The Isaacs-Navarro Conjecture for covering groups of the symmetric and alternating groups in odd characteristic

In this paper, we prove that a refinement of the Alperin-McKay Conjecture for $p$-blocks of finite groups, formulated by I. M. Isaacs and G. Navarro in 2002, holds for all covering groups of the symmetric and alternating groups, whenever $p$ is an odd prime

On defect groups for generalized blocks of the symmetric group

In a paper of 2003, B. K\"ulshammer, J. B. Olsson and G. R. Robinson defined $\ell$-blocks for the symmetric groups, where $\ell >1$ is an arbitrary integer. In this paper, we give a definition for the defect group of the principal $\ell$-block. We then check that, in the Abelian case, we have an analogue of one of M. Brou\'e's conjectures.Comment: 18 page

An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3

Our main object of study is a certain degree-one cohomology class of the space K of long knots in R^3. We describe this class in terms of graphs and configuration space integrals, showing the vanishing of some anomalous obstructions. To show that this class is not zero, we integrate it over a cycle studied by Gramain. As a corollary, we establish a relation between this class and (R-valued) Casson's knot invariant. These are R-versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in a different way from ours.Comment: 11 pages, 4 figure

Defect of characters of the symmetric group

Following the work of B. Külshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the -defect of characters of the symmetric group n , where is an arbitrary integer. We prove that the -defect is given by an analogue of the hook-length formula, and use it to prove, when , an -version of the McKay conjecture in