The long root geometry $A_{n,\lbrace 1,n\rbrace }(\mathbb{K})$ for the special linear group $\mathrm{SL}(n+1,\mathbb{K})$ admits an embedding in the (projective space of) the vector space of the traceless square matrices of order $n+1$ with entries in the field $\mathbb{K}$, usually regarded as the natural embedding of $A_{n,\lbrace 1,n\rbrace }(\mathbb{K})$. S. Smith and H. Völklein in [10] have proved that the natural embedding of $A_{2,\lbrace 1,2\rbrace }(\mathbb{K})$ is relatively universal if and only if $\mathbb{K}$ is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of $A_{2,\lbrace 1,2\rbrace }(\mathbb{K})$ which covers the natural one, but that information is not sufficient to exhaustively describe it. The “if” part of Smith-Völklein’s result also holds true for any $n$, as proved by Völklein in [13] in his investigation of the adjoint modules of Chevalley groups. In this paper we give an explicit description of the relatively universal embedding of $A_{n,\lbrace 1,n\rbrace }(\mathbb{K})$ which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to $\mathfrak{d}+n^2+2n$ where $\mathfrak{d}$ is the transcendence degree of $\mathbb{K}$ over its minimal subfield (if $\mathrm{char}(\mathbb{K}) = 0$) or the generating rank of $\mathbb{K}$ over $\mathbb{K}^p$ (if $\mathrm{char}(\mathbb{K}) = p > 0$). Accordingly, both the “if” and the “only if” part of Smith-Völklein’s result hold true for every $n \ge 2$.
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Keywords: Long root geometry, relatively universal embedding, adjoint module
Cardinali, Ilaria 1 ; Giuzzi, Luca 2 ; Pasini, Antonio 1
CC-BY 4.0
@article{ALCO_2026__9_1_231_0,
author = {Cardinali, Ilaria and Giuzzi, Luca and Pasini, Antonio},
title = {The relatively universal cover of the natural embedding of the long root geometry for the group $\mathrm{SL}(n+1,\mathbb{K})$},
journal = {Algebraic Combinatorics},
pages = {231--259},
year = {2026},
publisher = {The Combinatorics Consortium},
volume = {9},
number = {1},
doi = {10.5802/alco.473},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.473/}
}
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JO - Algebraic Combinatorics
PY - 2026
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PB - The Combinatorics Consortium
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Cardinali, Ilaria; Giuzzi, Luca; Pasini, Antonio. The relatively universal cover of the natural embedding of the long root geometry for the group $\mathrm{SL}(n+1,\mathbb{K})$. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 231-259. doi: 10.5802/alco.473
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