Generalized diagram categories and monoids, and their representations

Classical diagram categories and monoids, including the Temperley–Lieb, Brauer, and partition cases, arise as special instances of the category of two dimensional cobordisms and admit additional twists that produce a large new family of diagram categories and monoids. In this paper we introduce this family and develop a unified approach to their representation theory.

💡 Research Summary

The paper presents a unified framework for a broad family of diagram categories and monoids that generalize the classical Temperley‑Lieb, Brauer, and partition cases. The starting point is the two‑dimensional cobordism category Cob ∞, whose objects are circles and whose morphisms are surfaces built from pairs of pants, caps, cups, and swaps. By fixing a rational function p/q and interpreting its Taylor coefficients as evaluation data for closed surfaces, the authors obtain quotient categories Cob p/q. These quotients encode the “handle relation” that determines how handles (dots) are evaluated, and the resulting categories contain, as special cases, all the familiar diagram categories.

From Cob p/q the authors extract endomorphism monoids for each natural number n, yielding a spectrum of diagram monoids: the partition monoid P_ac(n), the (decorated) root Brauer monoid RoB_{a0,a1,r}(n), the Brauer monoid B_{a1,r}(n), the root monoid Ro_{a0,r}(n), the symmetric group S_r(n), and their planar versions (pP, Mo, TL, pRo, pS). Parameters a = (a_i) record the scalar value assigned to a closed component of genus i, while r controls the allowed handle types. By varying these parameters one obtains a large lattice of new monoids that include the classical ones as the special case a_i = δ (a constant) and r = 1.

A central novelty is the introduction of “twistings”. Given a monoid S, a map Φ : S×S→ℕ satisfying a cocycle‑type identity, an additive commutative monoid M, and a distinguished element q∈M, one defines a twisted product M×_q^Φ S whose multiplication records the statistic Φ. For diagram monoids the natural statistic is the number of floating components created when two diagrams are concatenated. The authors study “tight twistings” (Condition 3E.6) and prove a reduction theorem (Theorem 3F.1): for planar diagram monoids (and more generally finite monoids with trivial H‑classes) a tight twisted version has exactly the same simple‑module indexing and the same simple dimensions as either the untwisted monoid or the corresponding “0‑twisted” monoid. Thus, in this wide class the twist does not create new simple modules; it only determines whether floating components behave as in the classical case or are annihilated.

The representation‑theoretic backbone is sandwich cellularity, a refinement of Graham–Lehrer cellularity introduced by Brown. In Section 2 the authors show that all generalized diagram algebras admit a compatible cellular basis that respects the sandwich decomposition (shirt‑belt‑pants picture). This provides a uniform tool for indexing simples, relating Green’s relations to representation theory, and comparing different parameter choices.

To obtain concrete dimensions, the paper systematically develops Schur–Weyl dualities for the generalized diagram monoids. By constructing commuting actions of a diagram algebra and a classical group (or Lie algebra) on a tensor power V^{⊗n} of a fixed G‑module V, the authors identify the diagram algebra as the centralizer of the group action. This yields explicit decomposition multiplicities and, via classical character theory, closed formulas for the dimensions of simple modules. Sections 5 and 6 treat the semisimple regime: a typical‑highest‑weight phenomenon is proved, showing that for large n most of the weight space mass concentrates in a narrow window of highest weights. Consequently, the dimensions of simple modules follow a binomial‑type growth (e.g., for Temperley‑Lieb the dimensions are given by Catalan‑type numbers). Asymptotic formulas and sums of dimensions are derived.

In the non‑semisimple regime (Section 7) the authors show that while extensions, truncations, and exceptional low‑rank phenomena appear, the dominant growth of simple dimensions remains close to the semisimple prediction. They discuss how the cellular structure controls the Loewy layers and how twistings affect the radical.

Overall, the paper achieves three intertwined goals: (1) it defines a large, parameter‑controlled family of diagram categories and monoids via cobordism quotients; (2) it proves that a broad class of twistings is representation‑theoretically mild, preserving simple‑module data; (3) it unifies the computation of dimensions through Schur–Weyl dualities and typical‑weight analysis. This work not only subsumes the classical Temperley‑Lieb, Brauer, and partition algebras but also provides a robust framework for studying new diagrammatic algebras arising from topological and combinatorial modifications.