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Constructing Koszul filtrations: existence and non-existence for G-quadratic algebras
Emily Berghofer, Lisa Nicklasson, Peder Thompson, Thomas Westerb\"ack
https://arxiv.org/abs/2602.06490 https://arxiv.org/pdf/2602.06490 https://arxiv.org/html/2602.06490
arXiv:2602.06490v1 Announce Type: new
Abstract: Given a standard graded algebra over a field, we consider the relationship between G-quadraticity and the existence of a Koszul filtration. We show that having a quadratic Gr\"obner basis implies the existence of a Koszul filtration for toric algebras equipped with the degree reverse lexicographic term order and for algebras defined by binomial edge ideals. We also resolve a conjecture of Ene, Herzog, and Hibi by constructing an example where this implication fails. These results are underpinned by algorithms we develop for constructing Koszul filtrations. We also demonstrate the utility of these algorithms on the pinched Veronese algebra.
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Polynomials in $c$-free random variables with applications to free denoising
Adrian Celestino, Franz Lehner, Kamil Szpojankowski
https://arxiv.org/abs/2603.21372 https://arxiv.org/pdf/2603.21372 https://arxiv.org/html/2603.21372
arXiv:2603.21372v1 Announce Type: new
Abstract: We study distributions of polynomials in conditionally free (c-free) random variables, a notion of independence for two-state noncommutative probability spaces introduced by Bozejko, Leinert and Speicher. To this end we establish recursive relations between the joint Boolean cumulants of c-free random variables, analogous to previously found recursions for Boolean cumulants of free random variables. The algebraic reformulation of these recursions on the free associative algebra provides an effective formal machinery for the computation of the moment generating functions and thus the distributions of arbitrary self-adjoint polynomials in c-free random variables. As an application of a recent observation, our approach can be used to determine conditional expectations of the form $E[a|P(a,b)]$, where $P(a,b)$ is a self-adjoint polynomial in free (in the sense of Voiculescu) random variables $a,b$. We illustrate this with an example where $P(a,b)=i[a,b]$. Finally we define orthogonal projections that formally play the role of conditional expectations in the framework of c-freeness and share some properties with the conditional expectations of free variables. In particular they can be used to re-derive by purely algebraic methods the formula of Popa and Wang for the $\Sigma$-transform for the c-free multiplicative convolution.
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Minimal and intrinsic topologies on monoids of elementary embeddings
J. de la Nuez Gonzalez, Zaniar Ghadernezhad, Paolo Marimon, Michael Pinsker
https://arxiv.org/abs/2603.28419 https://arxiv.org/pdf/2603.28419 https://arxiv.org/html/2603.28419
arXiv:2603.28419v1 Announce Type: new
Abstract: To every $\omega$-categorical structure $M$ one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group $\mathrm{Aut}(M)$ of its automorphisms and the topological monoid $\mathrm{EEmb}(M)$ of its elementary embeddings, both equipped with the topology of pointwise convergence $\tau_{\mathrm{pw}}$. We investigate the relation of $\tau_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $\tau_{\mathrm{pw}}$ is minimal, i.e.~does not admit any strictly coarser Hausdorff semigroup topology.
A common method to prove minimality of $\tau_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $\tau_{\mathrm{Z}}$. We show that $\tau_{\mathrm{pw}}$ differs from $\tau_{\mathrm{Z}}$ on $\mathrm{EEmb}(M)$ whenever $\mathrm{Aut}(M)$ has non-trivial centre. We then provide general conditions on the behaviour of algebraic closure on $M$ that imply minimality of $\tau_{\mathrm{pw}}$. These condition cover, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{Aut}(M)$, we describe the minimal $T_1$ semigroup topologies on the automorphism groups of model-theoretically simple one-based $\omega$-categorical structures with weak elimination of imaginaries. We conclude by proving that the metric pointwise topology $\tau_{\mathrm{mpw}}$ is minimal, equals $\tau_{\mathrm{Z}}$, and is strictly coarser than $\tau_{\mathrm{pw}}$, on $\mathrm{EEmb}(M)$ for the real and the rational Urysohn space and sphere.
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Introducing pixelation with applications
J. Daisie Rock
https://arxiv.org/abs/2603.25432 https://arxiv.org/pdf/2603.25432 https://arxiv.org/html/2603.25432
arXiv:2603.25432v1 Announce Type: new
Abstract: Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$.
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