Small sum of group elements acting as rank 1 matrix. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:33:40Z http://mathoverflow.net/feeds/question/59807 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59807/small-sum-of-group-elements-acting-as-rank-1-matrix Small sum of group elements acting as rank 1 matrix. Klim Efremenko 2011-03-28T07:22:53Z 2011-03-31T16:14:10Z <p>I am interested in constructing small (as possible) group $G$ with large dimensional irreducible representation $\rho,V$ such that exist three elements of $g_1,g_2,g_3\in G$ such that for some $c_1,c_2,c_3\in C$ the matrix $c_1\rho(g_1)+c_2\rho(g_2)+c_3\rho(g_3)$ has rank one.($3$ elements here is arbitrary it may be any constant number)</p> <p>In fact I know that if there is only two elements then $|G|\geq 2^{n}$, where $n=dim V$. It is easily floows from the post: <a href="http://mathoverflow.net/questions/57806/irreducible-representation-flipping-two-elements" rel="nofollow">http://mathoverflow.net/questions/57806/irreducible-representation-flipping-two-elements</a></p> <p>My question is how to construct such irreducible representation of dimension $n>>log|G|$?</p> <p>Example when $|G|>2^n$ is symmetric group $S_n$ acting on $n$ elements induces reps on $F^n$. If $\rho$ is $n-1$ dimensional irreducible sub-representation then $id-(1,2)$ acts as rank one matrix. </p>