# Comment on Volume 13(1), article R86 (October 12, 2006)

Double Crystals of Binary and Integral Matrices
Marc A. A. van Leeuwen

Comment by the author, October 13, 2006:

On the day of publication of this paper we found out, after Christian Krattenthaler had kindly brought their work to our attention, that the publication [DaKo] by Danilov and Koshevoi describes operations and many facts that are essentially equivalent to the ones presented in our paper. We regret not having included any reference to their work in our paper.

Here is a brief indication of the correspondence of our notions and results with theirs. What we call "integral matrices" are called "arrays" in [DaKo], and they are displayed differently; they use the conventional display of Cartesian coordinates, so formally our matrices are rotated a quarter turn clockwise with respect to their arrays, but in practice (notably for the correspondence with semistandard tableaux) it is better to view these notions as being upside down with respect to each other (so that rows remain rows), with an additional inversion of the order of the two indices (because in Cartesian coordinates the row index comes second). Our "binary matrices" correspond to the "Boolean arrays" of appendix C of [DaKo], and since those are made to correspond to row-strict tableaux, the two notions are best thought of as related by a quarter turn (fortunately, because the rules for moves are rotation symmetric in this case).

Our crystal operations on integral matrices correspond to the operations of [DaKo] Part I section 3, and those on our binary matrices to those of their mentioned appendix. The following points of our paper have their direct counterparts in [DaKo]: the encoding of tableaux by integral or binary matrices, the commutations theorems (part I, section 4), the symmetric group action of our theorem 2.4 (appendix A), the decomposition theorem 3.1.3 (part I, section 6), the relation with jeu de taquin, the relation with Greene's poset invariant (appendix B; although we use a complementary poset with respect to [DaKo], they transpose the invariant, so ours comes out the same as theirs).

## Reference

[DaKo]
V. I. Danilov and G. A. Koshevoi, Arrays and combinatorics of Young tableaux, Uspehi Math. Nauk 60:2 (2005) 79-142 (in Russian); translation in Russian Math. Surveys 60:2 (2005), 269-334.