Hook Length and Jeu De Taquin History

Historical Remarks for
Hook Length and Jeu de Taquin Properties

Here are some rudimentary historical comments.
Please send additions and corrections to me at rap)at(email.unc.edu.

Hook Length Posets:

(1) In his 1970 Harvard thesis, Stanley introduced a generating function for P-partitions on a given poset. In this thesis (an abridged version appeared as Memoirs of the AMS 119 (1972)) he considered the question of whether this generating function factors nicely. He did not explicitly introduce the terminology "hook length poset," but did refer to "hook lengths" in this context. He proved that the PPGF's for shapes factor in this manner and conjectured that the PPGF's for shifted shapes also did so. He was also aware that the 'double-tailed diamond' posets dk(1) (defined on the d-complete definition page) had this property.

(2) A.P. Hillman and R.M. Grassl gave a nice algorithmic proof of the hook length property for shapes in JCT A 21 (1976).

(3) A student of Stanley's, Emden Gansner, proved the shifted shape conjecture in his 1978 M.I.T. thesis.

(4) Bruce Sagan introduced the terminology of 'hook length poset' and gave a Hillman-Grassl style proof of the hook length property for trees and for shifted shapes in European J. Combins 3 (1982).

(5) Around the mid-1990's, Fomin and Stanley found a hook product enumeration formula for the number e(P) of order extensions of any poset P which can be formed by prepending one box to each of the first two rows of a Young diagram which has at least two rows. Such posets can now be seen to be members of the 4th class of d-complete posets. (In addition, around 1993 Fomin discovered a way to combine three posets P, Q, R with product enumeration formulas for e(P), e(Q), e(R) into a poset with |P|+|Q|+|R|+3 elements which also has a product enumeration formula. This result overlaps with our results in some ways.)

(6) David Behrman investigated small non-d-complete posets which have the hook length property in a 1996 UNC Masters' project. (His email address is dmbehr0@pop.uky.edu.)

(7) In the late 1990's, Bruce Sagan and Will Brockman have proved the hook length property for many posets with a Hillman-Grassl type algorithm. These posets appear to be a subset of the d-complete posets consisting of many or all of those whose slant irreducible components are shapes and/or shifted shapes.

(8) We do not know of other results or research concerning hook length posets. (References welcome!)

Jeu de Taquin Posets:

(1) M.-P. Schutzenberger considered this procedure for sliding labels around on the order diagram of a poset in the 1960's.

(2) In the early 1970's Schutzenberger was able to prove that the jdt process was well-defined on shapes.

(3) G.P. Thomas published his own proof of this fact in 1977 in Discrete Mathematics (volume 17).

(4) A student and a former student of Richard Stanley independently proved that jdt was well-defined on shifted shapes: Dale Worley, 1984 M.I.T. thesis and Bruce Sagan, JCT A 45 (1987).

(5) At any time anyone could have noticed and easily proved that the jdt process is well-defined on trees and on 'double-tailed diamonds' dk(1).

(6) Sarah Wilmesmeier (now Sarah Bergmann) investigated small non-d-complete posets which have the jeu de taquin property in a 1997 UNC Masters' project. (Her email address is sarahw@nortel.ca.)

(7) Apparently jdt was not known to hold on any other infinite classes of posets and possibly not even on any particular other posets. It seems that jdt in and of itself on general posets has been studied in very few, if any, papers. (References welcome!)

Move on to Compare JDT to Evacuation and Promotion

or return to Recent Research Table of Contents.