Abstract We give a fermionic formula for R-matrices of exterior powers of the vector representations of $$U_q(\widehat{ \mathfrak {gl}}_N)$$ U q ( gl ^ N ) and relate it to the dynamical Weyl group of Tarasov–Varchenko and Etingof–Varchenko, via a Howe ( $$\mathfrak {gl}_N,\mathfrak {gl}_M)$$ gl N , gl M ) -duality. In the limit $$N\rightarrow \infty $$ N → ∞ we obtain R-matrices for Fock spaces. As a consequence of our result we obtain a dynamical action of the Weyl group on integrable $$U_q\mathfrak {gl}_M$$ U q gl M -modules, extending the known action on zero weight spaces. In an Appendix by Anfisa Gurenkova it is shown that the latter property also holds if we replace $$\mathfrak {gl}_M$$ gl M by a general symmetrizable Kac–Moody algebra.

Abstract We introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain “ubiquity” and “rigidity” properties that in combination render them very useful in the study of general wound unipotent groups. As an illustration of their utility, we present two applications: We prove that nonsplit smooth unipotent groups over (infinite) fields finitely generated over $$\textbf{F}_p$$ F p have infinite first cohomology; and we show that every commutative p-torsion wound unipotent group over a field of degree of imperfection 1 is the maximal unipotent quotient of a commutative pseudo-reductive group, thus partially answering a question of Totaro.