## EVENT DETAILS AND ABSTRACT

**Colloquium**
**Title:** A q-analogue of the algebra of symmetric functions in infinitely many variables

**Speaker:** Professor Edward Frenkel

**Speaker Info:** University of California, Berkeley

**Brief Description:**

**Special Note**:

**Abstract:**

It is well-known that there is a natural Hopf algebra structure
on the ring of symmetric functions in infinitely many variables. This
structure has a clear representation theoretic meaning if one thinks of
the ring of symmetric functions as the direct limit of Rep GL(N), the
Grothendieck groups of (polynomial) representations of the group GL(N).
The symmetric functions appear if one considers characters of these
representations. We will introduce a q-analogue of this Hopf algebra: the
direct limit of Grothendieck groups of finite-dimensional representations
of the affine quantum group of GL(N). It turns out that this Hopf algebra
is isomorphic to a familiar Hopf algebra, namely, the Hall algebra of an
infinite linear (resp., cyclic) quiver if q is generic (resp., root of
unity). In addition, to each representation one can attach its
"q-character", so one obtains an algebra of "q-symmetric functions". The
q-characters have already found interesting applications in representation
theory and mathematical physics.

**Date:** Tuesday, December 4, 2001

**Time:** 4:00pm

**Where:** Lunt 105

**Contact Person:** Prof. Kari Vilonen

**Contact email:** vilonen@math.northwestern.edu

**Contact Phone:** 847-491-5557

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