2020/05/17 by Vinoth Nandakumar, Nandakumar, Vinoth, Gufang Zhao +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2005.08248
Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of \mathfraksl2 using singular blocks of category O for \mathfraksln. In earlier work, we construct a positive characteristic analogue using blocks of representations of \mathfraksln over a field k of characteristic p > n, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical \mathfrakslk-action, following Sussan's approach, by considering more singular blocks of modular representations of \mathfraksln. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of \mathfraksln.