Extensions between Cohen-Macaulay modules of Grassmannian cluster categories

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 ${\mathrm{Ext}}^i(L_I,L_J)$ for arbitrary rank 1 modules $L_I$ and $L_J$. An explicit combinatorial algorithm is given for computation of ${\mathrm{Ext}}^i(L_I,L_J)$ when $i$ is odd, and for $i$ even, we show that ${\mathrm{Ext}}^i(L_I,L_J)$ 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 ${\mathrm{Ext}}^i(L_I,L_J)$ for any $i>0$.

References in corpus (2)

• Total positivity, Grassmannians, and networks
• Dimer models and cluster categories of Grassmannians