Abstract
We define a new category analogous to ${\bf FI}$ for the $0$-Hecke algebra $H_n(0)$ called the $0$-Hecke category, $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We establish a new type of representation stability in this setting and prove it is implied by being a finitely generated $\mathcal{H}$-module. We then provide examples of $\mathcal{H}$-modules and discuss further desirable properties these modules possess.
Highlights
The category FI of finite sets and injections, first defined in [3], and its variants have been of great interest recently
Being a finitely generated FI-module implies many desirable properties that are often very difficult to prove on their own about sequences of symmetric group representations, such as representation stability and polynomial growth. The study of this combinatorial category and its modules has been fruitful, providing tools to prove a variety of stability results about spaces such as Hi(Confn(M ); Q) the cohomology of configuration space of n distinct ordered points on a connected, oriented manifold M and many others [3]
This paper provides a more systematic approach to studying natural sequences of representations of the 0-Hecke algebra
Summary
The category FI of finite sets and injections, first defined in [3], and its variants have been of great interest recently. Noetherianity up to symmetry is important in [26, 35, 36, 37], where the authors explore various manifestations of this idea to prove finite generation results for various representations of categories and twisted commutative algebras. These ideas are present in [3, 6, 20, 34, 40, 42] and many other recent papers.
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