Let G_n(R) be the semi-direct product of the symmetric group S_n by the Steinberg group St_n(R)of a ring R. We first prove that G_n(R) has a Coxeter-type presentation. The canonical morphism from St_n(R) to GL_n(R) extends to a group homomorphism F from G_n(R) to GL_n(R). We next determine the kernel of F for n = infinity. We also give an expression for the generator of the algebraic K-group K_2(Z) of the integers in terms of permutation matrices.
Publié le : 1998-07-05
Classification:
Steinberg group,
algebraic K-theory,
symmetric group,
central extension,
Coxeter,
MSC 19C09, 19C99, 20B30, 20F55,
[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT]
@article{hal-00124660,
author = {Kassel, Christian and Reutenauer, Christophe},
title = {Une variante \`a la Coxeter du groupe de Steinberg},
journal = {HAL},
volume = {1998},
number = {0},
year = {1998},
language = {fr},
url = {http://dml.mathdoc.fr/item/hal-00124660}
}
Kassel, Christian; Reutenauer, Christophe. Une variante à la Coxeter du groupe de Steinberg. HAL, Tome 1998 (1998) no. 0, . http://gdmltest.u-ga.fr/item/hal-00124660/