The Structure of the Grothendieck Rings of Wreath Product Deligne Categories and their Generalisations

Abstract

Given a tensor category C over an algebraically closed field of characteristic zero, we may form the wreath product category Wn(C). It was shown in [Ryb17] that the Grothendieck rings of these wreath product categories stabilise in some sense as n → ∞. The resulting "limit" ring, G Z ∞ (C), is isomorphic to the Grothendieck ring of the wreath product Deligne category St(C) as defined by [Mor12]. This ring only depends on the Grothendieck ring G(C). Given a ring R which is free as a Z-module, we construct a ring G Z ∞ (R) which specialises to G Z ∞ (C) when R = G(C). We give a description of G Z ∞ (R) using generators very similar to the basic hooks of [Har16]. We also show that G Z ∞ (R) is a λ-ring wherever R is, and that G Z ∞ (R) is (unconditionally) a Hopf algebra. Finally we show that G Z ∞ (R) is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in (W ⊗ Z R) × , where W is the ring of Big Witt Vectors.Contents 2 alternative way of specifying a partition λ is to give the numbers m i = |{r | λ r = i}| which count the number of parts of λ of size i; in this case we write λ = (1 m1 2 m2 · · · ) (thus |λ| = λ 1 +λ 2 +· · · = 1m 1 +2m 2 +3m 3 +· · · ). When it is unclear which partition we are considering, we write m i (λ) for the number of parts of size i in the partition λ. The length of λ, denoted l(λ), is the number of nonzero parts of λ; l(λ) = m 1 + m 2 + · · · . We will make use of the quantities z λ = i m i !i mi and ε λ = (−1) i (i−1)mi .We write λ ⊢ n to mean that λ is a partition of n, and P = {λ | λ ⊢ n, n ∈ N ≥0 } is the set of all partitions.Definition 2.1. Let λ = (λ 1 , λ 2 , . . . , λ l ) be a partition, and assume n is a natural number such that n ≥ |λ| + λ 1 . We write λ[n] for the partition (n − |λ|, λ 1 , λ 2 , . . . , λ l ).Note that the inequality on n guarantees that this sequence is weakly decreasing. Thus λ[n] is the partition of n obtained by adding a first part to λ (or top row when the partition is depicted as a Young diagram in English notation) of the appropriate size.The main algebraic tool we use is the ring of symmetric functions, denoted Λ. It is defined via a graded inverse limit of rings. Consider the rings of invariants R n = Z[x 1 , x 2 , · · · , x n ] Sn , with the symmetric group S n acting by permutation of variables. There are homomorphisms R n+1 → R n defined by setting x n+1 = 0, and the graded inverse limit defined by these is Λ. We refer the reader to [Mac95] for details.As a commutative Z-algebra, Λ is freely generated by the elementary symmetric functions e i , as well as the complete symmetric functions h i . In particular Λ = Z[e 1 , e 2 , . . .] = Z[h 1 , h 2 , . . .] (by convention h 0 = e 0 = 1). There is an automorphism ω of Λ defined by ω(e i ) = h i . It turns out that ω is an involution: ω(h i ) = e i .Another important family of elements of Λ are the power-sum symmetric functions p n which do not generate Λ over Z, however, Q ⊗ Z Λ = Q[p 1 , p 2 , . . .]. The relations between the e i , h i , and p i are encapsula...

Stable Centres I: Wreath Products

“…In that case, g is not necessarily abelian and the universal enveloping algebra is the correct construction. The interested reader is directed to [Ryb19a] and [Ryb19b].…”