arXiv:2011.03606v1 [math.RT] 6 Nov 2020 A COMBINATORIAL TRANSLATION PRINCIPLE FOR THE SYMPLECTIC GROUP HENRY LI AND RUDOLF TANGE Summary. Let k be an algebraically closed field of characteristic p> 2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups. Introduction The present paper concerns the symplectic group and the Brauer algebra. In the companion paper [16] the analogous results for the general linear group and the walled Brauer algebra are obtained. Let Sp n , n =2m, be the symplectic group over an algebraically closed field k of characteristic p> 2, and let V be the natural module. Since Williamson [18] disproved Lusztig’s conjecture for SL n and p bigger than any linear bound in n, it has become more interesting to determine decomposition numbers of reductive groups for special sets of weights. In the present paper we do this for Sp n and dominant weights for which p is bigger than the greatest hook length. The Brauer algebra is a cellular algebra and an interesting problem is to determine its decomposition numbers. In characteristic 0 this was first done in [13] and in [5] an alternative proof was given which included the analogous result for the walled Brauer algebra. In [15] the decomposition numbers of the Brauer algebra were determined in characteristic p>r. All these results are in terms of certain cap (or cap-curl) diagrams. In characteristic 0 there is a well-known relation between certain represen- tations of Sp n and the representations of the Brauer algebra B r (−n), given by the double centraliser theorem for their actions on V ⊗r . In characteristic p such a connection doesn’t follow from the double centraliser theorem and requires more work. This was done in [8] by means of the symplectic Schur functor. In the present paper we determine the Weyl filtration multiplicities in the indecomposable tilting modules T (λ) and the decomposition numbers for the induced modules ∇(λ) of Sp n when λ has greatest hook length less than p. Using the symplectic Schur functor we then obtain from the first multiplicities the decomposition numbers of the Brauer algebra under the assumption that p is bigger than the greatest hook length in the largest partition involved. Since we use the transposed labels, our description of the decomposition numbers 1