Representations of Symmetric and Unipotent Groups

This talk will serve as an introduction to representation theory, using the symmetric group and unipotent groups as our key examples. Recall that a representation of a group G is a set of linear operators on a vector space W, defined by a homomorphism from G into the automorphisms of W. It turns out that all such homomorphisms admit a more or less explicit description.
We will begin with the simplest (yet important) example: the action of the symmetric group S_n on the tensor product $V \otimes V \otimes ... \otimes V$ ($n$ copies of an arbitrary vector space V). We will briefly review the representation theory of the symmetric groups S₂ and S₃, and discuss the role of conjugacy classes for representations of general finite groups. Then, we will move on to the symmetric group S_n and describe the Gelfand-Zetlin basis in terms of semistandard Young tableaux.
Next, we will discuss the structure of representations of unipotent groups in Hilbert spaces, starting with the three-dimensional Heisenberg group and then moving to the example of upper-triangular matrices. It turns out that such representations also admit a description somewhat reminiscent of that for finite groups; however, instead of conjugacy classes within the group itself, one must consider coadjoint orbits (the orbits of the group's action on the co-tangent space at the identity). We will examine what these orbits look like for the Heisenberg group and for upper-triangular matrices. We will also discuss the concept of polarizing linear forms, which allows for the explicit construction of unipotent group representations as L²(X) for a suitable space X with an action of the original group.
Finally, I will briefly outline (with the aid of tables) the classification of coadjoint orbits of dimensions 2, 4, and 6 for upper-triangular matrices—a result from my most recent paper.