Linearly independent products of rectangularly complementary Schur functions (Q1856332)

Schur functions are of great importance due to their fundamental role in the representation theory of the symmetric group. Results concerning them often have interesting combinatorial descriptions from the Littlewood-Richardson rule of the 1930's to the saturation theorem. It is these functions that are the focus here. If we have a rectangular chessboard consisting of an even number of squares, we say a puzzle piece of shape \(p\) (where \(p\) is a partition) is self-complementary if two pieces of shape \(p\) cover the board completely with no overlap. In this beautifully written paper it is shown that squares of Schur functions \(S_p\) are linearly independent of all other products \(S_qS_r\) where \(q\) and \(r\) are two puzzle pieces which cover the board completely with no overlap. A similar result holds for boards with an odd number of squares. It is conjectured that all such products are independent, and this plus a stronger conjecture are verified for boards upto size \(8\times 8\) and \(7\times 9.\) Proofs are also included for boards of size \(1\times b\) and \(2\times b\). Finally natural generalisations which fail are discussed.