Relations between character formula of classical groups, and the generating functions of P-partitions of d-complete posets

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D-complete posets are defined by R. P. Proctor in relation with the generalized Weyl groups of Kac-Moody algebra. R. P. Proctor showed that d-complete posets are obtained from 15 irreducible d-complete ones. We studied the generating functions of (P. w)-partitions of 15 irreducible d-complete posets and showed that generating functions of any d-complete posets are obtained from those of irreducible ones. Further we also showed that the ordinary one variable generating functions can extended to certain multi-variable generating functions. Along the proof of those formulas, we find a lot of interesting determinants and Pfaffians which are extensions of classical ones. We also use (k, l)-hook Schur functions and its formulas to evaluate those determinants and Pfaffians. We also found that there are several evidences that we can expect that there must be a kind of k-rim hook tableaux for general d-complete posets and a kind of hook formula must hold for these rim hook tableaux. The simplest case corresponds to the classical formula, i.e., Young's lattice, which corresponds to the ordinary Young semi-standard tableaux. The second simplest case is the shifted shapes. There are famous symmetric functions, i.e., Schur Q-functions, associated with shifted shapes. It is also interesting theme to study similar generating functions of posets. We also study the (P. w)-partitions of height at most n, which can be considered as a generalization of ordinary plane partitions. Usually these generating functions do not have hook formulas, but the value of these generating functions at q=-1 might be very interesting. In this way we found a lot of interesting formulas and the study is still in progress.