This unique book provides the first introduction to crystal base theory from the combinatorial
point of view. Crystal base theory was developed by Kashiwara and Lusztig from the …

We lift the parabolic quantum Bruhat graph (QBG) into the Bruhat order on the affine Weyl
group and into Littelmann's poset on level-zero weights. We establish a quantum analog of …

We establish the equality of the specialization of the Macdonald polynomial at with the
graded character of a tensor product of “single-column” Kirillov–Reshetikhin modules for …

We define cylindric versions of skew Macdonald functions P λ/μ (q, t) for the special cases q=
0 or t= 0. Fixing two integers n> 2 and k> 0 we shift the skew diagram λ/μ, viewed as a …

Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s,
we define deterministic dynamics on the space of pairs of skew Young tableaux is a …

The charge is an intricate statistic on words, due to Lascoux and Schützenberger, which
gives positive combinatorial formulas for Lusztigʼs t-analogue of weight multiplicities and …

The alcove model of the first author and Postnikov uniformly describes highest weight
crystals of semisimple Lie algebras. We construct a generalization, called the quantum …

We introduce a q-weighted version of the Robinson-Schensted (column insertion) algorithm
which is closely connected to q Whittaker functions (or Macdonald polynomials with t=0) and …

Abstract The Ram–Yip formula for Macdonald polynomials (at t= 0) provides a statistic which
we call charge. In types A and C it can be defined on tensor products of Kashiwara …

Kirillov–Reshetikhin (KR) crystals are colored directed graphs encoding the structure of
certain finite-dimensional representations of affine Lie algebras. A tensor product of column …