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VI-modules and the Lannes-Schwartz conjecture: the work of Putman, Sam, and Snowden

Banff International Research Station for Mathematical Innovation and Discovery

The object of "generic representation theory" is to describe families of \(\text{GL}_n(\mathbb{F}_q)\)-representations in characteristic \(p\), similar to the way that an FI-module captures a whole family of \(S_n\)-representations. However a basic finiteness property of generic representations, the Lannes-Schwartz conjecture of 1994, was never resolved. Putman, Sam, and Snowden proved this conjecture last year by understanding VI-modules and VIC-modules, which are \(\text{GL}_n(\mathbb{F}_q)\) analogues of FI-modules. At the same time, the methods they introduced provide the strongest tools we have for proving finiteness properties for twisted commutative algebras like FI, FI\(_d\), VI, etc. I'll give an accessible overview of generic representations and describe the innovations of Putman, Sam, and Snowden, which show us that the Lannes-Schwartz conjecture was not really about characteristic-\(p\) representations at all. I'll also explain how their methods provide "user-friendly" tools usable by non-experts.

Mathematics; Commutative rings and algebras; Category theory; homological algebra; Commutative algebra

video/mp4

Author affiliation: Stanford University

2016-10-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59376

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/