Budget Amount *help 
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥1,900,000 (Direct Cost: ¥1,900,000)

Research Abstract 
Dcomplete posets are defined by R. P. Proctor in relation with the generalized Weyl groups of KacMoody algebra. R. P. Proctor showed that dcomplete posets are obtained from 15 irreducible dcomplete ones. We studied the generating functions of (P. w)partitions of 15 irreducible dcomplete posets and showed that generating functions of any dcomplete posets are obtained from those of irreducible ones. Further we also showed that the ordinary one variable generating functions can extended to certain multivariable 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 krim hook tableaux for general dcomplete 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 semistandard tableaux. The second simplest case is the shifted shapes. There are famous symmetric functions, i.e., Schur Qfunctions, 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.
