Open Access
2014 Embeddings of local fields in simple algebras and simplicial structures
Daniel Skodlerack
Publ. Mat. 58(2): 499-516 (2014).


We give a geometric interpretation of Broussous-Grabitz embedding types. We fix a central division algebra $D$ of finite index over a non-Archimedean local field $F$ and a positive integer $m$. Further we fix a hereditary order $\mathfrak{a}$ of $\operatorname{M}_m(D)$ and an unramified field extension $E|F$ in $\operatorname{M}_m(D)$ which is embeddable in $D$ and which normalizes $\mathfrak{a}$. Such a pair $(E,\mathfrak{a})$ is called an embedding. The embedding types classify the $\operatorname{GL}_m(D)$-conjugation classes of these embeddings. Such a type is a class of matrices with non-negative integer entries. We give a formula which allows us to recover the embedding type of $(E,\mathfrak{a})$ from the simplicial type of the image of the barycenter of $\mathfrak{a}$ under the canonical isomorphism, from the set of $E^\times$-fixed points of the reduced building of $\operatorname{GL}_m(D)$ to the reduced building of the centralizer of $E^\times$ in $\operatorname{GL}_m(D)$. Conversely the formula allows to calculate the simplicial type up to cyclic permutation of the Coxeter diagram.


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Daniel Skodlerack. "Embeddings of local fields in simple algebras and simplicial structures." Publ. Mat. 58 (2) 499 - 516, 2014.


Published: 2014
First available in Project Euclid: 21 July 2014

zbMATH: 1344.20068
MathSciNet: MR3264509

Primary: 12J25 , 17C20 , 20E42

Keywords: buildings , Embeddings types , non-archimedean local fields , simple algebras

Rights: Copyright © 2014 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.58 • No. 2 • 2014
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