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Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G0 is a torus and p does not divide the index (G : G0). [ABSTRACT FROM AUTHOR]

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) - n, where μ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL2) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer. [ABSTRACT FROM AUTHOR]

We classify the irreducible complex characters of the symplectic groups Sp2n(q) and the orthogonal groups [image omitted], Spin2n+1(q) of degrees up to the bound D, where D = (qn - 1)q4n-10/2 for symplectic groups, D = q4n-8 for orthogonal groups in odd dimension, and D = q4n-10 for orthogonal groups in even dimension. [ABSTRACT FROM AUTHOR]

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves. [ABSTRACT FROM AUTHOR]

Given two affine algebraic varieties X and Y such that X ×  ≅ Y × , it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1. [ABSTRACT FROM AUTHOR]

This article provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the general linear group in terms of their Jordan blocks. This result was originally due to Gerstenhaber. Finally, we obtain a proof of the Bala-Carter Theorem, which holds even in certain bad characteristics. This proof requires the prior knowledge of the number of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening. [ABSTRACT FROM AUTHOR]