We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces.

Source : oai:HAL:hal-01207602v1

Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)

Section: Proceedings

Published on: January 1, 2014

Submitted on: November 21, 2016

Keywords: representation theory,infinite symmetric group,centralizer algebras,partition algebras,symmetric functions,Banach spaces,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

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