Decomposition numbers for perverse sheaves

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group

Parity sheaves and tilting modules

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Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (for representations in odd characteristic). In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence.Comment: 31 page

Generic singularities of nilpotent orbit closures

According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(\mathbb C)$-variety whose normalization is ${\mathbb A}^2$, an $Sp_4(\mathbb C)$-variety whose normalization is ${\mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowy's work for the regular nilpotent orbit.Comment: 56 pages (5 figures). Minor corrections. Accepted in Advances in Mat

NATURE, DYNAMICS AND COPING STRATEGIES IN THE FACE OF PARADOXES IN A DIGITAL TRANSFORMATION: A RECORDS MANAGEMENT CASE STUDY

As a change in a setting displaying scarcity of resources, plurality of choices and technological change, Digital Transformation implies paradoxes of change. Following this, we wonder how paradoxes and Digital Transformation interact and unfold. We investigate the case of a records management company engaging with digitizing and digitalizing of their offers. We implement 23 hours of semi-directive interviews, two site visits and written sources analyses that we code thematically. We find that paradoxes of Digital Transformation can be managed leveraging specificities of digital technologies. Organizing and performing paradoxes are addressed through temporal and geographical splitting strategies, relying on external and internal skills, before developing the resources to hire. National-level reflection complements the strategy. Performing is demonstrated via expertise-oriented online and offline communication supports. The paradox of belonging, fuelled by splitting strategies is the most difficult to manage, addressed through reframing discourses, posters representing the human stories behind the file and expertise-raising actions. On the basis of these dynamics, we suggest an exploratory model