On a relation between the basic representation of the affine Lie algebra \(\widehat\sl\) and a Schur--Weyl representation of the infinite symmetric group

We prove that there is a natural grading-preserving isomorphism of \(\sl\)-modules between the basic module of the affine Lie algebra \(\widehat\sl\) (with the homogeneous grading) and a Schur--Weyl module of the infinite symmetric group \(\sinf\) with a grading defined through the combinatorial not...

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Bibliographic Details
Published inarXiv.org
Main Authors Tsilevich, Natalia, Vershik, Anatoly
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.03.2014
Subjects
Algebra
Combinatorial analysis
Grading
Group theory
Isomorphism
Lie groups
Modules
Quantum theory
Representations
Traveling salesman problem
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Summary:We prove that there is a natural grading-preserving isomorphism of \(\sl\)-modules between the basic module of the affine Lie algebra \(\widehat\sl\) (with the homogeneous grading) and a Schur--Weyl module of the infinite symmetric group \(\sinf\) with a grading defined through the combinatorial notion of the major index of a Young tableau, and study the properties of this isomorphism. The results reveal new and deep interrelations between the representation theory of \(\widehat\sl\) and the Virasoro algebra on the one hand, and the representation theory of \(\sinf\) and the related combinatorics on the other hand.
ISSN:2331-8422