---- > [!definition] Definition. ([[symmetric power]]) > Let $R$ be a [[ring]] and let $M$ be an $R$-[[module]]. Denote by $\mathbb{T}^{\ell}(M)$ the $\ell$th [[tensor product of modules|tensor power]] of $M$. > > The **$\ell$th symmetric power** of $M$ is a new $R$-[[module]] $\mathbb{S}^{\ell}(M)$ satisfying the [[universal property]] that every $R$-[[symmetric multilinear map|symmetric multilinear map]] $\varphi:M^{\ell} \to P$ factors through $\mathbb{S}^{\ell}(M)$ via a unique $R$-[[linear map]] $\overline{\varphi}$ > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFkA9AHW5gYYgAvqXSZc+QigCM5KrUYs2ABWGiQGbHgJEy0+fWatEIXgBk6AWwBGUOjz4CABAAp2ASmHyYUAObwiUAAzACcISyQAJmocCCRZBSM2XhgADyw4HDgnAEInXggaGBCGLDAYYF56ELQACywhEGoGOmt+ZXFtKRAQrF9anDVgsIjEMhBYqOpDJRNeAHcffyaQFraGDq1JNl7+wZFh8KRxycQEmeNTbmq6rC8hIA > \begin{tikzcd} > M^\ell \arrow[d, "i"'] \arrow[r, "\varphi"] & P \\ > \mathbb{S}^\ell (M) \arrow[ru, "\exists ! \overline{\varphi}"'] & > \end{tikzcd} > \end{document} > ``` > > where $i$ is a [[symmetric multilinear map]]. This defines $\mathbb{S}^{\ell}(M)$ [[terminal objects are unique up to a unique isomorphism|up to isomorphism]], if it exists. > > Indeed, symmetric powers *do* exist in $R$-$\mathsf{Mod}$. Let $W \subset \mathbb{T}^{\ell}(M)$ be the [[submodule]] generated by all pure tensors of the form $m_{\sigma(1)} \otimes \dots \otimes m_{\sigma(\ell)} - m_{\sigma'(1)} \otimes \dots \otimes m_{\sigma'(\ell)}$ > where $\sigma,\sigma'$ range over the [[symmetric group]] $S_{\ell}$. Then $\mathbb{S}^{\ell}(M) \cong \frac{\mathbb{T}^{\ell}(M)}{W}.$ > Here the map $i$ is of course the composition $M^{\ell} \xrightarrow{\otimes} \mathbb{T}^{\ell}(M) \xrightarrow{\pi} \frac{\mathbb{T}^{\ell}(M)}{W}$ > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```