P_G \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t^*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of P_G {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g _n ) _n∈ω of nonzero elements of G, there is n ∈ ω such that

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

A Higgs bundle on a Hermitian symmetric space,II

Let M : = Γ\G/K be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice G ì G{\Gamma \subset G}. Let V be a complex irreducible representation of G. We give a Hodge decomposition of the cohomology of the Γ-module V in terms of the cohomologies of automorphic vector bundles on M associated to the Lie algebra cohomologies H*(\mathfrak p⁺ ,V){H*({\mathfrak p}^{+} ,V)}.     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M