Although we have the character table for S4, we're still missing the ac-tual representation matrices (apart from the 1-dim cases), and I really don't know how we'd go about finding these.. The Character Table of S4 Let's use our inner tensor products to fill in the character table of . We start by listing out the conjugacy classes along with their sizes: Now we have the same three representations as in the character table of : the trivial, the signum, and the complement of the signum in the defining representation.
Character Polynomials From Stanley's Positivity Problems in Algebraic Combinatorics Problem 12: Give a combinatorial interpretation of the row sums of the character table for Sn (combinatorial proof of non-negativity) Sn = permutations of n things Contains n! elements. The character table of the alternating group A4 is easier. The eight cyclic permutations of order 3 in S4 is the union of two conjugacy classes in A4, each with two elements.