Abstract
We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category $\mathrm{\underline{Re}p}(S_t)$, to the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions.
Highlights
In [Del07], Deligne introduced a linear monoidal category Rep(St) that interpolates between the categories of representations of the symmetric groups
The endomorphism algebras of the partition category are the partition algebras first introduced by Martin ([Mar94]) and later, independently, by Jones ([Jon94]) as a generalization of the Temperley–Lieb algebra and the Potts model in statistical mechanics
The partition algebras are in duality with the action of the symmetric group on tensor powers of its permutation representation; that is, the partition algebras generate the commutant of this action
Summary
In [Del07], Deligne introduced a linear monoidal category Rep(St) that interpolates between the categories of representations of the symmetric groups. Categorification, partition category, Deligne category, Heisenberg category, symmetric group, partition algebra, monoidal category. Deligne’s category Rep(St) is the additive Karoubi envelope of the partition category, we have an induced faithful linear monoidal functor. The Grothendieck ring of Heis is isomorphic to a central reduction Heis of the universal enveloping algebra of the Heisenberg Lie algebra This was conjectured by Khovanov in [Kho, Conj. (a) Replacing the role of the symmetric group with wreath product algebras, one should be able to define an embedding, analogous to Ψt, relating the G-colored partition algebras of [Blo03], the wreath Deligne categories of [Mor, Kno07], and the Frobenius Heisenberg categories of [RS17, Sav19]. We let N denote the additive monoid of nonnegative integers
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