Growth rates of the number of indecomposable summands in tensor powers

Coulembier, Kevin; Ostrik, Victor; Tubbenhauer, Daniel

doi:10.21203/rs.3.rs-2456982/v1

Cited by 2 publications

(3 citation statements)

References 38 publications

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“…(c) The second statements in Theorem 1 and Theorem 2 were already observed in [COT23] (in the setting of [COT23] the Perron-Frobenius dimension agrees with the usual dimension), but the (finer) asymptotic behavior appears to be new.…”

Section: Remarkmentioning

confidence: 63%

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Asymptotics in finite monoidal categories

Lacabanne,

Tubbenhauer,

Vaz

2023

Proc. Amer. Math. Soc. Ser. B

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Section: Remarkmentioning

confidence: 63%

“…More generally, the paper [CEO23b] studies, working in certain tensor categories, the growth rates of summands of categorical dimension prime to the underlying characteristic. The paper [COT23] studies the growth rate of all summands, while [KST22] studies the Schur-Weyl dual question.…”

Section: Remarkmentioning

confidence: 99%

Asymptotics in finite monoidal categories

Lacabanne,

Tubbenhauer,

Vaz

2023

Proc. Amer. Math. Soc. Ser. B

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“…Asymptotic properties of tensor powers of a modular representation (or, more generally, an object of a symmetric tensor category) is an interesting and mysterious subject about which rather little is known. It has recently been studied in the papers [B2,BS,CEO,EK,COT]. The goal of this paper is to continue this study.…”

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confidence: 98%

Asymptotic properties of tensor powers in symmetric tensor categories

Coulembier¹,

Etingof²,

Ostrik³

2023

Preprint

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Let G be a group and V a finite dimensional representation of G over an algebraically closed field k of characteristic p > 0. Let d n (V ) be the number of indecomposable summands of V ⊗n of nonzero dimension mod p. It is easy to see that there exists a limit δ(V ) := lim n→∞ d n (V ) 1/n , which is positive (and ≥ 1) iff V has an indecomposable summand of nonzero dimension mod p. We show that in this case the numberand moreover this holds for any symmetric tensor category over k of moderate growth. Furthermore, we conjecture that in fact log(c(V ) −1 ) = O(δ(V )) (which would be sharp), and prove this for p = 2, 3; in particular, for p = 2 we show that c(V ) ≥ 3 − 4 3 δ(V )+1 . The proofs are based on the characteristic p version of Deligne's theorem for symmetric tensor categories obtained in [CEO]. We also conjecture a classification of semisimple symmetric tensor categories of moderate growth which is interesting in its own right and implies the above conjecture for all p, and illustrate this conjecture by describing the semisimplification of the modular representation category of a cyclic p-group. Finally, we study the asymptotic behavior of the decomposition of V ⊗n in characteristic zero using Deligne's theorem and the Macdonald-Mehta-Opdam identity. 1.4. Conjectural classification of semisimple symmetric tensor categories and its application to Conjecture 1.2 1.5. Cyclic groups 1.6. Organization of the paper 1.7. Acknowledgements 2. Characteristic zero 2.1. Finite groups in the non-modular case 2.2. The asymptotic formula 2.3. Tensor powers of a representation of a connected reductive group 2.4. Computation of C V (s) 3. Positive characteristic 3.1. A lower bound for c n (V ) 3.2. An upper bound on dimensions of simple objects in a semisimple symmetric tensor category 3.3. Theorem 3.2 implies Theorem 1.1 3.4. Proof of Theorem 3.2 for p = 2 3.5. Proof of Theorem 3.2 for p > 2 3.6. Conjecture 1.3 implies Conjecture 1.2 4. A conjectural classification of semisimple symmetric tensor categories in positive characteristic 4.1. The conjecture 4.2. Examples 4.3. Rank 2 simple Lie algebras 4.4. Invariantless simple Lie algebras in Ver p 5. Semisimplification of the representation category of a cyclic group 5.1. Fusion rules for representations of Z/p n . 5.2. Fusion rules for the semisimplification of the representation category of Z/p n . 5.3. Categorifications of K p References

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Growth rates of the number of indecomposable summands in tensor powers

Abstract: In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.

Cited by 2 publications

References 38 publications

Asymptotics in finite monoidal categories

Asymptotics in finite monoidal categories

Asymptotic properties of tensor powers in symmetric tensor categories

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