On the modular generalized Springer correspondence for disconnected groups
We study the construction of a modular generalized Springer correspondence for a possibly disconnected complex reductive algebraic group.
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arXiv subjects
Kostas I. Psaromiligkos
Publications and source records attributed to Kostas I. Psaromiligkos.
Character sheaves in characteristic $p$ have nilpotent singular support
We prove that character sheaves have nilpotent singular support in any characteristic, partially extending the work of Mirkovic, Vilonen and independently Ginzburg to positive characteristic. We do this by introducing a category of tame perverse sheaves and studying its properties.
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The Lafforgue variety and irreducibility of induced representations
We construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative unital algebra $R$ over any field $k$, provided that the center $Z(R)$ is finitely generated and $R$ is finitely generated as a $Z(R)$-module. Our main technical tool is a generalization of the Hilbert scheme for non-commutative algebras, which may be of independent interest. Applying our construction in the case of Hecke algebras of Bernstein components, we derive a characterization for the irreducibility of induced representations in terms of the vanishing of a generalized discriminant on the Bernstein variety. We explicitly compute the discriminant in the case of an Iwahori-Hecke algebra of a split reductive $p$-adic group.
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Branchwidth is (1,g)-self-dual
A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that the branchwidth of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most g is an (1,g)-self-dual parameter. This is the first proof that branchwidth is an additively self-dual width parameter.
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Directed Lovász Local Lemma and Shearer's Lemma
Moser and Tardos (2010) gave an algorithmic proof of the lopsided Lovász local lemma (LLL) in the variable framework, where each of the undesirable events is assumed to depend on a subset of a collection of independent random variables. For the proof, they define a notion of a lopsided dependency between the events suitable for this framework. In this work, we strengthen this notion, defining a novel directed notion of dependency and prove LLL for the corresponding graph. We show that this graph can be strictly sparser (thus the sufficient condition for LLL weaker) compared with graphs that correspond to other extant lopsided versions of dependency. Thus, in a sense, we address the problem "find other simple local conditions for the constraints (in the variable framework) that advantageously translate to some abstract lopsided condition" posed by Szegedy (2013). We also give an example where our notion of dependency graph gives better results than the classical Shearer lemma. Finally, we prove Shearer's lemma for the dependency graph we define. For the proofs, we perform a direct probabilistic analysis that yields an exponentially small upper bound for the probability of the algorithm that searches for the desired assignment to the variables not to return a correct answer within $n$ steps. In contrast, the method of proof that became known as the entropic method, gives an estimate of only the expectation of the number of steps until the algorithm returns a correct answer, unless the probabilities are tinkered with.
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An interactive version of the Lovász local lemma
Assume we are given (finitely many) mutually independent variables and (finitely many) "undesirable" events, each depending on a subset of the variables of at most $k$ elements, called the scope of the event. Assume that the probability of a variable belonging to the scope of an occurring event is bounded by $q$. We prove that if $ekq \leq 1$ then there exists at least one assignment to the variables for which none of the events occurs. Examples are given where the criterion $ekq \leq 1$ is applicable, whereas that of the classical version of the Lovász local lemma is not. The proof of the result is through an interactive, private-coin implementation of the algorithm by Moser. The original implementation, which yields the classical result, finds efficiently, but probabilistically, an assignment to the events that avoids all undesirable events. Interestingly, the interactive implementation given in this work does not constitute an efficient, even if probabilistic, algorithm to find an assignment as desired under the weaker assumption $ekq \leq 1$. We can only conclude that under the hypothesis that $ekq \leq 1$, the interactive protocol will produce an assignment as desired within $n$ rounds, with probability high with respect to $n$; however, the provers' choices remain non-deterministic. Plausibly finding such an assignment is inherently hard, as the situation is reminiscent, in a probabilistic framework, of problems complete for syntactic subclasses of TFNP.
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Acyclic Edge Coloring through the Lovász Local Lemma
We give a probabilistic analysis of a Moser-type algorithm for the Lovász Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $Δ$ has an acyclic proper edge coloring with at most $\lceil 3.74(Δ-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(Δ-1)$. The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.
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An alternative proof for the constructive Asymmetric Lovász Local Lemma
We provide an alternative constructive proof of the Asymmetric Lovász Local Lemma. Our proof uses the classic algorithmic framework of Moser and the analysis introduced by Giotis, Kirousis, Psaromiligkos, and Thilikos in "On the algorithmic Lovász Local Lemma and acyclic edge coloring", combined with the work of Bender and Richmond on the multivariable Lagrange Inversion formula.
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