Iwahori-Hecke algebras acting on tensor space by \(q\)-deformed letter permutations and \(q\)-partition algebras
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 structure on \(V^{\otimes r}\), which coincides with the one arising by restricting the natural diagonal action of \(GL_n(R)\) on \(V^{\otimes r}\) to \(\mathfrak{S}_n\leq GL_n(R)\). Since \(\mathcal{H}_n\) does not have a Hopf coproduct allowing it to act on the tensor product \(V^{\otimes r}\), this action of \(R\mathfrak{S}_n\) on \(V^{\otimes r}\) does not carry over directly to \(\mathcal{H}_n =\mathcal{H}_{R,q}(R\mathfrak{S}_n)\).\N\NAlso, the embedding of \(\mathfrak{S}_n\) into \(GL_n(R)\) has no \(q\)-analogue replacing \(GL_n(R)\) by some quantized version of general linear groups. It follows from section~3 of the paper under review, there exists a more complicated \(\mathcal{H}_n\)-action on \(V^{\otimes r}\) which specializes to the letter action of \(R\mathfrak{S}_n\) on \(V^{\otimes r}\) by taking \(q=1\). This result is based on the observation of Halverson and Ram, that \(V^{\otimes r} \in _{R\mathfrak{S}_n}\mathrm{Mod}\) can be obtained by iterated restriction to \(\mathfrak{S}_{n-1}\leq\mathfrak{S}_{n}\) and induction to \(\mathfrak{S}_{n}\) of the trivial \(R\mathfrak{S}_{n}\)-module.\N\NBy Theorem~3.11 of the paper under review, this construction can easily be extended to the Iwahori-Hecke algebra \(\mathcal{H}=\mathcal{H}_{R,q}(R\mathfrak{S}_n)\) and exploited to define an action of \(\mathcal{H}\) on \(V^{\otimes r}\) and decompose tensor space into a direct sum of \(q\)-permutation modules. The main goal of the paper under review concerns the endomorphism algebra of this \(\mathcal{H}\)-action. In the classical situation of the \(\mathfrak{S}_n\)-action, this is isomorphic to the partition algebra introduced provided \(n\geq 2r\). In general, the endomorphism algebra \(\mathrm{End}_{R\mathfrak{S}_n}(V^{\otimes r})\) is an epimorphic image of some partition algebra.\N\NThe authors define that the \(q\)-partition algebra \(\mathcal{P}_{R,q}(n,r)\) is the endomorphism ring of the \(\mathcal{H}_{R,q}(R\mathfrak{S}_n)\)-module \(V^{\otimes r}\). Note that this construction works for arbitrary commutative rings \(R\) and arbitrary units \(q\in R\). Indeed all constructions are stable under base change. Thus one can work over ring \(R=\mathbb{Z}[t,t^{-1}]\) of integral Laurent polynomials in the variable \(t\) and then prove the results for arbitrary rings \(R\) and units \(q\) by specialisation. In particular, one obtains\N\N\[\mathcal{P}_{R,q}(n,r)\cong R\otimes_{\mathbb{Z}[t,t^{-1}]}\mathcal{P}_{\mathbb{Z}[t,t^{-1}],q}(n,r).\]\N\NHalverson and Thiem defined a \(q\)-analogue \(Q_r (n, q)\) of the partition algebra \(P_r(n)\) to be the endomorphism algebra of a certain \(\mathbb{C}GL_n(q)\)-module \(\mathfrak{T}_Q^r\). This is obtained by iterating \(r\) times Harish-Chandra restriction and induction between \(GL_{n-1}(q)\) and \( GL_n(q)\) applied to the trivial \(GL_n(q)\)-module \(_{GL_n(q)}{\mathbb{C}}\). Thus \(Q_r(n,q)\) is defined only for prime powers \(q\). The main result of the paper under review is if \(q\in\mathbb{C}\) is a prime power and \(n,r\in\mathbb{N}\), then \[ \mathcal{P}_{\mathbb{C},q}(n,r)\cong Q_r(n,q). \]
scientific article; zbMATH DE number 7905969
scientific article