Affine quiver Schur algebras and $p$-adic $GL_n$

Vanessa Miemietz; Catharina Stroppel

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arXiv · 1601.07323

Affine quiver Schur algebras and $p$-adic $GL_n$

Vanessa Miemietz·Catharina Stroppel

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Abstract

In this paper we consider the (affine) Schur algebra introduced by Vign\'eras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a $p$-adic field, a large part of the unipotent block over fields of characteristic different from $p$. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of $\mathrm{GL}_2(\mathbb{Q}_5)$ in characteristic $3$

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BibTeX RIS

Vanessa Miemietz, Catharina Stroppel. 2016-01-27. Affine quiver Schur algebras and $p$-adic $GL_n$. https://arxiv.org/abs/1601.07323

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