A materialisation of Frobenius determinantal formula in prime characteristic

The determinantal formula gives an expression of Specht modules by permutational modules in the Grothendieck ring of modules over the symmetric group Sn. It was shown by Zelevinskii in "Resolutions, dual pairs and character formulas" (J. Funktsional. Anal. i Prilozhen. Vol. 21 p. 74-75) that for each Specht module over a field of zero characteristic there is a coresolution by permutational modules that realises this expression. We construct permutational coresolutions for Specht modules over a field of positive characteristic.