Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.40602

Permutation twisted cohomology, reduced and immersed

Juan Omar Gómez·Sam K. Miller
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we introduce a `reduced' form of permutation twisted cohomology as introduced by Balmer--Gallauer and further developed by Miller. This ring is bigraded by a well-chosen choice of endotrivial complex, rather than multigraded by a family of endotrivial complexes. The Balmer spectrum of the small derived category of permutation modules openly immerses into the homogeneous spectrum of this ring. We also show that for $p$-groups, Miller's `remixed' twisted cohomology ring also admits the Balmer spectrum as an open subspace, and confirm in both cases it can be upgraded to have Dirac scheme structure; furthermore, we prove that the two open covers witnessing this are in general different.

Explore related subjects

math.RTmath.AGmath.ATmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Juan Omar Gómez, Sam K. Miller. 2026-10-29. Permutation twisted cohomology, reduced and immersed. https://arxiv.org/abs/2610.40602

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Archimedean period relations for Rankin-Selberg convolutions

We formulate and prove the archimedean period relations for Rankin-Selberg convolutions of $\text{GL}(n)\times \text{GL}(n)$ and $\text{GL}(n)\times \text{GL}(n-1)$, for all generic cohomological representations. As a consequence, we prove the non-vanishing of the archimedean modular symbols. This extends the earlier results in [LLS24] for essentially tempered representations of $\text{GL}(n)\times\text{GL}(n-1)$.

math.RT↗
arXiv2026

Rational structures on quivers and a generalization of Gelfand's equivalence

We introduce the notion of rational structure on a quiver and associated representations to establish a coherent framework for studying quiver representations in separable field extensions. This notion is linked to a refinement of the notion of $K$-species, which we term étale $K$-species. We establish a categorical anti-equivalence between $K$-rational quivers and étale $K$-species, which extends to an equivalence of their respective representation categories. For $K$-rational quivers there is a canonical notion of base change, which suggests a corresponding notion of base change for (étale) $K$-species which we elaborate. As a primary application, we generalize Gelfand's celebrated equivalence between certain blocks of Harish-Chandra modules for $\mathrm{SL}_2(\mathbb{R})$ and representations of the Gelfand quiver to a rational setting. To this end, we define a $\mathbb{Q}$-rational structure on the Gelfand quiver and its representations. A key technical tool, which we call unipotent stabilization, is developed to construct the functor from rational Harish-Chandra modules to nilpotent rational quiver representations. We prove that this functor is an equivalence. A similar result is established for the cyclic quiver. Both results hold over every field $E$ of characteristic $0$, for suitable $E$-rational structures on the two quivers; for $E=\mathbb{C}$ they recover Gelfand's equivalence. In this rational framework the defining relation of the Gelfand quiver acquires a descent-theoretic meaning: its Galois orbit is stable, it descends to a single relation on the associated $\mathbb{Q}$-species, and a $\mathbb{Q}$-rational representation satisfies it if and only if the composite of the two arrows through the vertex $-$ is a $\mathbb{Q}$-rational endomorphism of the space at the central vertex. This allows us to recast our results in the language of $\mathbb{Q}$-species.

math.RT↗
arXiv2026

Perverse filtration on the cohomology of affine Springer fibers

We show that the top cohomology of any affine Springer fiber, as a Weyl group representation, contains a large part of the total cohomology of certain Springer fibers. The main ingredient of the proof is the construction of a ``perverse filtration'' on the pure part of the cohomology of affine Springer fibers.

math.RT↗