Geometric constructions of the irreducible representations of GL m ( C )

Content Provider	Semantic Scholar
Author	Sambin, Nicola
Copyright Year	2010
Abstract	In presenting the contents and the spirit of his 1997 article Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups V. Ginzburg asserts that recent developments and discoveries " have made representation theory, to a large extent, part of algebraic geometry ". In this brief work we are not able to describe all the deep reasons underneath this conclusion but at least we may corroborate it by showing how some representation-theoretic results follow, at times quite easily, when placed in the appropriate geometric context. In other words we give some examples of geometric constructions of the irreducible representations of the general linear group which make possible the use of certain geometrical methods to gain informations such as the dimensions, the characters of these representations, beyond obviously an explicit realization of themselves. The choice of the constructions we exhibit has been suggested by an article of J. Kamnitzer [K]. In particular this thesis deals with a Borel-Weil type construction and Ginzburg's work contained in his Representation Theory and Complex Geometry. The former dates back to the early 1950s in its original version and was extended by R. Bott in 1957; the latter is more recent and due to V. Ginzburg. Consequently the thesis is fundamentally divided into two parts besides a short chapter in which we recall the basics of representation theory for the general linear group, mainly to fix notation used throughout these pages. In chapter two we describe a realization of a family of irreducible representations of the general linear group on the space of global sections of certain line bundles defined on varieties of flags (of a given vector space) which in our case are not supposed to be complete, in this differing from the usual Borel-Weil construction. In fact we follow more closely the approach in Fulton's book Young Tableaux which provides a constructive version of the Borel-Weil theorem. Namely in the original Borel-Weil construction we gain any irreducible representation of highest weight taken among the dominant weights on the space of global sections of a line bundle, depending on the weight, on the variety of complete flags, whereas in our case also the flag variety varies together with the weight. The pro of this 3 4 choice is essentially that all the constructions become more " visible " , that is for example we are able in this setting to give an explicit formula …
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Language	English
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Content Type	Text
Resource Type	Article