A Journey Through Representation Theory by Caroline Gruson & Vera Serganova

Author:Caroline Gruson & Vera Serganova
Language: eng
Format: epub
ISBN: 9783319982717
Publisher: Springer International Publishing

for all and . One can easily check that the actions of GL(V) and in the space commute. We will show that GL(V) and form a dual pair.

Theorem 2.4.

(Schur–Weyl duality) Let and denote the set of all Young diagrams with n boxes such that the number of rows of is not bigger than m. Then

where is the irreducible representation of associated to and

is an irreducible representation of . Moreover, and are isomorphic if and only if .

Remark 2.5.

If and is an arbitrary Young tableau of shape , then the image of the Young symmetrizer in is a simple GL(V)-module isomorphic to .

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