Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 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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