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Using the dcQMD transport model, the isoscalar and isovector in-mediumpotentials of the $\Delta$(1232) baryon are studied and information regardingtheir effective strength is obtained from a comparison to experimental pionproduction data in heavy-ion collisions below 800 MeV/nucleon impact energy.The best description is achieved for an isoscalar potential moderately moreattractive than the nucleon optical potential and a rather small isoscalarrelative effective mass m$^*_\Delta \approx$ 0.45. For the isovector componentonly a constraint between the potential's strength at saturation and theisovector effective mass difference can be extracted, which depends onquantities such as the slope of the symmetry energy and the neutron-protoneffective mass difference. These results are incompatible with the usualassumption, in transport models, that the $\Delta$(1232) and nucleon potentialsare equal. The density dependence of symmetry energy can be studied using thehigh transverse momentum tail of pion multiplicity ratio spectra. Results arehowever correlated with the value of neutron-proton effective mass difference.This region of spectra is shown to be affected by uncertain model ingredientssuch as the pion potential or in-medium correction to inelastic scatteringcross-sections at levels smaller than 10$\%$. Extraction of precise constraintsfor the density dependence of symmetry energy above saturation will requireexperimental data for pion production in heavy-ion collisions below 800MeV/nucleon impact energy and experimental values for the high transversemomentum tail of pion multiplicity ratio spectra accurate to better than 5$\%$.

In-medium $\Delta(1232)$ potential, pion production in heavy-ion collisions and the symmetry energy