By Hahl H., Salzmann H.
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Les ? ‰l? ©ments de math? ©matique de Nicolas Bourbaki ont pour objet une pr? ©sentation rigoureuse, syst? ©matique et sans pr? ©requis des math? ©matiques depuis leurs fondements. Ce troisi? ?me quantity du Livre sur les Groupes et alg? ?bres de Lie, neuvi? ?me Livre du trait? ©, poursuit l ? ©tude des alg?
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Extra resources for 16-dimensional compact projective planes with a large group fixing two points and two lines
Then f˜a M N if and only if e˜ a N M. 4). 1(i). 4 Let M N ∈ n -mod be irreducible. Then f˜a M f˜a N if and only if M N . Similarly, providing a M a N > 0, e˜ a M e˜ a N if and only if M N . 1 The map ch K n -mod →K n -mod is injective. Crystal operators 50 Proof We need to show that the characters of the irreducible modules in n -mod are linearly independent in K RepI n . Proceed by induction on n, the case n = 0 being trivial. 13) for some irreducible modules L ∈ n -mod. Choose any a ∈ F . We will show by downward induction on k = n 1 that cL = 0 for all L with a L = k.
A set of nodes of this form will be called a skew shape. The number of nodes in / will be denoted / . The number of rows occupied by / − 1 will be denoted by L / . A skew shape is called a skew hook if it is connected and does not have two boxes on the same diagonal (equivalently, if the residues of the nodes of the shape form a segment of integers). We know that V / is an irreducible n−k k -module. On restriction to Sk ⊂ n−k k it becomes a (not necessarily irreducible) Sk -module. Let / be the character of this Sk -module.
5) for each m ≥ 1. Obviously an contains xk − a n! for each k = 1 n, whence each algebra m an is finite dimensional. 2, L an is the unique irreducible m an -module up to isomorphism. Let Lm an denote a projective cover of L an in the category m an -mod (for convenience, we also define L0 an = 0 an = 0). 1 For each m ≥ 1, m an Lm an ⊕n! 7) L2 a where Rm an and Lm an are considered as n -modules by inflation. 4 Covering modules 41 By an argument involving lifting idempotents (see for example [La]) we may assume that these surjections agree with the decompositions m an Lm an ⊕n!
16-dimensional compact projective planes with a large group fixing two points and two lines by Hahl H., Salzmann H.