Download PDF by Salzmann H.: 16-dimensional compact projective planes with a collineation

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By Salzmann H.

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Extra resources for 16-dimensional compact projective planes with a collineation group of dimension >= 35

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The jth bs with < [t] and and prove the suppose Unless (j+l)th aq > standard. as a l i n e a r By i n d u c t i o n , co~]ination salne r e s u l t that columns, for e t. the e n t r i e s t is s t a n d a r d , have we may of s t a n d a r d some Since in t are adjacent entries assume that polytabloids etz = (sgnz) e t in i n c r e a s i n g pair order of columns, say a I < a 2 <... < a r, b I < b 2 <... 6 = [(sgn <.. 2 . < It] from our for ~ ~ 1 • induction Since hypothesis. of the S p e c h t module the standard ~-tableaux.

4 EXAMPLE If t = 1 3 4 5 and T = 2 2 1 1 2 {t}@T then 1 = 2 2 1 1 + 2 1 2 1 + 2 1 1 2 + 1 2 2 1 + 1 2 1 2 + 1 1 2 2 1 1 1 1 1 and 1 {t}(123)@ T = 2 1 1 1 + 1 1 2 1 + 1 1 1 2 + 2 1 2 1 + 2 1 1 2 + 1 1 2 2 2 Notice different ponding 2 that the w a y tableaux to w r i t e whose rows 1 down contain i {t}@ T is s i m p l y the same numbers 1 to s u m a l l as the the corres- r o w o f T. 5 2 that T

Irreducible every from Curtis is a s p l i t t i n g classes Since sees than prefer we quote and Reiner the that are i s o m o r p h i c , F is i s o m o r p h i c of i n e q u i v a l e n t regular all Rather and Reiner field Curtis ber over probably which Curtis shall result. 6 a field the p - r a n k gives The dimension o f the i r r e d u c i b l e of characteristic o f the G r a m m a t r i x with p can be respect representation calculated D ~ of by evaluating to the s t a n d a r d basis o f S ~.

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16-dimensional compact projective planes with a collineation group of dimension >= 35 by Salzmann H.


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