39 research outputs found
On characters of Chevalley groups vanishing at the non-semisimple elements
Let G be a finite simple group of Lie type. In this paper we study characters
of G that vanish at the non-semisimple elements and whose degree is equal to
the order of a maximal unipotent subgroup of G. Such characters can be viewed
as a natural generalization of the Steinberg character. For groups G of small
rank we also determine the characters of this degree vanishing only at the
non-identity unipotent elements.Comment: Dedicated to Lino Di Martino on the occasion of his 65th birthda
Subgroups Of Simple Algebraic Groups Containing Regular Tori, And Irreducible Representations With Multiplicity 1 Non-Zero Weights
Our main goal is to determine, under certain restrictions, the maximal closed connected subgroups of simple linear algebraic groups containing a regular torus. We call a torus regular if its centralizer is abelian. We also obtain some results of independent interest. In particular, we determine the irreducible representations of simple algebraic groups whose non-zero weights occur with multiplicity 1
Remarks on singular Cayley graphs and vanishing elements of simple groups
Let Γ be a finite graph and let A(Γ) be its adjacency matrix. Then Γ is singular if A(Γ) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs Cay(G,H) when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character χ of G for which ∑h∈Hχ(h)=0. At this stage, we focus on the case when H is a single conjugacy class hG of G; in this case, the above equality is equivalent to χ(h)=0 . Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element h∈G is called vanishing if χ(h)=0 for some irreducible character χ of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs