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Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]

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]

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G 2 ( q ), where q is a power of a prime p > 3. [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]

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ϵ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type. [ABSTRACT FROM AUTHOR]

In this article we prove Broué's conjecture on isotypies for the case of principal blocs of Sp(4, Fq)〈σ〉 : a split cyclic extension of the symplectic group of dimension 4 over a finite field Fq for q odd. The core of the proof is to lift to the group Sp(4, Fq)〈σ〉 an isotypy of the group Sp(4, Fq). For this purpose, we build a convenient version of a theorem of lifting isotypies. [ABSTRACT FROM AUTHOR]