International Mathematics Research Notices

doi: 10.1093/imrn/rnt040pmid: N/A

doi: 10.1093/imrn/rnt038pmid: N/A

doi: 10.1093/imrn/rnt039pmid: N/A

Bringmann, Kathrin; Mahlburg, Karl; Mellit, Anton

doi: 10.1093/imrn/rns008pmid: N/A

We introduce a new family of directed, multi-state bootstrap percolation models that naturally occur as the convolution of classical bootstrap percolation models as well as generalized k-cross models studied by Gravner, Holroyd, Liggett, and the first two authors. We prove bounds for the probability of indefinite growth by relating the percolation process to sequences of random variables that characterize the percolation growth combinatorics. The corresponding stochastic processes are of independent interest, and we prove a general bound for the limiting density of their probability distributions. We prove these bounds using new results for the convexity and monotonicity of linear operators; these results are of independent interest and the techniques also apply to other stochastic processes with forbidden patterns. In the simplest case of the new multi-state percolation models, we prove a stronger result that gives the precise asymptotic behavior for the limiting probability density of the corresponding stochastic process. This follows from the surprising appearance of Ramanujans mock theta functions, whose cuspidal asymptotics are closely connected to the limiting probabilities.

Burungale, Ashay; Hsieh, Ming-Lun

doi: 10.1093/imrn/rns019pmid: N/A

Let p>2 be an ordinary prime for a CM field . Katz and HidaTilouine constructed the p-adic Hecke L-function attached to a p-ordinary CM-type and a branch character. In this note, we prove that the -invariant of this p-adic Hecke L-function always vanishes when p is unramified in .

Keller, Bernhard; Nicols, Pedro

doi: 10.1093/imrn/rns009pmid: N/A

Using techniques due to DwyerGreenleesIyengar, we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, in certain situations, to deduce the existence of a canonical t-structure on the perfect derived category of a dg algebra. From this, we can obtain a bijection between hearts of t-structures and sets of the so-called simple-minded objects for some dg algebras (including Ginzburg algebras associated to quivers with potentials). In three appendices, we elucidate the relation between Milnor colimits and homotopy colimits and clarify the construction of t-structures from sets of compact objects in triangulated categories as well as the construction of a canonical weight structure on the unbounded derived category of a nonpositive dg category.

Gothen, Peter B.; Oliveira, Andr G.

doi: 10.1093/imrn/rns020pmid: N/A

We give a description of the singular fiber of the Hitchin map on the moduli space of L-twisted Higgs pairs of rank 2 with fixed determinant bundle, when the corresponding spectral curve has any singularity of type Am1. In particular, we prove directly that this fiber is connected.

Fresse, Lucas; Melnikov, Anna

doi: 10.1093/imrn/rns018pmid: N/A

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of a reductive Lie algebra. Orbital varieties arise in representation theory as the geometric analogs of the enveloping algebras primitive ideals. They are also related to the irreducible components of the Springer fibers. In type A, the orbital varieties have a convenient parameterization by standard Young tableaux, in particular, one has a natural duality on the set of orbital varieties induced by the Young tableau transposition. In this paper, we provide several combinatorial criteria to guarantee that an orbital variety is smooth or contains a dense orbit for the action of the Borel subgroup B. We point out that, in all the cases that we can compute, the smooth orbital varieties and the orbital varieties which have a dense B-orbit correspond via the aforementioned duality.