The affine preservers of non-singular matrices

When K is an arbitrary field, we study the affine automorphisms of M_n(K) that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n>2 or #K>2. We include a short new proof of the more general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank. We also find that the group of affine transformations of M_2(F_2) that stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of quadratic forms over F_2, we construct explicit isomorphisms between it, the symplectic group Sp_4(F_2) and the symmetric group S_6.Comment: 13 pages, very minor corrections from the first versio

Supersymmetric Yang-Mills Theory as Higher Chern-Simons Theory

We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.Comment: v2: 25 pages, conventions improved, typos fixed, published versio

Crossings and nestings in set partitions of classical types

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections that interchange crossings and nestings. For types B and C, they generalize a construction by Kasraoui and Zeng for type A, whereas for type D, we were only able to construct a bijection between non-crossing and non-nesting set partitions. On the other hand we generalize a bijection to type B and C that interchanges the cardinality of the maximal crossing with the cardinality of the maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type A. Using a variant of this bijection, we also settle a conjecture by Soll and Welker concerning generalized type B triangulations and symmetric fans of Dyck paths.Comment: 22 pages, 7 Figures, removed erroneous commen