Skew hook formula for $d$-complete posets

Abstract

Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P\setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides an alternate proof of Peterson--Proctor's hook formula.

References in corpus (2)

• Hook formulas for skew shapes I. $q$-analogues and bijections
• Schubert Classes in the Equivariant K-Theory and Equivariant Cohomology of the Lagrangian Grassmannian

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