Polytope symmetries of Feynman integrals

Feynman integrals appropriately generalized are $\mathsf A$-hypergeometric functions. Among the properties of $\mathsf A$-hypergeometric functions are symmetries associated with the Newton polytope. In ordinary hypergeometric functions these symmetries lead to linear transformations. Combining tools of $\mathsf A$-hypergeometric systems and the computation of symmetries of polytopes, we consider the associated symmetries of Feynman integrals in the Lee-Pomeransky representation. We compute the symmetries of $\mathtt n$-gon integrals up to $\mathtt n=8$, massive banana integrals up to 5-loop, and on-shell ladders up to 3-loop. We apply these symmetries to study finite on-shell ladder integrals up to 3-loop.