Schur–Weyl dualities for the rook monoid: an approach via Schur algebras

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra....

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Bibliographic Details
Published inSemigroup forum Vol. 109; no. 1; pp. 38 - 59
Main Authors M. André, Carlos A., Legatheaux Martins, Inês
Format Journal Article
LanguageEnglish
Published New York Springer US 2024
Springer Nature B.V
Subjects
Algebra
Mathematics
Mathematics and Statistics
Monoids
Research Article
Symmetry
Tensor spaces
Schur–Weyl duality
Schur algebras
Representation theory of associative algebras
Rook monoid
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Summary:The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon’s Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach.
ISSN:0037-1912
1432-2137
DOI:10.1007/s00233-024-10434-w