Budget Amount *help 
¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000)

Research Abstract 
[A partila order on the symmetric groups defined by 3cycles] We define a partial order on the symmetric group S_n of degree n by x【less than or equal】y iff y=a_1【triple bond】a_kx with i(y)=i(x)+2k where a_1,【triple bond】, a_k are 3cycles of increasing or decreasing consecutive three letters and i(*) is the nmber of inversions of the element * of S_n, on the analogy of the weak Bruhat Order. Whether an even parmutaion is comparable to the identity or not in this ordering is considered. It is shown that all of the even permutations of degree n which map 1 to n or n1 are comparable to the identity. [The varieties of subspaces stable under a nilpotent transformation] Let f : V → V be a nilpotent linear transformation of a vector space V of type V=λ,i.e. the size of Jordan blocks λ_1【greater than or equal】λ_2【greater than or equal】【triple bond】【greater than or equal】λ_ι. For an fstable subspace W of V,i.e. f(W) ⊂ W, the types of W and V/W are those of the maps fw : W → W and fv/w : V/W → V/W induced by f, respectively. For partitions ν and μ we investigate the set S(λ,ν,μ)={W ⊂ V;f(W) ⊂ W,type W=ν,type V/W=μ} and the singular locus of the Zariski closure X(λ,ν,μ) of S(λ,ν,μ) in the grassmaniann of subspaces of V of dimension ν. We show that S(λ,ν,μ) is nonsingular and its connected components are rational varieties ; generic vectors are introduced, which define the generic points of the irreducible components of X(λ,ν,μ) whose Plucker coordinates are fairly simple to express their defining equations. We describe explicitly the coordinate ring of an affine openset of X(λ,(d),μ) with the singular locus of codimension two.
