Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

The representation dimension of Hecke algebras and symmetric groups

We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and “most” of the algebras of types B and D, we also establish upper bounds. Moreover, we establish bounds for the representation dimension of group algebras of some symmetric groups.     

A note on geodesic connectedness of Gödel type spacetimes

In this note we reduce the problem of geodesic connectedness in a wide class of Gödel type spacetimes to the search of critical points of a functional naturally involved in the study of geodesics in standard static spacetimes. Then, by using some known accurate results on the latter, we improve previous results on the former.     

A semi-recursion for the number of involutions in special orthogonal groups over finite fields

Let I(n) be the number of involutions in a special orthogonal group SO(n,Fq) defined over a finite field with q elements, where q is the power of an odd prime. Then the numbers I(n) form a semi-recursion, in that for m>1 we haveI(2m+3)=(q2m+2+1)I(2m+1)+q2m(q2m−1)I(2m−2). We give a purely combinatorial proof of this result, and we apply it to give a universal bound for the character degree sum for finite classical groups defined over Fq.     

Kac–Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory Cw of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cw is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that Cw yields a cluster algebra structure on the coordinate ring C[N(w)] of the unipotent group N(w):=N∩(w−1N₋w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify C[N(w)] with a subalgebra of , the graded dual of the universal enveloping algebra U(n) of n. Let S^? be the dual of Lusztig?s semicanonical basis S of U(n). We show that all cluster monomials of C[N(w)] belong to S^?, and that S^?∩C[N(w)] is a C-basis of C[N(w)]. Moreover, we show that the cluster algebra obtained from C[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C[Nw] of regular functions on the unipotent cell Nw of the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of C[Nw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.     

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SLn(q) in the non-defining characteristic ?, describe restrictions of irreducible modules from GLn(q) to SLn(q), classify complex irreducible characters of SLn(q) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SLn(q).     

Basic sets in defining characteristic for general linear groups of small rank

Let q be a power of some prime number p. Let be a connected reductive group defined over the field with q elements and let F be the corresponding Frobenius map. In this note, we give methods to find relations between the restrictions on semisimple elements of the irreducible characters of . As illustration, we explicitly determine a p-basic set for , and .     

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C^?-algebras do not contain nontrivial idempotents.Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle? sphere.     

Singular linear space and its applications

As a generalization of attenuated spaces, the concept of singular linear spaces was introduced in [K. Wang, J. Guo, F. Li, Association schemes based on attenuated spaces, European J. Combin. 31 (2010) 297–305]. This paper first gives two anzahl theorems in singular linear spaces, and then discusses their applications to the constructions of Deza digraphs, quasi-strongly regular graphs, lattices and authentication codes.     

The classification problem for S-local torsion-free abelian groups of finite rank

Suppose that n?2 and that S, T are sets of primes. Then the classification problem for the S-local torsion-free abelian groups of rank n is Borel reducible to the classification problem for the T-local torsion-free abelian groups of rank n if and only if S⊆T.     

On special types of nonholonomic contact elements

Our starting point has been a recent clarification of the role of semiholonomic contact elements in the theory of submanifolds of Cartan geometries, Kolá? and Vitolo (2010) [5]. We deduce some further properties of the iterated contact elements by using the general concept of contact (n,F)-element for a regular subcategory F of the category of nonholonomic r-jets. Special attention is paid to the incidence relation of contact F-elements of different dimensions.     

Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.     

Discrete complex analysis on isoradial graphs

We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic measures, Green?s functions and Poisson kernels to their continuous counterparts. Among other applications, the results can be used to establish universality of the critical Ising and other lattice models.     

Jacquet modules and Dirac cohomology

We show that Dirac cohomology of the Jacquet module of a Harish-Chandra module is a Harish-Chandra module for the corresponding Levi subgroup. We obtain an explicit formula of Dirac cohomology of the Jacquet module for most of the principal series, based on our determination of Dirac cohomology of irreducible generalized Verma modules with regular infinitesimal characters.     

Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian manifolds.     

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.     

Reduced class groups grafting relative invariants

Let (X,T) be a regular stable conical action of an algebraic torus on an affine normal conical variety X defined over an algebraically closed field of characteristic zero. We define a certain subgroup of Cl(X//T) and characterize its finiteness in terms of a finite T-equivariant Galois descent of X. Consequently we show that the action (X,T) is equidimensional if and only if there exists a T-equivariant finite Galois covering such that is cofree. Moreover the order of is controlled by a certain subgroup of Cl(X). The present result extends thoroughly the equivalence of equidimensionality and cofreeness of (X,T) for a factorial X. The purpose of this paper is to evaluate orders of divisor classes associated to modules of relative invariants for a Krull domain with a group action. This is useful in studying on equidimensional torus actions as above. The generalization of R.P. Stanley?s criterion for freeness of modules of relative invariants plays an important role in showing key assertions.