irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory. Low-dimensional representations Symmetric groups The lowest-dimensional representations of the symmetric groups can be described explicitly, and over arbitrary fields. The smallest two degrees in characteristic zero are described here: Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix. For , there is another irreducible representation of degree 1, called the sign representation or alternating character, which takes a permutation to the one by one matrix with entry ±1 based on the sign of the permutation. These are the only one-dimensional representations of the symmetric groups, as one-dimensional representations are abelian, and the abelianization of the symmetric group is C2, the cyclic group of order 2. For all n, there is an n-dimensional representation of the symmetric group of order n!, called the , which consists of permuting n coordinates. This has the trivial subrepresentation consisting of vectors whose coordinates are all equal. The orthogonal complement consists of those vectors whose coordinates sum to zero, and when , the representation on this subspace is an -dimensional irreducible representation, called the standard representation. Another -dimensional irreducible representation is found by tensoring with the sign representation. An exterior power of the standard representation is irreducible provided . For , these are the lowest-dimensional irreducible representations of Sn – all other irreducible representations have dimension at least n. However for , the surjection from S4 to S3 allows S4 to inherit a two-dimensional irreducible representation. For , the exceptional transitive embedding of S5 into S6 produces another pair of five-dimensional irreducible representations. Alternating groups The representation theory of the alternating groups is similar, though the sign representation disappears. For , the lowest-dimensional irreducible representations are the trivial representation in dimension one, and the -dimensional representation from the other summand of the permutation representation, with all other irreducible representations having higher dimension, but there are exceptions for smaller n. The alternating groups for have only one one-dimensional irreducible representation, the trivial representation. For there are two additional one-dimensional irreducible representations, corresponding to maps to the cyclic group of order 3: and . For , there is just one irreducible representation of degree , and this is the smallest degree of a non-trivial irreducible representation. For the obvious analogue