Oscillating Tableaux , S p × S q - modules , and Robinson - Schensted - Knuth correspondence

The Robinson-Schensted-Knuth correspondence (RSK, see [8] and Corollary 2.5 below) is a bijection between pairs of semi-standard Young tableaux of the same shape and matrices with nonnegative integer entries with prescribed column and row sums. This correspondence plays an important role in the representation theory of the symmetric group and general linear groups, and in the theory of symmetric functions. It is possible (see [2, 3, 4, 5, 10]) to construct an analogue of the RSK for oscillating tableaux, i.e., sequences of Young diagrams α = (α(0), . . . , α(k)) such that each α(i) and α(i+1) differ by a horizontal strip. We present a new approach to the RSK correspondence for oscillating tableaux. First, we show that the number of oscillating tableaux of a given weight and shape is equal to the multilplicity of the corresponding irreducible representation in a certain naturally defined Sp × Sq-module. This allows us to recover the enumerative results from [4, 10, 11, 12] (see Section 4). In Section 5, we extend this construction to oscillating supertableaux. In