The representation theory of the increasing monoid

Abstract

We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.

Profunctors between posets and Alexander duality

“…Parts b and c are then immediate from a. The domain in (10) above is the same as UHom(P, Q) op and so we have the map (11) Γ :…”

“…Remark 2.13. The small maps Hom S (N, N) are in bijection with the tame (small) increasing monoid of [10] by sending a small map f to f + id − 1.…”

“…In [15] Nagel and Römer show that ideals in the infinite polynomial ring invariant for the increasing monoid, have an essentially finite Grobner basis, thereby generalizing previous results for the symmetric group. In [10] Güntürkün and Snowden studies in depth the representation theory of the increasing monoid. Note that the injective order preserving maps g : N → N are in bijection with order preserving maps f : N → N by g = f + id − 1.…”

Shift modules, strongly stable ideals, and their dualities

“…Incidence algebras 39 Appendix B. Equivalence with modules over incidence algebras 40 References Proof. Given a homogeneous (for the N ∞ 0 -grading) syzygy in F p (21) α i,u x a i,u (i | u).…”

“…The increasing monoid Hom inj (N, N) of injective order-preserving maps f : N → N is in one-one correspondence with Hom(N, N) by mapping f to f + id N − 1. The increasing monoid has been used to study k[x N ] in [34], [26], see also [21]. The use differs however sharply from ours, as Hom inj (N, N) is used there to act on k[x N ], while we use Hom(N, N) in a distinct and intrinsic way by diagram (1).…”