[Paper Review] On the representation theory of finite J-trivial monoids

Tom Denton, Florent Hivert+2 more|arXiv (Cornell University)|Oct 17, 2010

Advanced Combinatorial Mathematics29 references21 citations

TL;DR

This paper establishes a combinatorial framework for the representation theory of finite $χ$-trivial monoids by showing that key algebraic invariants—such as the Cartan matrix, quiver, and projective modules—can be computed purely through counting elements in the monoid itself. The central contribution is a constructive, field-independent method to decompose the identity into orthogonal idempotents using Möbius inversion and recursive lifting, enabling $O(n^2)$ computation of representation-theoretic data.

ABSTRACT

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both from a theoretical and computational point of view, and with an implementation in Sage. Motivated by ongoing work on related monoids associated to Coxeter systems, and building on well-known results in the semi-group community (such as the description of the simple modules or the radical), we describe how most of the data associated to the representation theory (Cartan matrix, quiver) of the algebra of any J-trivial monoid M can be expressed combinatorially by counting appropriate elements in M itself. As a consequence, this data does not depend on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M| and m is the number of generators. Along the way, we construct a triangular decomposition of the identity into orthogonal idempotents, using the usual Möbius inversion formula in the semi-simple quotient (a lattice), followed by an algorithmic lifting step. Applying our results to the 0-Hecke algebra (in all finite types), we recover previously known results and additionally provide an explicit labeling of the edges of the quiver. We further explore special classes of J-trivial monoids, and in particular monoids of order preserving regressive functions on a poset, generalizing known results on the monoids of nondecreasing parking functions.

Motivation & Objective

• To develop a combinatorial, field-independent method for computing representation-theoretic invariants of finite $χ$-trivial monoids.
• To generalize the representation theory of the $0$-Hecke algebra by treating it as a special case of $χ$-trivial monoids.
• To provide an algorithmic framework for computing the Cartan matrix, quiver, and projective modules using only monoid elements and their structure.
• To construct a triangular decomposition of the identity into orthogonal idempotents via Möbius inversion and lifting in the semi-simple quotient.
• To extend known results on nondecreasing parking functions and order-preserving regressive functions to broader classes of $χ$-trivial monoids.

Proposed method

• Use the $χ$-trivial property to define a partial order on the monoid such that $xy \leq x$ and $xy \leq y$ for all $x,y$.
• Leverage the semi-simple quotient (a lattice) to apply Möbius inversion and construct orthogonal idempotents in the quotient algebra.
• Implement an algorithmic lifting step to lift these idempotents from the semi-simple quotient back to the full monoid algebra.
• Express the Cartan matrix and quiver data combinatorially by counting elements satisfying specific order and idempotent conditions in the monoid.
• Construct a recursive branching rule for idempotents using linear extensions of meet semi-lattices and prefix trees of valid diagrams.
• Define demipotents $C_D = L_D R_D$ for each valid $+/-\,$ diagram $D$, with $L_D$ and $R_D$ built recursively from generators and projections.

Experimental results

Research questions

• RQ1 Can the representation theory of $χ$-trivial monoids be fully described using only combinatorial data from the monoid itself?
• RQ2 Is the Cartan matrix and quiver of a $χ$-trivial monoid computable without linear algebra, relying only on element counting?
• RQ3 Can the identity of the monoid algebra be decomposed into orthogonal idempotents via a constructive, recursive method based on the monoid's order structure?
• RQ4 Does the recursive branching construction of idempotents in the monoid of order-preserving regressive functions yield a complete, primitive orthogonal decomposition?
• RQ5 To what extent do the representation-theoretic invariants of $χ$-trivial monoids depend on the ground field?

Key findings

• The Cartan matrix of any $χ$-trivial monoid is upper unitriangular and can be computed by counting elements in the monoid, independent of the ground field.
• The quiver of the algebra is fully determined by counting pairs of elements satisfying specific order and idempotent conditions, with explicit edge labeling provided for the $0$-Hecke algebra.
• A decomposition of the identity into orthogonal idempotents is constructed via Möbius inversion in the semi-simple quotient followed by algorithmic lifting, achieving $O(n^2)$ complexity where $n = |M|$.
• For the monoid of order-preserving regressive functions on a meet semi-lattice $L$, the Cartan matrix is upper unitriangular, and the idempotents $C_D$ form a complete, orthogonal set.
• The demipotents $C_D$ defined via the branching rule are directly idempotent in all tested cases (up to 8 elements), suggesting they form a set of primitive orthogonal idempotents.
• The method recovers known results on the $0$-Hecke algebra and nondecreasing parking functions, while extending them to a broader class of monoids with a unified combinatorial framework.