For a simple Lie algebra g and an irreducible faithful representation π of g, we introduce the Schur polynomials of (g, π)-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of g-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of (g, π)-type with the coefficients being the Plücker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For g of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy.

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In 1978, M. Adler and J. Moser proved that there exists a unique change of variables that transforms the Adler-Moser polynomials into the polynomial tau functions of the KdV hierarchy. In this paper we exhibit this change of variables.

In this paper we compute explicitly the double ramification hierarchy and its quantization for the D 4 Dubrovin-Saito cohomological field theory obtained applying the Givental-Teleman reconstruction theorem to the D 4 Coxeter group Frobenius manifold, or equivalently the D 4 Fan-Jarvis-Ruan-Witten cohomological field theory (with respect to the non-maximal diagonal symmetry group J = Z/3Z). We then prove its equivalence to the corresponding Dubrovin-Zhang hierarchy, which was known to coincide with the D 4 Drinfeld-Sokolov hierarchy. Our techniques provide hence an explicit quantization of the D 4 Drinfeld-Sokolov hierarchy. Moreover, since the DR hierarchy is well defined for partial CohFTs too, our approach immediately computes the DR hierarchies associated to the invariant sectors of the D 4 CohFT with respect to folding of the Dynkin diagram, the B 3 and G 2 Drinfeld-Sokolov hierarchies. Contents 15 2.5. B 3 and G 2 double ramification hierarchies 17 3. Quantum double ramification hierarchy for the D 4 Dubrovin-Saito CohFT 18 3.1. Quantum double ramification hierarchy 18 3.2. Quantum systems of DR type and the quantum D 4 hierarchy 19 References 20 1. Drinfeld-Sokolov D 4 hierarchy A. du Crest de Villeneuve: LAREMA,