Iwahori-Hecke algebras acting on tensor space by \(q\)-deformed letter permutations and \(q\)-partition algebras
type: Item
instance of: scholarly article
title: Iwahori-Hecke algebras acting on tensor space by \(q\)-deformed letter permutations and \(q\)-partition algebras
author: Q6131625
author: Richard Dipper
published in: Journal of Algebra
publication date: 2024-09-03T00:00:00Z
review text: For rings \(A\) and \(B\) with identity, denote the category of left \(A\)-modules by \(_{A}\mathrm{Mod}\), of right \(B\)-modules by \(\mathrm{Mod}_{B}\) and \(A\)-\(B\)-bimodules by \(_{A}\mathrm{Mod}_{B}\). Let \(R\) be a commutative ring, \(V\) a free \(R\)-module of rank \(n\in\mathbb{N}\), and for \(r\in\mathbb{N}\) let \(V^{\otimes r}=V\otimes_R V\otimes_R\cdots\otimes_R V\). We know that if \(\mathcal{E}=\{e_1,\ldots,e_n\}\) is an \(R\)-basis for \(V\), then \[ \mathcal{E}^r=\{e_{i_1}\otimes\cdots\otimes e_{i_r}\in V^{\otimes r}|\,i_1,\ldots,i_r\in\{1,\ldots,n\}\} \] is an \(R\)-basis of \(V^{\otimes r}\).\N\NThe symmetric group \(\mathfrak{S}_r\) on \(r\) letters acts on \(V^{\otimes r}\) by place permutations permuting the tensor factors making \(V^{\otimes r}\) an \(R\mathfrak{S}_n\)-module. Then clealy \(R\mathfrak{S}_r\) centralizes the action of the group algebra \(RGL_n(R)\), where the general linear group \(GL_n(R)\) acts on \(V\) and hence diagonally on \(V^{\otimes r}\) in the usual way. In fact for infinite fields \(R\) it is well known that the actions of the group algebras \(RGL_n(R)\) and \(R\mathfrak{S}_r\) on \(V^{\otimes r}\) satisfy Schur-Weyl duality. That is their images in \(\mathrm{End}_R(V^{\otimes r})\) are mutual centralizing algebras of each other. It has been proved this holds under some mild restrictions for general commutative rings \(R\). Moreover it follows from a work by M. Jimbo that the action of \(R\mathfrak{S}_r\) on \(V^{\otimes r}\) is the special case \(q=1\) of a \(q\)-deformed action of the Iwahori-Hecke algebra \(\mathcal{H}_{R,q}(R\mathfrak{S}_r)\) on \(V^{\otimes r}\).\N\NOn the other hand defining \(w(e_{i_1}\otimes\cdots\otimes e_{i_r})= e_{i_{w1}}\otimes\cdots\otimes e_{i_{wr}}\) for \(1\leq i_1,\ldots,i_r\leq n\) and \(w\in\mathfrak{S}_n\), the symmetric group \(\mathfrak{S}_n\) on \(n\) letters acts by letter permutations on \(\mathcal{E}^{r}\). Extending this by linearity defines an \(R\mathfrak{S}_n\)-module…
reviewed by: Kamal Aziziheris
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 7905969
zbMATH Keywords: Iwahori-Hecke algebras
zbMATH Keywords: tensor space
zbMATH Keywords: set partitions
zbMATH Keywords: finite general linear groups
zbMATH Keywords: double centralizer
zbMATH Keywords: q-partition algebras
MaRDI profile type: Publication
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DOI: 10.1016/J.JALGEBRA.2024.05.050
description: scientific article; zbMATH DE number 7905969