ALGEBRAIC COMBINATORICS

Skew hook formula for $d$-complete posets via equivariant $K$-theory
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 541-571.

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 of $\left(P\setminus F\right)$-partitions for a $d$-complete poset $P$ and an order filter $F$ of $P$, 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. As the equivariant cohomology version, we derive a skew generalization of a combinatorial reformulation of Nakada’s colored hook formula for roots.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.54
Classification: 05A15, 06A07, 14N15, 19L47
Keywords: $d$-complete posets, hook formulas, $P$-partitions, Schubert calculus, equivariant $K$-theory
Naruse, Hiroshi 1; Okada, Soichi 2

1 Graduate School of Education, University of Yamanashi 4-4-37, Takeda, Kofu, Yamanashi 400-8510, Japan
2 Graduate School of Mathematics, Nagoya University Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
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Naruse, Hiroshi; Okada, Soichi. Skew hook formula for $d$-complete posets via equivariant $K$-theory. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 541-571. doi : 10.5802/alco.54. https://alco.centre-mersenne.org/articles/10.5802/alco.54/

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