Categorical actions and multiplicities in the Deligne category Rep(GLt)

JOURNAL OF ALGEBRA

卷:504

Article

Entova-Aizenbud, Inna¹ 

[1] Ben Gurion Univ Negev, Math Dept, Beer Sheva, Israel

关键词: Deligne categories;    Categorical actions;   

DOI  :  10.1016/j.jalgebra.2018.02.016

来源: Elsevier

【 摘 要 】

We study the categorical type A action on the Deligne category D-t = Rep(GL(t)) t is an element of C) and its abelian envelope Vt constructed in [13]. For t Deligne Z, this action categorifies an action of the Lie algebra sly on the tensor product of the Fock space a with ay, its restricted dual shifted by t, as was suggested by I. Losev. In fact, this action makes the category V-t the tensor product (in the sense of Losev and Webster, [20]) of categorical sly-modules Po! and Poll.'. The latter categorify a and al/ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described in [16,18]. When t Z, the Deligne category Dt is abelian semisimple, and the type A action induces a categorical action of sly x 51z. This action categorifies the sly x sly-module av, making A the exterior tensor product of the categorical sly-modules Pod, Polv. Along the way we establish a new relation between the Kazhdan Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in Vt. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free