[Paper Review] The numbers of support-tilting modules for a Dynkin algebra

Mustafa A. A. Obaid, S. Khalid Nauman+2 more|arXiv (Cornell University)|Mar 24, 2014

Algebraic structures and combinatorial models14 references3 citations

TL;DR

This paper computes the number of support-tilting modules over any Dynkin algebra of type An, Bn, Cn, Dn, and the exceptional types, establishing a unified formula using combinatorial structures such as the Catalan triangle, Pascal’s triangle, and Lucas triangle. It proves that these numbers correspond bijectively to generalized non-crossing partitions, providing a categorification of classical combinatorial results and deriving recurrence relations and closed-form expressions for all cases, including explicit formulas for an(∆n) and as(∆n).

ABSTRACT

The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper exhibits the number of support-tilting modules for any Dynkin algebra. Since the support-tilting modules for a Dynkin algebra correspond bijectively to the non-crossing partitions of the same type, the calculations presented here may also be considered as a categorification of results concerning non-crossing partitions.

Motivation & Objective

• To determine the number of support-tilting modules for all Dynkin algebras, including types An, Bn, Cn, Dn, and exceptional types.
• To establish a bijection between support-tilting modules and generalized non-crossing partitions of the corresponding Dynkin type.
• To unify the enumeration of support-tilting modules across Dynkin types using combinatorial sequences such as the Catalan triangle, increasing Pascal triangle, and Lucas triangle.
• To derive recurrence relations and closed-form formulas for the number of support-tilting modules of each support-rank s and total count a(∆n).
• To provide a complete categorification of known results on generalized non-crossing partitions via tilting theory in representation theory.

Proposed method

• Uses the concept of support-tilting modules, where a module T is support-tilting if it is tilting over its support algebra Λ(T).
• Applies the fact that the number of tilting modules over a Dynkin algebra depends only on its Dynkin type, not on the specific algebra.
• Employs the Bailey notation [t/s] = (s+t)/t * (t choose s) to express numbers in the D-type case.
• Derives recurrence relations via structural decomposition of the quiver, particularly by removing vertex 1 and analyzing the resulting algebras Λ′ and An−c.
• Uses induction and hook-length type formulas to prove identities such as ∑_{i=0}^s ai(∆n) = as(∆n+1) for An, Bn, and Dn.
• Validates results through known values for E6, E7, E8, F4, G2 and confirms consistency with earlier literature, correcting prior errors.

Experimental results

Research questions

• RQ1 What is the number of support-tilting modules of support-rank s for a Dynkin algebra of type An, Bn, Cn, or Dn?
• RQ2 How do the numbers of support-tilting modules relate to generalized non-crossing partitions of the corresponding Dynkin type?
• RQ3 Can a unified combinatorial formula be derived for the total number of support-tilting modules across all Dynkin types?
• RQ4 What recurrence relations govern the growth of support-tilting module counts across different ranks and types?
• RQ5 How do the formulas for D-type algebras differ from those for An and Bn/Cn, and what combinatorial structure underlies them?

Key findings

• For type An, the number of support-tilting modules of support-rank s is given by the Catalan triangle entry ]n+s/s[ = (n+s−2s+1)/(n+s−s+1) * (n+s choose s), with total count a(An) = 1/(n+2) * (2n+2 choose n+1).
• For types Bn and Cn, the number of support-tilting modules of support-rank s is (n+s-1 choose s), and the total count is a(Bn) = a(Cn) = (2n choose n).
• For type Dn, the number of support-tilting modules of support-rank s is [n+s−2/s] = (n+s−2)/s * (s choose s) for s ≥ 2, with total count a(Dn) = (2n−1 choose n−1).
• The number an−1(Dn) is given by the formula an−1(Dn) = [2n−3/n−1] = (3n−4)/(2n−3) * (2n−3 choose n−1), confirming a closed-form expression.
• The paper establishes the identity ∑_{i=0}^s ai(∆n) = as(∆n+1) for An, Bn, and Dn (n≥2), which leads to a recursive formula for the total count a(∆n).
• The total number of support-tilting modules for E6, E7, E8, F4, and G2 are computed as 833, 4160, 25080, 105, and 8 respectively, correcting earlier inaccuracies in the literature.