Dual Numbers, Weighted Quivers, and Extended Somos and Gale-Robinson Sequences

Abstract

We investigate a general method that allows one to construct new integer sequences extending existing ones. We apply this method to the classic Somos-4 and Somos-5, and the Gale-Robinson sequences, as well as to more general class of sequences introduced by Fordy and Marsh, and produce a great number of new sequences. The method is based on the notion of "weighted quiver", a quiver with a Z-valued function on the set of vertices that obeys very special rules of mutation.

Shadow sequences of integers, from Fibonacci to Markov and back

“…Moreover, for some "mysterious" reasons, the new sequence pα n q nPN turns out to be integer! This idea to construct dual integer sequences was suggested in [7,8] and applied to the Gale-Robinson and Somos sequences. Today I will go further and apply it to several other interesting sequences, making the method more universal.…”

“…This integrality persists thanks to a "miracle", called the Laurent phenomenon [2], the same miracle that guarantees integrality of the Somos and Gale-Robinson sequences. The complete proof is too technical to be reproduced here, a general statement can be found in [8].…”

An Expansion Formula for Decorated Super-Teichmüller Spaces

“…Steps towards defining super cluster algebras appeared in work of Ovsienko [15] and separately in the work of Li, Mixco, Ransingh, and Srivastava [12]. These initial steps were followed up by related work such as [16,17,21].…”

Cluster algebras with Grassmann variables

“…(2) a negative value of w(x i ) corresponds to the number of 2-path ξ 2 → x i → ξ 1 . This function was called the weight function in [17] where it was applied to integer sequences.…”

“…where ε := ξ 1 ξ 2 denotes the product of the odd variables (cf. [17]). Indeed, one obviously has (1 + ε) w k = (1 + w k ε) since ε 2 = 0.…”