no. 4
Bounds for $\mathrm{SL}_2$-indecomposables in tensor powers of the natural representation in characteristic $2$
Larsen, Michael Jeffrey1
1Department of Mathematics, Indiana University Rawles Hall Bloomington, IN 47405-5701 United States
Annals of Representation Theory, Volume 2 (2025) no. 4, pp. 575-598

Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the algebraic group $\mathrm{SL}_2$ over $K$, and $V$ be the natural representation of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta = \frac{3}{2} - \frac{\log 3}{2\log 2}$. Then there exists a smooth multiplicatively periodic function $\omega (x)$ such that $b_{2k}^{G,V} = b_{2k+1}^{G,V}$ is asymptotic to $\omega (k) k^{-\delta }4^k$. We also prove a lower bound of the form $c_W k^{-\delta }(\dim W)^k$ for $b_k^{G,W}$ for any tilting representation $W$ of $G$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/art.30
Classification: 20G05, 20C20

Larsen, Michael Jeffrey  1

1 Department of Mathematics, Indiana University Rawles Hall Bloomington, IN 47405-5701 United States
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Larsen, Michael Jeffrey. Bounds for $\mathrm{SL}_2$-indecomposables in tensor powers of the natural representation in characteristic $2$. Annals of Representation Theory, Volume 2 (2025) no. 4, pp. 575-598. doi: 10.5802/art.30
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