Categorification via blocks of modular representations of sl(n)

Vinoth Nandakumar, Gufang Zhao

Published 2016-12-21Version 1

Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$, where they use singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_m$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_m$ over a field $\textbf{k}$ of characteristic $p > 2$, with zero Frobenius character, and singular Harish-Chandra character; this is related to a categorification constructed by Chuang-Rouquier using representations of $\text{SL}_m(\textbf{k})$. The classes of the irreducible modules give a basis in the Grothendieck group, depending on $p$, that we call the "$p$-canonical weight basis". When $p \gg 0$, we show that the aforementioned categorification admits a graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. This equivalence is established using the positive characteristic localization theory developed by Riche and Bezrukavnikov-Mirkovi\'c-Rumynin. As a consequence, we obtain an abelian refinement of the [CKL] categorification; these results are closely related to the more general framework recently developed by Cautis-Koppensteiner and Cautis-Kamnitzer.