Generalized Schur-Weyl Duality

Student: Kashirkina Anna

Educational Programme: Mathematics (Bachelor)

Final Grade: 10

Year of Graduation: 2024

The classic Schur-Weyl duality establishes a connection between finite-dimensional polynomial representations of the general Lie algebra $\mathfrak{gl}(N)$ and representations of the symmetric groups $S_n$. L. Solomon (2002) suggested a generalization of the Schur-Weyl duality in which the symmetric group $S_n$ is replaced by a larger object, the symmetric inverse semigroup $R_n$, also called the rook monoid. On the other hand, A. Sergeev (1985) found an analog of the Schur-Weyl duality connecting polynomial representations of the `queer' Lie superalgebra $\mathfrak q(N)$ with representations of certain finite groups $G_n\supset S_n$. In the present work, we describe a generalization of the Sergeev duality in the spirit of Solomon's idea.

Full text (added June 2, 2024)

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