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Then we have A ∈ Cλ if and only if for all k ∈ {1, . . , n} we k have i=1 λi = n − rkK ((A − En )k ). b) The set C λ := µ λ Cµ ⊆ Gu is closed. Proof. a) For a Jordan block Jm (1) ∈ Km×m , for some m ∈ N, we have rkK ((Jm (1) − Em )k ) = m − k for all k ∈ {0, . . , m}. Thus for A ∈ Cλ , where n λ = [1a1 , . . , nan ] n, we get i=k+1 (i − k)ai = rkK ((A − En )k ) for all k ∈ {0, . . , n − 1}. Hence the rank vector [rkK ((A − En )k ); k ∈ {0, . . , n − 1}] ∈ Qn is determined by λ, and since the above conditions form a unitriangular system of n linear equations for [a1 , .

N ] being called the associated conjugate partition. n Hence we have λi = j=i aj (λ), for i ∈ {1, . . , n}. Moreover, we have Yλ = {[i, j] ∈ N2 ; i ∈ {1, . . , n}, j ∈ {1, . . , λi }} = {[i, j] ∈ N2 ; j ∈ {1, . . , n}, i ∈ {k ∈ {1, . . , n}; λk ≥ j}} = {[i, j] ∈ N2 ; j ∈ {1, . . , n}, i ∈ {1, . . , λj }}, implying that Yλ = {[i, j] ∈ N2 ; [j, i] ∈ Yλ }, and thus (λ ) = λ. c) Let λ = [λ1 , . . , λn ] n and µ = [µ1 , . . , µn ] n. Then µ is called to k k dominate λ, if for all k ∈ {1, .

Proof. 3]. 13) Exercise: Integral ring extensions. Let R ⊆ S be a ring extension. a) Show that an element s ∈ S is integral over R, if and only if there is an Rsubalgebra of S containing s, which is a finitely generated R-module. Conclude that R ⊆ S is a finite ring extension, i. e. S is a finitely generated R-algebra and integral over R, if and only if S is a finitely generated R-module. b) Show that the integral closure R := {s ∈ S; s integral over R} ⊆ S of R in S is a subring of S, and that R = R holds.

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Algebraic Groups by Jürgen Müller


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