Abstracts
for the Seminar

Spring 2017

**Speaker: **Darij Grinberg, University of Minnesota

**Title: **Function-field symmetric functions: in search of a $GF(q)[T]$-combinatorics

**Time:** 2:30 PM, Monday, February 27, 2017

**Place:** Malott 206

**Abstract:** The ring of symmetric functions is more than a ring: It also carries
two coalgebra structures and a notion of plethysm (and much more).
From an algebraic viewpoint, these structures can be combined to a
"plethory" in the sense of Borger and Wieland; roughly speaking, this
is a ring whose elements can be evaluated at "alphabets" (actual and
reals), as symmetric functions can. This allows some alternative
definitions of this ring; in particular, we can view the symmetric
functions as the representing object of the functor of big Witt
vectors. I introduce an analogue to big Witt vectors when the integers
are replaced by monic univariate polynomials over a finite field; this
analogue is a representable functor, and its representing object can
be viewed as a function-field analogue of the ring of symmetric
functions. Unfortunately, combinatorial structures (e.g., an e-, h- or
s-basis) in this analogue have so far proven elusive, but some
properties can be proven and some computations made.
(This is based on this paper, but the talk should be much more understandable and take fewer
detours.)

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