Extensions between Cohen-Macaulay modules of Grassmannian cluster categories

TL;DR

This paper investigates extensions between Cohen-Macaulay modules in Grassmannian cluster categories, providing explicit formulas, algorithms, and conditions for computing and understanding these extensions.

Contribution

It introduces explicit combinatorial methods and formulas for computing Ext groups between rank 1 Cohen-Macaulay modules, including periodicity and vanishing conditions.

Findings

Rank 1 modules are periodic with computable periods

Explicit formulas for Ext^i between rank 1 modules

Vanishing conditions for Ext^i for i > 0

Abstract

In this paper we study extensions between Cohen-Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine ${\rm Ext}^i(L_I, L_J)$ for arbitrary rank 1 modules $L_I$ and $L_J$. An explicit combinatorial algorithm is given for computation of ${\rm Ext}^i(L_I, L_J)$ when $i$ is odd, and for $i$ even, we show that ${\rm Ext}^i(L_I, L_J)$ is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of ${\rm Ext}^i(L_I, L_J)$ for any $i>0$.