Schur-Weyl dualities for the rook monoid: an approach via Schur algebras (Q6591553)

Let \(k\) be an infinite field and fix \(V=k^n\) for some natural number \(n\). The classical Schur-Weyl duality asserts that the actions of the general linear group \(\operatorname{GL}_n(k)\) and the symmetric group \(S_d\) on the tensor power \(V^{\otimes d}\) centralise each other. Here, \(\operatorname{GL}_n(k)\) acts on the left by matrix multiplication on \(V\) and on \(V^{\otimes d}\) by the diagonal action while \(S_d\) acts on the right on \(V^{\otimes d}\) by permuting the tensor factors. The centraliser algebra of \(S_d\) on \(V^{\otimes d}\) is the classical Schur algebra \(S(n, d)\) introduced implicitly by Schur in his PhD thesis and popularised by \textit{J. A. Green} [Lect. Notes Math. 848, 124--140 (1981; Zbl 0453.20029)].\N\NOther instances of Schur-Weyl duality have also been obtained for subgroups of the general linear group, for example, Schur-Weyl duality for orthogonal groups (on the left) and Brauer algebras (on the right) as in [\textit{S. Doty} and \textit{J. Hu}, Proc. Lond. Math. Soc. (3) 98, No. 3, 679--713 (2009; Zbl 1177.20051)] and for symmetric groups (on the left) and partition algebras (on the right) as in [\textit{T. Halverson} and \textit{A. Ram}, Eur. J. Comb. 26, No. 6, 869--921 (2005; Zbl 1112.20010)].\N\NAssume now that \(k\) is a field of characteristic zero. Consider \(\operatorname{GL}_{n-1}(k)\) as subgroup of \(\operatorname{GL}_n(k)\). Thus the action of \(\operatorname{GL}_n(k)\) on \(V\) restricts to an action of \(\operatorname{GL}_{n-1}(k)\) by fixing the last component of \(V=k^{n-1}\oplus k\). Consequently, the action of \(\operatorname{GL}_{n}(k)\) on \(V^{\otimes d}\) restricts to an action of \(\operatorname{GL}_{n-1}(k)\) on \(V^{\otimes d}\).\N\N\textit{L. Solomon} [J. Algebra 256, No. 2, 309--342 (2002; Zbl 1034.20056)] proved that \(\operatorname{GL}_{n-1}(k)\) is in Schur-Weyl duality with a rook monoid \(R_d\). That is, the rook monoid \(R_d\) also acts on \(V^{\otimes d}\) in such a way that extends the place permutation action of \(S_d\) and the actions of \(\operatorname{GL}_{n-1}(k)\) and \(R_{d}\) centralise each other. Here, the rook monoid \(R_d\) is the set of all bijective partial maps from \(\{1, \ldots, d\}\) to \(\{1, \ldots, d\}\) under the composition of partial functions.\N\NIn the paper under review, the authors show that the centraliser of \(R_d\) in \(V^{\otimes d}\) is an extended Schur algebra and then they reformulate Solomon's [loc. cit.] Schur-Weyl duality as a Schur-Weyl duality between extended Schur algebras and rook monoids. Extended Schur algebras arise as a finite product of classical Schur algebras and are a particular case of generalised Schur algebras in the sense of Donkin. In particular, since classical Schur algebras are semi-simple in characteristic zero, the extended Schur algebras studied in this paper are also semi-simple.