437 research outputs found
On isomorphisms of abelian Cayley objects of certain orders
AbstractLet m be a positive integer such that gcd(m,ϕ(m))=1 (ϕ is Euler's phi function) with m=p1⋯pr the prime power decomposition of m. Let n=p1a1⋯prar. We provide a sufficient condition to reduce the Cayley isomorphism problem for Cayley objects of an abelian group of order n to the prime power case. In the case of Cayley k-ary relational structures (which include digraphs) of abelian groups, this sufficient condition reduces the Cayley isomorphism problem of k-ary relational structures of abelian groups to the prime power case for Cayley k-ary relational structures of abelian groups. As corollaries, we solve the Cayley isomorphism problem for Cayley graphs of Zn (for the specific values of n as above) and prove several abelian groups (for specific choices of the ai) of order n are CI-groups with respect to digraphs
Quasisplit Hecke algebras and symmetric spaces
Let (G,K) be a symmetric pair over an algebraically closed field of
characteristic different of 2 and let sigma be an automorphism with square 1 of
G preserving K. In this paper we consider the set of pairs (O,L) where O is a
sigma-stable K-orbit on the flag manifold of G and L is an irreducible
K-equivariant local system on O which is "fixed" by sigma. Given two such pairs
(O,L), (O',L'), with O' in the closure \bar O of O, the multiplicity space of
L' in the a cohomology sheaf of the intersection cohomology of \bar O with
coefficients in L (restricted to O') carries an involution induced by sigma and
we are interested in computing the dimensions of its +1 and -1 eigenspaces. We
show that this computation can be done in terms of a certain module structure
over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above.Comment: 46 pages. Version 2 reorganizes the explicit calculation of the Hecke
module, includes details about computing \bar, and corrects small misprints.
Version 3 adds two pages relating this paper to unitary representation
theory, corrects misprints, and displays more equations. Version 4 corrects
misprints, and adds two cases previously neglected at the end of 7.
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