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Asymptotics for number of indecomposable components of tensor powers of the natural SL2-module in odd characteristic
Abstract
Let K be an algebraically closed field of odd characteristic p, let G= SL2(K), and let V be the natural representation of G. Let bk denote the number of G-indecomposable factors of V⊗ k, counted with multiplicity, and let δp=1−logp2 ((p+1)/2). Then there exists a smooth, strictly positive, multiplicatively p2-periodic function ω(t) such that bk is asymptotic to ω(k)k−δp2k. We also show that t−δω(t) arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight 3/2 theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on (0,p).
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BibTeX
@misc{lahiri2026asymptotics,
title = {{Asymptotics for number of indecomposable components of tensor powers of the natural $\mathrmSL_2$-module in odd characteristic}},
author = {Aranya Lahiri and Nai-Heng Sheu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37706},
eprint = {2609.37706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37706},
}
RIS
TY - UNPB
TI - Asymptotics for number of indecomposable components of tensor powers of the natural SL_2-module in odd characteristic
AU - Aranya Lahiri
AU - Nai-Heng Sheu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.37706
UR - https://arxiv.org/abs/2609.37706
AB - Let K be an algebraically closed field of odd characteristic p, let G= SL_2(K), and let V be the natural representation of G. Let b_k denote the number of G-indecomposable factors of V^⊗ k, counted with multiplicity, and let δ_p=1−log_p^2 ((p+1)/2). Then there exists a smooth, strictly positive, multiplicatively p^2-periodic function ω(t) such that b_k is asymptotic to ω(k)k^−δ_p2^k. We also show that t^−δω(t) arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight 3/2 theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on (0,p).
ER -