Generators for the representation rings of certain wreath products

Harman, Nate

doi:10.1016/j.jalgebra.2015.09.003

Cited by 6 publications

(15 citation statements)

References 16 publications

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“…By Theorem 7.14 of [Ryb17] it follows that G Z ∞ (C) is the Grothendieck ring of the wreath-product Deligne category S t (C) as introduced by [Mor12], with X #» λ corresponding to the simple objects (for generic t) of S t (C). It was proved in [Har16] that the Grothendieck ring of S t (C) has a filtration ("|λ|-filtration"), and a generating set was given (called "basic hooks"). In [Ryb17], the algebraic structure of G Q ∞ (C) was established.…”

Section: Upon Applying the Induction Functor Indmentioning

confidence: 99%

“…It was shown in [Har16] that the associated graded algebra of G Z ∞ (C) is a free polynomial algebra in basic hooks, which are defined as…”

Section: Structure Of the Rational Limiting Grothendieck Ring Of Wreamentioning

confidence: 99%

“…It was shown in [Har16] that G(S t (C)) is a filtered ring with associated graded ring isomorphic to a free polynomial algebra in certain elements called "basic hooks", indexed by Z >0 × I(C), where I(C) is the set of isomorphism classes of simple objects in C. We show an analogous result for our case, indexed by Z >0 × I, where I is a Z-basis of R. If U ∈ R we define the elements e n (U ) by giving an expression for the generating where e 0 (U ) = 1. This allows us to state one family of relations between the elements e n (U ); for U, V ∈ R,…”

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confidence: 92%

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

Ryba

2019

International Mathematics Research Notices

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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...

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Section: Upon Applying the Induction Functor Indmentioning

confidence: 99%

“…It was shown in [Har16] that the associated graded algebra of G Z ∞ (C) is a free polynomial algebra in basic hooks, which are defined as…”

Section: Structure Of the Rational Limiting Grothendieck Ring Of Wreamentioning

confidence: 99%

mentioning

confidence: 92%

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

Ryba

2019

International Mathematics Research Notices

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“…Proof. Firstly, the multiplication in G ∞ (C) is seen to be associative by considering The filtration is essentially the same as the |λ|-filtration defined in Definition 2.7 of [Har16]. In particular, the associated graded algebra (with basis induced from X #» λ ) has structure constants equal to those of the ring of symmetric functions with the Schur function basis.…”

Section: Definition and Basicmentioning

confidence: 99%

“…We show that this multiplication exhibits a certain stability property which allow us to define a "limiting Grothendieck ring", G ∞ (C) (here C = R − mod). This ring is the Grothendieck ring of the wreath product Deligne categories S t (C) introduced in [Mor12] and considered in [Har16]. When C is the category of finite-dimensional vector spaces over the field k (characteristic zero and algebraically closed), we recover the original Deligne category Rep(S t ).…”

Section: Introductionmentioning

confidence: 99%

Stable Grothendieck rings of wreath product categories

Ryba

2018

J Algebr Comb

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Let k be an algebraically closed field of characteristic zero, and let C = R − mod be the category of finite-dimensional modules over a fixed Hopf algebra over k. One may form the wreath product categories Wn(C) = (R ≀ Sn) − mod whose Grothendieck groups inherit the structure of a ring. Fixing distinguished generating sets (called basic hooks) of the Grothendieck rings, the classification of the simple objects in Wn(C) allows one to demonstrate stability of structure constants in the Grothendieck rings (appropriately understood), and hence define a limiting Grothendieck ring. This ring is the Grothendieck ring of the wreath product Deligne category St(C). We give a presentation of the ring and an expression for the distinguished basis arising from simple objects in the wreath product categories as polynomials in basic hooks. We discuss some applications when R is the group algebra of a finite group, and some results about stable Kronecker coefficients. Finally, we explain how to generalise to the setting where C is a tensor category. 12 8. The Ring Structure of G ∞ (C) 13 8.

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