Crystals and sign-reversing involutions for set-valued skew tableaux and decorated states of the five-vertex model

Abstract: We construct bijections among semistandard set-valued tableaux of skew shape, marked Gelfand-Tsetlin patterns, and decorated states of the five-vertex model with boundary data determined by the skew shape. The bijections preserve monomial weight, identify tableau excess with the number of marks and nontrivial bumps, and intertwine the ordinary type A crystal operators with local transformations of decorated states. We prove that each fiber determined by a fixed maximum tableau is a graded Boolean lattice whose rank function is the excess. Its weighted generating function has a product formula. The sign-reversing involution changes one Boolean coordinate on every tableau that it does not fix, and we determine its commutation with all raising operators and with lowering operators except at the two colors involving the changed entry. We construct the dual involution associated with minimum entries and prove corresponding Boolean structures for fibers determined by a fixed minimum tableau and for fibers with prescribed minimum and maximum tableaux.