Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:57:15Z http://mathoverflow.net/feeds/question/66311 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66311/picking-n-so-that-certain-schur-functors-of-the-standard-representation-of-s-n Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent John Wiltshire-Gordon 2011-05-28T20:58:55Z 2011-05-28T20:58:55Z <p>Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$.</p> <p>Let $\lambda$ range over all partitions of some natural number $k$. We may ask for the smallest $n$ making the characters of the representations $\mathbb{S}_{\lambda} V_n$ linearly independent. A little experimentation indicates that $n=k$.</p> <p>If instead $\lambda$ ranges over all partitions of natural numbers less than or equal to $k$, it seems that $n=2k$.</p> <p>Are these patterns correct?</p>