Betti numbers of fixed point sets and multiplicities of indecomposable summands

Let G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set X-Cn and the multiplicities of indecomposable summands of M considered as a kC(n)-module are related via a localization theorem in equivariant cohomology, where C-n is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.


Frobenius action on Carter subgroups
Ercan, Gülin (World Scientific Pub Co Pte Lt, 2020-08-01)
Let G he a finite solvable group and H be a subgroup of Aut(G). Suppose that there exists an H-invariant Carter subgroup F of G such that the semidirect product FH is a Frobenius group with kernel F and complement H. We prove that the terms of the Fitting series of C-G (H) are obtained as the intersection of C-G (H) with the corresponding terms of the Fitting series of G, and the Fitting height of G may exceed the Fitting height of C-G (H) by at most one. As a corollary it is shown that for any set of prime...
Chirality of real non-singular cubic fourfolds and their pure deformation classification
Finashin, Sergey (Springer Science and Business Media LLC, 2020-02-22)
In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.
On a Fitting length conjecture without the coprimeness condition
Ercan, Gülin (Springer Science and Business Media LLC, 2012-08-01)
Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p (n) q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.
Betti Numbers of Smooth Schubert Varieties and the Remarkable Formula of Kostant, Macdonald, Shapiro, and Steinberg
Akyıldız, Ersan (2012-01-01)
The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.
On equivariant Serre problem for principal bundles
Biswas, Indranil; Dey, Arijit; Poddar, Mainak (World Scientific Pub Co Pte Lt, 2018-08-01)
Let E-G be a Gamma-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of Gamma, where G and Gamma are complex linear algebraic groups. Suppose X is contractible as a topological Gamma-space with a dense orbit, and x(0) is an element of X is a Gamma-fixed point. We show that if Gamma is reductive, then E-G admits a Gamma-equivariant isomorphism with the product principal G-bundle X x rho E-G(x(0)), where rho : Gamma -> G is a homomorphism between algebraic...
Citation Formats
S. Öztürk, “Betti numbers of fixed point sets and multiplicities of indecomposable summands,” JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY, pp. 165–171, 2003, Accessed: 00, 2020. [Online]. Available: