On the decategorification of some higher actions in Heegaard Floer homology

We decategorify the higher actions on bordered Heegaard Floer strands algebras from recent work of Rouquier and the author, and identify the decategorifications with certain actions on exterior powers of homology groups of surfaces. We also suggest an interpretation for these actions in the language of open -closed TQFT, and we prove a corresponding gluing formula.