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.     

Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that Hom_FΣn(S^λ,S^μ) can have arbitrarily large dimension as n and p grow, where S^λ and S^μ are Specht modules for the symmetric group Σ_n. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.     

On maximal curves with Frobenius dimension 3

Frobenius dimension is one of the most important birational invariants of maximal curves. In this paper, a characterization of maximal curves with Frobenius dimension equal to 3 is provided. Our main tool is the Natural Embedding Theorem for maximal curves. As an application, maximal curves with Frobenius dimension 3 defined over the fields with 16 and 25 elements are completely classified.     

Representations of metabelian groups

A knowledge of the simple representation theory of finite abelian groups is useful for understanding the representations of solvable groups, since these provide the one-dimensional representations. The representation theory of metabelian groups (those G with abelian commutator subgroup G′) would seem to be a natural next level.In this paper we shall show that these representations, too, may be simply described in several ways: they are induced from linear representations of some explicity defined subgroups; their degrees may be calculated from a knowledge of the subgroups of G; these degrees depend only on the kernel of the representation (in fact, only on the intersection of this kernel with G′). As an application of these results, we can calculate for metabelian groups a certain measure of group-commutativity studied in an earlier paper [4].     

Finite groups with non-nilpotent maximal subgroups

In this paper, the structure of a finite group under group theoretic restrictions on its non-nilpotent subgroups has been investigated.     

Kodaira dimension of symmetric powers

We compute the plurigenera and the Kodaira dimension of the th symmetric power of a smooth projective variety . As an application we obtain genus estimates for the curves lying on .

     

On varieties of maximal Albanese dimension

We study the birational geometry of varieties of maximal Albanese dimension in this article. Given a non-birational, generically finite, and surjective morphism f: X → Y between varieties of maximal Albanese dimension, we show that the plurigenera P _m (X) and P _m (Y) for some m?≥ 2 could be equal only in very restrictive situations. We also prove that the 5-th pluricanonical map of a variety of maximal Albanese dimension always induces the Iitaka model.     

Fuzzy symmetric groups

We study fuzzy symmetric subgroups and obtain some properties of fuzzy symmetric subgroups of symmetric groups.     

Representations of free lattice-ordered groups

We show for any uncountable cardinal that the free groupG of rank has a linear right ordering on which the natural action of the free lattice-ordered groupF of rank is faithful and pathologically 2-transitive. As a consequence, we obtain results on the root system of prime subgroups ofF . This generalizes previous results of McCleary which required the generalized continuum hypothesis and to be regular.