[Paper Review] Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers

Lutz Hille|arXiv (Cornell University)|May 22, 2015

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper establishes a geometric realization of Catalan numbers via polytopes constructed from root systems of type $ȑA_n$, showing that the volumes of specific polytopes—defined using tilting modules and support tilting modules—exactly count the number of such modules. It proves that the volume of the polytope $P(Q)$ equals the number of 2-support tilting modules, which is ${2n \choose n}$, and that the volume of $P^+(Q)$ equals the Catalan number $C_n$, using integral volume normalization and connections to Dyck paths and rooted trees.

ABSTRACT

It is well known that the number of tilting modules over a path algebra of type A_n coincides with the Catalan number C(n). Moreover, the number of support tilting modules of type A_n is the Catalan number C(n+1). We show that the convex hull of all roots of a root system of type A_n is a polytope with integral volume (n + 1)C(n+1). Moreover, we associate to the set of tilting modules and to the set of support tilting modules certain polytopes and show that their volumes coincide with the number of those modules, respectively. Finally, we show that these polytopes can be defined just using the root system and relate their volumes, so that we can derive the above results in a new way.

Motivation & Objective

• To establish a geometric interpretation of the number of tilting and support tilting modules over path algebras of type $ȑA_n$ using polytopes.
• To show that the volumes of certain polytopes constructed from root systems and tilting modules coincide with the counts of these modules.
• To demonstrate that the volume of the polytope $P(Q)$ equals the number of 2-support tilting modules, which is ${2n \choose n}$, using integral volume normalization.
• To unify combinatorial interpretations of Catalan numbers via polytopes, rooted trees, and Dyck paths in the context of representation theory.

Proposed method

• Construct three series of polytopes—$C(Q)$, $C^{+}(Q)$, $C^{\mathrm{clus}}(Q)$—as convex hulls of roots in the root system of type $ȑA_n$.
• Define a second series of polytopes—$P(Q)$, $P^{+}(Q)$, $P^{\mathrm{clus}}(Q)$—as unions of simplices associated with tilting modules and their generalizations.
• Use integral volume normalization, where the volume of a simplex with an integral basis is 1, so that the volume of a lattice polytope is an integer.
• Prove that the volumes of $P^{+}(Q)$, $P^{\mathrm{clus}}(Q)$, and $P(Q)$ equal the number of tilting, support tilting, and 2-support tilting modules, respectively.
• Establish a bijection between tilting sequences and elements of the symmetric group $S_n$, showing that the volume of $P^{+}(Q)$ is $n!$.
• Use Dyck paths and generalized paths to interpret the volume of $P(Q)$ as ${2n \choose n}$, linking to the combinatorics of bracketings and 2-support tilting modules.

Experimental results

Research questions

• RQ1 How can the number of tilting modules over a path algebra of type $ȑA_n$ be geometrically realized via polytopes?
• RQ2 What is the relationship between the volumes of polytopes defined from root systems and the counts of tilting modules?
• RQ3 Can the volume of a polytope constructed from tilting modules be shown to equal the number of such modules using integral volume normalization?
• RQ4 How do Dyck paths and generalized paths relate to the combinatorics of 2-support tilting modules and the volume of $P(Q)$?
• RQ5 Is there a uniform geometric construction that explains the Catalan number counts for tilting modules using only the root system of type $ȑA_n$?

Key findings

• The volume of the polytope $P^{+}(Q)$, constructed from tilting modules, is exactly $n!$, which matches the number of tilting sequences.
• The volume of $P^{\mathrm{clus}}(Q)$ equals the Catalan number $C_{n+1}$, which counts the number of support tilting modules over $ȑA_n$.
• The volume of $P(Q)$, corresponding to 2-support tilting modules, is ${2n \choose n}$, which is the number of such modules.
• The polytope $C(ȑA_n)$, the convex hull of all roots in the root system of type $ȑA_n$, has integral volume $(n+1)C_{n+1} = {2n \choose n}$.
• The volumes of the polytopes $P(Q)$ and $C(ȑA_n)$ coincide, establishing a geometric correspondence between root systems and tilting module counts.
• The construction shows that the number of 2-support tilting modules is counted by the number of paths from $(0,0)$ to $(2n,0)$ with steps $(1,1)$ and $(1,-1)$ that stay non-negative or non-positive in segments, leading to the binomial coefficient ${2n \choose n}$.