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Subrepresentations of General Representations of Quivers

Subrepresentations of General Representations of Quivers SUBREPRESENTATIONS OF GENERAL REPRESENTATIONS OF QUIVERS WILLIAM CRAWLEY-BOEVEY We use the Schubert calculus to extend Schofield's results about general representations of quivers [4] from the case when the base field has characteristic zero to arbitrary characteristic. Let K be an algebraically closed field, and let Q be a finite quiver with vertex set {\,...,n} and Ringel form <«,£> = £ «*A- E «<A <=1 arrows n n for a,/?elR . For <xeN , we denote by Rep(a)= 0 Horn (K\K**) arrows the variety of representations of Q of dimension vector a, and for JC e Rep (a) we write K for the corresponding representation. We write Gr I a n dimensional subspaces in K , and for a,{ieN , we set Rep Gr I I = \ (x, V) e Rep (a) x Gr I 11 V is a subrepresentation of K a sub-bundle of the trivial vector bundle Rep (a) x Grl I -> Grl 11. Let /:RepGrQ j •Rep (a) be the projection. We write /? c* a if/is dominant, that is, if the general representation of dimension a has a subrepresentation of dimension /?. If y,Se N , then ext (y, S) = min | dim http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Subrepresentations of General Representations of Quivers

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/28.4.363
Publisher site
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Abstract

SUBREPRESENTATIONS OF GENERAL REPRESENTATIONS OF QUIVERS WILLIAM CRAWLEY-BOEVEY We use the Schubert calculus to extend Schofield's results about general representations of quivers [4] from the case when the base field has characteristic zero to arbitrary characteristic. Let K be an algebraically closed field, and let Q be a finite quiver with vertex set {\,...,n} and Ringel form <«,£> = £ «*A- E «<A <=1 arrows n n for a,/?elR . For <xeN , we denote by Rep(a)= 0 Horn (K\K**) arrows the variety of representations of Q of dimension vector a, and for JC e Rep (a) we write K for the corresponding representation. We write Gr I a n dimensional subspaces in K , and for a,{ieN , we set Rep Gr I I = \ (x, V) e Rep (a) x Gr I 11 V is a subrepresentation of K a sub-bundle of the trivial vector bundle Rep (a) x Grl I -> Grl 11. Let /:RepGrQ j •Rep (a) be the projection. We write /? c* a if/is dominant, that is, if the general representation of dimension a has a subrepresentation of dimension /?. If y,Se N , then ext (y, S) = min | dim

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jun 1, 1996

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