A diagram algebra for Soergel modules corresponding to smooth Schubert varieties (Q2796510)

Consider the BGG category \({\mathcal O}\) associated to the general linear Lie algebra \({\mathfrak gl}_n\) and its principal block \({\mathcal O}_0\). Given a pair of (orthogonal) parabolic subalgebras of \({\mathfrak gl}_n\), the author associates a Serre quotient of a Serre subcategory of \({\mathcal O}_0\). In related work of the author [Sel. Math., New Ser. 22, No. 2, 669--734 (2016; Zbl 1407.17012)], these subquotients of \({\mathcal O}_0\) are used to categorify some representations of the quantized Lie superalgebra \({\mathfrak gl}(1|1)\). In the paper under review, these subquotient categories are shown to be equivalent to (graded) module categories of certain diagram algebras. This equivalence is then used to compute the endomorphism rings of two special functors.NEWLINENEWLINEThe diagram algebras are explicitly constructed following the ideas used by \textit{J. Brundan} and \textit{C. Stroppel} [Mosc. Math. J. 11, No. 4, 685--722 (2011; Zbl 1275.17012)] to construct generalized Khovanov algebras. It is directly shown that these diagram algebras are graded cellular and properly stratified.NEWLINENEWLINEThe approach used by Brundan and Stroppel to define the multiplication in the diagram algebras will not work in this setting. The author overcomes this obstruction by making use of morphisms between Soergel modules, which are modules for the complex polynomial ring \({\mathbb C}[x_1, \dots, x_n]\). Each module is associated to a permutation (of \(n\) objects), but they are not well understood in general. The computation of the relevant Soergel modules is aided by the fact that they are all cyclic, that is, the corresponding Schubert variety is rationally smooth in the full flag variety. A key portion of the paper (of independent interest) is a study of these particular Soergel modules. Each relevant Soergel module is shown to be isomorphic to \({\mathbb C}[x_1,\dots, x_n]/I\) for an explicit ideal \(I\) generated by symmetric polynomials. From this, the author obtains a description of the morphisms between the relevant Soergel modules.NEWLINENEWLINEThe ideas in the paper are nicely developed with a thorough discussion of the tools and combinatorics being used, along with numerous examples.