arXiv subjects

Rose Berry

Publications and source records attributed to Rose Berry.

3 recordsLinked to original sources

arXiv2026

Affine Hecke and Schur algebras of type A without a square root of q

We provide an affine cellular structure on the extended affine Hecke algebra and affine $q$-Schur algebra of type $A_{n-1}$ that is defined over $\mathbb{Z}\left[q^{\pm1}\right]$, that is, without an adjoined $q^{\frac{1}{2}}$. This is with an eye to applications in the representation theory of $\mathrm{GL}_n(F)$ for a $p$-adic field $F$ over coefficient rings in which $p$ is invertible but does not have a square root, which have been a topic of recent interest. This is achieved via a renormalisation of the known affine cellular structure over $\mathbb{Z}\left[q^{\pm\frac{1}{2}}\right]$ at each left and right cell, which is chosen to ensure that the diagonal intersections remain subalgebras and that the left and right cells remain isomorphic. We furthermore show that the affine cellular structure on the Schur algebra has idempotence properties which imply finite global dimension, an important ingredient for the applications to representations of $p$-adic groups.

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arXiv2025

The Derived $l$-Modular Unipotent Block of $p$-adic $\mathrm{GL}_n$

For a non-Archimedean local field $F$ of residue cardinality $q=p^r$, we give an explicit classical generator $V$ for the bounded derived category $D_{fg}^b(\mathsf{H}_1(G))$ of finitely generated unipotent representations of $G=\mathrm{GL}_n(F)$ over an algebraically closed field of characteristic $l\neq p$. The generator $V$ has an explicit description that is much simpler than any known progenerator in the underived setting. This generalises a previous result of the author in the case where $n=2$ and $l$ is odd dividing $q+1$, and provides a triangulated equivalence between $D_{fg}^b(\mathsf{H}_1(G))$ and the category of perfect complexes over the dg algebra of dg endomorphisms of a projective resolution of $V$. This dg algebra can be thought of as a dg-enhanced Schur algebra. As an intermediate step, we also prove the analogous result for the case where $F$ is a finite field.

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arXiv2024

The Derived Unipotent Block of $p$-Adic $\mathrm{GL}_2$ as Perfect Complexes over a dg Schur Algebra

For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$.

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