# 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.