Categorical actions and multiplicities in the Deligne category Rep_(GLt) Academic Article uri icon

abstract

  • We study the categorical type A action on the Deligne category D t = R e p _ ( G L t ) ( t ∈ C ) and its “abelian envelope” V t constructed in [13] . For t ∈ Z , this action categorifies an action of the Lie algebra sl Z on the tensor product of the Fock space F with F t ∨ , 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 sl Z -modules Pol and P o l t ∨ . The latter categorify F and F t ∨ 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 D t is abelian semisimple, and the type A action induces a categorical action of sl Z × sl Z . This action categorifies the sl Z × sl Z -module F ⊠ F ∨ , making D t the exterior tensor product of the categorical sl Z -modules Pol , P o l ∨ . Along the way we establish a new relation between the Kazhdan–Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in V t .

publication date

  • June 15, 2018