Polyvector representations of $\operatorname{GL}_n$
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 69-97
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In the present paper we characterise $\bigwedge^n(\operatorname{GL}(n,R))$
over any commutative ring $R$ as the connected component
of the stabiliser of Plücker ideal. This folk theorem is
classically known for algebraically closed fields and should
be also well-known in general. However, we are not aware of any
obvious reference, so we produce a detailed proof which follows
a general scheme developed by W. C. Waterhouse. The present paper
is a technical preliminary for a subsequent paper, where we
construct decomposition of transvections in polyvector
representations of $\operatorname{GL}_n$.
@article{ZNSL_2006_338_a1,
author = {N. A. Vavilov and E. Ya. Perelman},
title = {Polyvector representations of $\operatorname{GL}_n$},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {69--97},
publisher = {mathdoc},
volume = {338},
year = {2006},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a1/}
}
N. A. Vavilov; E. Ya. Perelman. Polyvector representations of $\operatorname{GL}_n$. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 69-97. http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a1/