2005 Fiscal Year Final Research Report Summary
Construction of Markov chane via algebraic approach for random generation of partition tables
| Project/Area Number |
15540138
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Tokai University |
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Principal Investigator |
WATANABE Junzo Tokai University, School of Science, Professor, 理学部, 教授 (40022727)
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| Co-Investigator(Kenkyū-buntansha) |
TORIGOE Norio Tokai University, School of Science, Assistant professor, 理学部, 講師 (40297180)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Hilbert function / Lefschetz property / Young diagram / Weyl duality / polynomial ring / symmetric function / 対称群 |
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| Research Abstract |
Let A=【symmetry】^c_<i=0>A_i be a zero-dimensional graded Gorenstein algebra over a field and let ×A→End(A) be the regular representation. Let z∈A be a linear form. Suppose that the nilpotent matirx ×z∈End(V) decomposes into Jordan blocks of sizes {f_1,…,f_s}. We call the module U_i=0:z^<f_i-1>+(z)/0:z^<f_i>+(z) ith central simple module of (A,z). We proved the following result. Theorem (1) Each U_i has a symmetric Hilbert fucntion. (2) If each U_i has the strong Lefschetz property, for all i, then A has the strong Lefschetz property.
In this theorem if we drop the condition "Gorenstein", but add the conditions that (1) U_i has a symmetric Hilbert function, for all i and (2) A has a symmetric Hilbert function, then we may deduce the same result.
This has many applications. For example it can be proved that a complete intersection ideal generated by power sums of consecutive degrees in a polynomial ring has the strong Lefschetz property.
In the complete intersection A=K[x_1,x_2,…,x_n]/(x^d_1,…,x^d_n) put L=x_1+…+x_n. Then the central simple module module U_i of (A,L) is an S_n-module. When d=2, U_i is spanned by Specht polynomials of degree (i-1). When n=2, U_i is one-dimensional. In either case U_i is an irreducible S_n-module. Using this and the fact that L is a strong Lefschetz element, it is possible to decompose A into irreducible S_k-modules.
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