2002 Fiscal Year Final Research Report Summary
Relations between character formula of classical groups, and the generating functions of P-partitions of d-complete posets
| Project/Area Number |
13640022
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tottori University |
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Principal Investigator |
ISHIKAWA Masao Tottori University, Faculty of Education and Regional Sciences, Associate Professor, 教育地域科学部, 助教授 (40243373)
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| Project Period (FY) |
2001 – 2002
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| Keywords | combinatorics / plane partitions / poset / Pfaffian / generating functions |
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| Research Abstract |
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.
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