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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2405.16015 |
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| _version_ | 1866929358087651328 |
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| author | Larsen, Michael J. |
| author_facet | Larsen, Michael J. |
| contents | 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 $δ= \frac 32 - \frac{\log 3}{2\log 2}$. Then there exists a smooth multiplicatively periodic function $ω(x)$ such that $b_{2k}^{G,V} = b_{2k+1}^{G,V}$ is asymptotic to $ω(k) k^{-δ}4^k$. We also prove a lower bound of the form $c_W k^{-δ}(\dim W)^k$ for $b_k^{G,W} $ for any tilting representation $W$ of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for $\mathrm{SL}_2$-indecomposables in tensor powers of the natural representation in characteristic $2$ Larsen, Michael J. Representation Theory 20G05 (Primary) 20C20 (Secondary) 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 $δ= \frac 32 - \frac{\log 3}{2\log 2}$. Then there exists a smooth multiplicatively periodic function $ω(x)$ such that $b_{2k}^{G,V} = b_{2k+1}^{G,V}$ is asymptotic to $ω(k) k^{-δ}4^k$. We also prove a lower bound of the form $c_W k^{-δ}(\dim W)^k$ for $b_k^{G,W} $ for any tilting representation $W$ of $G$. |
| title | Bounds for $\mathrm{SL}_2$-indecomposables in tensor powers of the natural representation in characteristic $2$ |
| topic | Representation Theory 20G05 (Primary) 20C20 (Secondary) |
| url | https://arxiv.org/abs/2405.16015 |