The study of polynomial-time algorithms for groups has only recently begun. Only a few of the basic, elementary concepts or results of group theory are presently known to have polynomial-time versions. For example, ]G] can be found in polynomial time (Sims [22], Furst, Hopcroft, and Luks [9]), as can the centralizer of a normal subgroup of G (Luks [17]) and a composition series for G (Luks [16]...

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sln, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V . We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold ten...

Based on the multiplicative rule of Schubert classes obtained in [Du3], we present an algorithm computing the product of two arbitrary Schubert classes in a flag variety G/H, where G is a compact connected Lie group and H ⊂ G is the centralizer of a one-parameter subgroup in G. Since all Schubert classes on G/H constitute an basis for the integral cohomology H∗(G/H), the algorithm gives a metho...

In this paper we present a new key establishment protocol based on the decomposition problem in non-commutative groups which is: given two elements w, w1 of the platform group G and two subgroups A, B ⊆ G (not necessarily distinct), find elements a ∈ A, b ∈ B such that w1 = awb. Here we introduce two new ideas that improve the security of key establishment protocols based on the decomposition p...

If G is a Polish group and Γ is a countable group, denote byHom(Γ,G) the space of all homomorphisms Γ → G. We study properties of the group π(Γ) for the generic π ∈ Hom(Γ,G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric...

In this article we give an self contained existence proof for J. Conway’s sporadic simple group Co1 [4] using the second author’s algorithm [14] constructing finite simple groups from irreducible subgroups of GLn(2). Here n = 11 and the irreducible subgroup is the Mathieu group M24. From the split extension E of M24 by a uniquely determined 11-dimensional GF(2)M24module V we construct the centr...

For a finite group G and a prime p dividing the order of G, Donald and Flanigan conjecture that the group algebra FpG can be deformed into a semisimple (hence rigid) algebra. We demonstrate that this implies that for some element g of G, the centralizer CG(g) of g in G has a normal subgroup of index p. The method is to observe that the Donald–Flanigan deformation must be a jump. This implies th...

Let G be a connected unipotent group over a finite field Fq. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(Fq) and give an explicit description of L-packets in terms of the so-called “admissible pairs” for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension...

We consider the action of mapping class groups, by preand postcomposition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-cove...