Representation stability for sequences of 0-Hecke modules

Abstract

We define a new category analogous to $\mathbf{FI}$ 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.

51 pages

References in corpus (6)

• Introduction to twisted commutative algebras
• Syzygies of bounded rank symmetric tensors are generated in bounded degree
• Stability in the homology of unipotent groups
• Ordered set partitions and the 0-Hecke algebra
• Syzygies of secant ideals of Plücker-embedded Grassmannians are generated in bounded degree
• The spin-Brauer diagram algebra

Cited by in corpus (1)

• A counterexample to a Gröbner approach for noetherianity of the twisted commutative algebra $\mathrm{Sym}({\mathrm{Sym}}^2({\mathbf{C}}^{\infty }))$