Frobenius reciprocity on the space of functions invariant under a group action
| Autor: | Suksumran, Teerapong, Udomworarat, Tanakorn |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | This article studies connections between group actions and their corresponding vector spaces. Given an action of a group $G$ on a nonempty set $X$, we examine the space $L(X)$ of scalar-valued functions on $X$ and its fixed subspace: $$ L^G(X) = \{f\in L(X)\colon f(a\cdot x) = f(x) \textrm{ for all }a\in G, x\in X\}. $$ In particular, we show that $L^G(X)$ is an invariant of the action of $G$ on $X$. In the case when the action is finite, we compute the dimension of $L^G(X)$ in terms of fixed points of $X$ and prove several prominent results for $L^G(X)$, including Bessel's inequality and Frobenius reciprocity. Comment: 17 pages |
| Databáze: | arXiv |
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