_ A Suggestion on Mathematical Materials for Freshman Education Vol.16 ~ Concepts of Representations of Finite Groups ~ by Kenshi KIDA *1 ( received on Nov. 28, 2017 & accepted on Jan. 11, 2018 ) 4 1 x++jT'%¤Y*36¤3.r|'!+LQ*$%+cyK*69 ¥UNP'%,¤i*62+9\5q¥ Abstract As an introduction to the representation theory of finite groups, we explain basic matters of matrix representations, characters of groups, and orthogonality relations. We treat cases of symmetric groups as concrete examples. Keywords: Representation of Finite Group, Character of Group, Orthogonality Relation fgZk}*%¤¤¤N)(+Mv*$%g/¥yw&,¤x++T5]+X *$%+9) ¥ +w&,¤;EB?:G+@D>'a-76}+^ ( ) ( ) ⎩ ⎨ ⎧ ≠ = = j i j i j i 0 1 δ 9O''6¥ vN C I+ n }+Hlet¤)8" n }+~[Y$69 ( ) C , n GL '6¥ G 4 ( ) C , n GL +J0+`bVS () a A a → : A 9 G +'¤ n 9!+}v'¥+' ( ) ()() () ( ) () 1 1 , , − − = = = a A a A E e A b A a A ab A p5$¥ * A `bVS&6'¤79mh)¢ £'¥ G + $+ () a A a → : A ' () a B a → : B *i%¤~[Y P #% () () ( ) G a P a A P a B ∈ = −1 ¡u u=F>G Wus Liberal Arts Education Center, Takanawa Campus, Associate Professor ~有限群の表現論の基礎概念~ - 65 - 東海大学紀要情報通信学部 Vol.10,No.2,2017,pp.65-73 トピックス
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AC?;< Vol.xx, No.xx , 20xx, pp.xxx -xxx トピックス...z fg od Rg 東海大学紀要 Vol.xx, No.xx , 20xx, pp.xxx -xxx 1 x + +c {n _ 貴田 研司 A Suggestion on Mathematical Materials
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Vol.xx, No.xx , 20xx, pp.xxx -xxx
1
A Suggestion on Mathematical Materials for Freshman Education Vol.16
~ Concepts of Representations of Finite Groups ~
by
Kenshi KIDA*1 ( received on Nov. 28, 2017 & accepted on Jan. 11, 2018 )
Abstract
As an introduction to the representation theory of finite groups, we explain basic matters of matrix representations, characters of groups, and orthogonality relations. We treat cases of symmetric groups as concrete examples.
Keywords: Representation of Finite Group, Character of Group, Orthogonality Relation
As an introduction to the representation theory of finite groups, we explain basic matters of matrix representations, characters of groups, and orthogonality relations. We treat cases of symmetric groups as concrete examples.
キーワード:有限群の表現,群の指標,直交関係 Keywords: Representation of Finite Group, Character of Group, Orthogonality Relation
1. はじめに
大学初年次において,群,環,体などの代数系について学ぶ.本論文では,有限群の表現論の入り口の部分
についての解説をおこなう 1)2)3)4).
この論文では,クロネッカーのデルタと呼ばれる次の記号
jiji
ji 01
を使うこととする.
複素数体 C 上の n 次の一般線形変換群,すなわち n 次の正則行列がつくる群を C,nGL とする.群 G か
ら C,nGL の中への準同型写像 aAa:A を G の表現といい, n をその次数という.このとき
11,, aAaAEeAbAaAabA
が成り立つ.
特に A が同型写像であるとき,これを忠実な表現(faithful representation)という.
群 G の 2 つの表現 aAa:A と aBa:B に対して,正則行列 P があって
GaPaAPaB 1
*1 高輪教養教育センター 准教授 Liberal Arts Education Center, Takanawa Campus, Associate Professor