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On Kulkarni's theorems on degree reduction for polynomial modules

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JournalMathematical Proceedings of the Cambridge Philosophical Society
DatePublished - 1 May 2003
Issue number2
Volume134
Number of pages9
Pages (from-to)229-237
Original languageEnglish

Abstract

In a recent paper [13], Kulkarni proves that, for i > 0, one has
Exti
G(M
();r())%Exti
G(M;r(=))
{ an isomorphism of spaces of extensions of rational modules. Here G is a general
linear group scheme, ; are partitions and M is a polynomial G-module. The modules
(), r() and r(=) are respectively the Weyl module labelled by , the
induced module labelled by and the skew module labelled by =. A similar result
is given in which the roles of and r are interchanged.
Our purpose here is to set this result in the context of the representation theory
of an arbitrary reductive group. Specically, we prove a simple result which is valid
for general reductive groups and derive Kulkarni's results from this. For convenience
we work over an algebraically closed eld k.
This paper is dedicated to Idun Reiten on the occasion of her 60th birthday.

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