2006 Fiscal Year Final Research Report Summary
A research on hook length posets from combinatorics and representation theory
| Project/Area Number |
15540028
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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 | WAKAYAMA UNIVERSITY |
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Principal Investigator |
TAGAWA Hiroyuki Wakayama University, Faculty of Education, Associate Professor, 教育学部, 准教授 (80283943)
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| Project Period (FY) |
2003 – 2006
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| Keywords | hook length poset / hook length formula / d-complete poset / Coxeter group |
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| Research Abstract |
The purpose of this research is a strict classification of hook length posets, a clarification of combinatorial and representative structure of hook length posets. Chiefly, the following results were obtained.
1. We introduced a poset (called a leaf poset) which is an extension of the d-complete poset, and we showed that all leaf posets are hook length posets. As a corollary of the above result, we found many identities with respect to Schur functions, which are analogues or extensions of Cauchy's identity. Also, we proved that all leaf posets are multivaliable hook length posets. Here, we call a hook length poset a multivaliable hook length poset if its hook length formula can be extended to a multivaliable formula. Moreover, we found a composition method of a hook length poset by using a known (multivaliable) hook length poset. Any hook length poset with at most seven elements is constructed by this composition method.
2. We proved several identities of Cauchy-type determinant and Schur-type Pfaffian, which was conjectured by Soichi Okada in 2003.
3. It is known that a (lambda-) minuscule element of a Coxeter group is a fully commutative element, and a fully commutative element of a symmetric group is equal to a 321-avoiding permutation. For a Coxeter group, we introduced a fully covering element which was an extension of 321-avoiding permutations, and we proved that the Coxeter groups whose fully commutative elements coincide with their fully covering elements are the Coxeter groups of type A, D, E.
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Research Products
(2 results)
