Abstract
We introduce a type $A$ crystal structure on decreasing factorizations of fully-commu\-tative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.
Highlights
Grothendieck polynomials were introduced by Lascoux and Schutzenberger [LS82, LS83] as representatives for the Schubert classes in the K-theory of the flag manifold
Gλ(x1, . . . , xm; β) = β|μ|−|λ|Mλμ sμ(x1, . . . , xm), μ where Mλμ is the number of highest-weight set-valued tableaux of weight μ in the crystal SVTm(λ). Their approach recovers a Schur expansion formula for Grassmannian Grothendieck polynomials given by Lenart [Len[00], Theorem 2.2] in terms of flagged increasing tableaux
We provide a new insertion algorithm, which we call -insertion, from decreasing factorizations on fully-commutative elements in the 0-Hecke monoid to pairs of semistandard Young tableaux of the same shape, which intertwines with crystal operators the electronic journal of combinatorics 27(2) (2020), #P2.29
Summary
Grothendieck polynomials were introduced by Lascoux and Schutzenberger [LS82, LS83] as representatives for the Schubert classes in the K-theory of the flag manifold. Xm), μ where Mλμ is the number of highest-weight set-valued tableaux of weight μ in the crystal SVTm(λ) Their approach recovers a Schur expansion formula for Grassmannian Grothendieck polynomials given by Lenart [Len[00], Theorem 2.2] in terms of flagged increasing tableaux. We provide a new insertion algorithm, which we call -insertion, from decreasing factorizations on fully-commutative elements in the 0-Hecke monoid to pairs of (transposes of) semistandard Young tableaux of the same shape (see Definition 27 and Theorem 35), which intertwines with crystal operators the electronic journal of combinatorics 27(2) (2020), #P2.29.
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