Skew hook formula for $d$-complete posets

TL;DR

This paper extends Peterson and Proctor's hook formula for $d$-complete posets to skew cases using excited diagrams, providing new proofs via equivariant $K$-theory techniques.

Contribution

It introduces a skew generalization of the hook formula for $d$-complete posets using excited diagrams and algebraic geometry methods.

Findings

Derived a skew hook formula for $d$-complete posets.

Connected the formula to equivariant $K$-theory of Kac--Moody flag varieties.

Provided an alternative proof of the original hook formula.

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.