OFFSET
0,3
COMMENTS
K-Knuth equivalence on words is the K-theoretic analog for Knuth equivalence on words. Two words are said to be Knuth equivalent if one can be obtained from the other via a finite series of applications of the Knuth relations:
xzy ~ zxy, (x < y < z)
yxz ~ yzx, (x < y < z).
In the K-theoretic version, two words are said to be K-Knuth equivalent if one can be obtained from the other via a finite series of applications of the K-Knuth relations:
xzy ~ zxy, (x < y < z)
yxz ~ yzx, (x < y < z)
x ~ xx,
xyx ~ yxy.
In 2006, Buch et al. introduced a new combinatorial algorithm called Hecke insertion, which is a K-theoretic analog of the well-known Schensted algorithm for the insertion of a word into a semistandard Young tableau. The Hecke insertion algorithm results in a strictly increasing tableau. An important difference between Knuth equivalence and K-Knuth equivalence is that, while insertion equivalence via the Schensted algorithm (resp. the Hecke algorithm) implies Knuth equivalence (resp. K-Knuth equivalence), the converse holds for the standard version but not for the K-theoretic version. In other words, two words can be K-Knuth equivalent but insert into different tableaux via the Hecke insertion algorithm.
LINKS
A. Buch and M. Samuel, K-Theory of Minuscule Varieties, arXiv:1306.5419 [math.AG], 2003.
Christian Gaetz et al. K-Knuth Equivalence for Increasing Tableaux, preprint arXiv:1409.6659 2015.
R. Patrias and P. Pylyavskyy, K-Theoretic Poirer-Reutenauer Bialgebra, arXiv:1409.6659 [math.CO], 2014.
H. Thomas and A. Yong, A Jeu de Taquin Theory for Increasing Tableaux, with Applications to K-Theoretic Schubert Calculus, Algebra Number Theory 3 (2009), no. 2, 121-148.
H. Thomas and A. Yong, A Jeu de Taquin Theory for Increasing Tableaux, with Applications to K-Theoretic Schubert Calculus, arXiv:0705.2915 [math.CO], 2007.
EXAMPLE
For n = 2, there are 3 K-Knuth classes, each with one tableau. The tableaux representing the classes are the minimal tableaux of partition shapes (2), (1,1), and (2,1).
(A minimal tableau is a tableau in which each box is filled with the smallest positive integer that will make the filling a valid strictly increasing tableau.)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Michelle Mastrianni, Aug 25 2015
STATUS
approved