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darij grinberg
darij grinberg
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Given a S$\mathbb{S}$-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the S$\mathbb{S}$-module? In Loday Vallette's book on OperadsLoday/Vallette's book on Operads, they say that if P$P$ is the Schur functor, look at P(V)$P(V)$, where V$V$ is an n$n$-dimensional vectorspace. The n multilinear$n$-multilinear part of P(V)$P(V)$ is isomorphic to P(n)$P(n)$, as a S_n modulean $S_n$-module. Can someone explain what is meant by the 'multilinear part'?

Given a S-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the S-module? In Loday Vallette's book on Operads, they say that if P is the Schur functor, look at P(V), where V is an n-dimensional vectorspace. The n multilinear part of P(V) is isomorphic to P(n), as a S_n module. Can someone explain what is meant by the 'multilinear part'?

Given a $\mathbb{S}$-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the $\mathbb{S}$-module? In Loday/Vallette's book on Operads, they say that if $P$ is the Schur functor, look at $P(V)$, where $V$ is an $n$-dimensional vectorspace. The $n$-multilinear part of $P(V)$ is isomorphic to $P(n)$, as an $S_n$-module. Can someone explain what is meant by the 'multilinear part'?

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ani
ani
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bijection between S-modules and Schur functors

Given a S-module, we can construct a Schur functor, which is an endo functor in the category of vectorspaces. But given a Schur functor, how can we get back the S-module? In Loday Vallette's book on Operads, they say that if P is the Schur functor, look at P(V), where V is an n-dimensional vectorspace. The n multilinear part of P(V) is isomorphic to P(n), as a S_n module. Can someone explain what is meant by the 'multilinear part'?