WEDNESDAY Feb. 14 : | Combinatorics Seminar |

Title: | Representation Theory of Sn, Part I. |

Gabriel Necoechea
Kansas State University A (complex) linear representation of a group is a group homomorphism pi: G -> GL(V), where V is a complex vector space. An important tool in studying representations of finite groups is character theory, which tells us that we need only consider conjugacy classes when trying to count the irreducible representations (=irreps) of a group. For G = Sn, the conjugacy classes are in bijection with partitions of n, and we can extract lots of information about the irreps of Sn from these partitions. In particular, by Specht's construction, each irrep of Sn can be thought of as a vector space over the standard young tableaux associated to a partition L. Such a vector space is called the Specht Module associated to L. By construction, the dimension of the Specht Module associated to L is #{standard young tableaux of shape L}, which is given by the Hook Length Formula. In this talk, we will define representations, irreducible representations, and characters of representations. We will see that characters, which are certain C-valued class functions on G, characterize irreducible representations. Then we will turn attention to the specific case of Sn, where we will construct Specht modules. An attempt will be made to present many examples. | |

Time and Place: | 12:30 PM CW 129 |