Do the first two columns of the character table of the permutation group $S\_n$ (the conjugacy classes () and (1 2) - that's a fairly standard order) always suffice to identify an irrep? (E.g. take $S\_6$: four irreps have dimension 5 but the second entry is 1,-1,3 or -3.) If yes, can this be generalized to other groups? How many entries can be equal for two different irreps anyway? (E.g. $S\_6$ again, trivial and standard irrep have two equal entries out of eleven.)



How to recognize different types of irreps
For $S\_{16}$ there appear (I asked GAP) to be two pairs of non-self-conjugate partitions where the value of the corresponding characters are zero on transpositions.


I haven't tried to figure out what the partitions are. 


**Edit.** Suzuki ("The values of irreducible characters of the symmetric group", Arcata conference on representations of finite groups, Part II, Proceedings of symposia in pure mathematics 47, 1986) gives a simple criterion for an irreducible character of a symmetric group to take the value zero on transpositions. If the corresponding Young diagram has boxes with coordinates $(x,y)$ with the main diagonal $x=y$ then the criterion is that $\sum (x-y)=0$. This is obviously satisfied by self-conjugate partitions, but he also gives the example of $6+3+2+2+2$, which gives a pair of degree $112112$ characters for $S\_{15}$ that I missed when asking GAP.


