(q,t)-hook formula for Birds and Banners

TL;DR

This paper proves Okada's $(q,t)$-hook formula conjecture for birds and banners, specific types of $d$-complete posets, using symmetric functions and Gasper's identity for VWP-series.

Contribution

It provides the first proof of the conjecture for birds and banners, expanding the understanding of $(q,t)$-hook formulas in $d$-complete posets.

Findings

Proof of the conjecture for birds and banners

Application of Gasper's identity for VWP-series ${}_{12}W_{11}$

Advancement in combinatorial representation theory

Abstract

We study Okada's conjecture on $(q,t)$-hook formula of general $d$-complete posets. Proctor classified $d$-complete posets into 15 irreducible ones. We try to give a case-by-case proof of Okada's $(q,t)$-hook formula conjecture using the symmetric functions. Here we give a proof of the conjecture for birds and banners, in which we use Gasper's identity for VWP-series ${}_{12}W_{11}$.