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数理解析研究所講究録 (2006), 1466: 93-98 Issue Date 2006-01 URL http://hdl.handle.

1 A remark on Glauberman-Watanabe cor Titlea normal defect group(cohomology Th and Related Topics) Author(s) 田阪, 文規 Citation 数理解析研究所講究録 (2006), 1466: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 A remark on Glauberman-Watanabe corresponding blocks with a normal defect group ftasaka@g.math.s.chiba-u.ac.jp $G$ $G$ $S$ $G$ S $G$ $S$ ( ) 1 1 (Glauberman ) [18] $S$ centralise defect group Brauer category (p-)block 1 1 (Watanabe ) [18] Glauberman Watanabe blocks isotypy blocks normal defect group biocks (Glauberman $\text{}$ [5] trivial $)$ module source bimodule splended[9] [5] Morita $[10]_{\text{}}$ (p $ 7 [6]$ ) Watanabe blocks normal defect group Glauberman Morita bimodule $\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}_{\text{}}$ block algebra bimodule induction bimodule (Glauberman [8] Puig [17] ) $p$ $(\mathcal{o}, K, k)$ $p$-modular system. 0 $p$ 1(Glauberman [4]). $(G,S)$ ( $G$ $S$ $G$ $G$ ) $(G)^{S}arrow \mathrm{i}\mathrm{r}\mathrm{r}(g^{s})$ $\pi(g, S)$ : Irr (1) $T$ $S$ $\mathrm{i}\mathrm{r}\mathrm{r}(g)^{s}$ $\pi(g, T)$ $\mathrm{i}\mathrm{r}\mathrm{r}(g^{t})^{s}$ $\pi(g, S)=\pi(G^{T}, S/T)\circ\pi(G, T)$ (2) $S$ restriction $\pm 1_{0})_{\text{}}$ $\chi$ q $\pi(g, S)(\chi)$ $\chi$ ( SS $G$

3 $\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\rfloor)$ ) 94 1 Glauberman $G$ 2(Watanabe[18]). SS block defect group $P$ (1) $G$ S (2) block $w(b)$ :Glauberman $w(b)$ (3) block $w(b)$ $P$ defect group $w(b)$ Brauer category (4) blocks $w(b)$ perfect isometry ( isotypy) 2(2 ) block (Glauberman-) Watanabe $G$ $P$ Pp \mathrm{s}$ $S$ centralise ($\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{b}\mathrm{e} sit- $G$ ( $G$ ). $S$ statement $(\downarrow)$ ( 1 ) block algebra primitive interior -algebra $G$ defect multiplicity module twisted group $k$ $\mathrm{a}^{\tau}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}_{0}$ $V(\overline{G}:=N_{G}(P_{\gamma})/P$ $\overline{g}^{s}$ $P_{\gamma}$ defect pointed $\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}_{\text{}}$ twisted group algebra restriction ( $V $ ) ( 1) $V$ $\mathcal{o}$ simple projective module lift $K$ $\overline{g}$ $\overline{g}^{s}$ twisted group algebra defect 0 character twited group algebra t restriction $(_{q}$ 1) -. defect 0 character ( covering group Glauberman $($ $[6])_{\text{}}$ cf. $[3]_{0}$ defect [2] ) $V$ (projevtive module) restriction $V $ $(W$ ) $K$ character twisted gromp algebra ( ) Cartan $k$ $k$ $p$ $p$ $W$ $G_{\beta}^{S}$ pointed group $V $ Puig $\beta$ $f$ $f\mathcal{o}gbf$

4 95 $G^{S}- \mathrm{a}1\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}$ primitive interior, ( ) $(\mathcal{o}gb)_{\beta^{\text{}}}f\mathcal{o}gb$ $\mathcal{o}[g^{s}\rangle\langle G]$ -module $M$ $G$ 3. SG block $S$ centralise normal $(\mathcal{o}gb)\beta$ $\mathcal{o}g^{s}w(b)$ defect group $P$ -algebra - primitive interior (1) $w(b)$ source algebra. splendid Morita [9] $[5]_{0}$. $w(b)$. -module (2) $M$ $w(b)$ Morita category $\mathcal{o}g^{s}$-module category restriction functor $M$ $qn\pm 1$ ( $n$ ) $\mathcal{o}[g^{s}\cross G]$-module $\mathcal{o}g$ tensor functor Glauberman Morita bimodule $[5]_{0}$ dreen normal defect $M$ Morita bimodule -module $M$ 0 -module $\mathcal{o}[g\cross 1]$ $[G^{S}\cross 1]$ primitive interior -algebra $G$ (Puig Morita ([15]Prop.6.5)) $\mathcal{o}[g\cross G^{S}]$-module $M$ -module $\mathcal{o}[g^{s}\cross G^{S}]$ G8s subalgebra $\overline{g}^{s}$ Puig defect multiplicity module restriction $M$ dual -module $\mathcal{o}[g\cross G]$ $\mathcal{o}g^{s}w(b)$ $\mathcal{o}g^{s}w(b)$ -module ( induction module $\mathcal{o}[g^{s}\cross G]$ $\overline{g}$ defect multiplicity module induction module (cf. [1]) 4. SS block $G$ $S$ centralise defect group $\mathcal{o}[g\rangle\langle G]- \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{e}\mathcal{o}gb$ $\mathcal{o}[g^{s}\cross G^{S}]-$ $\triangle P$ restriction vertex $\mathrm{d}_{\text{}}$ ( $[G^{S}\cross G^{S}]- \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{e}\mathcal{o}g^{s}w(b)$ $\mathcal{o}g^{s}w(b)$ 0 $\triangle P$ G]q induction vertex O[G $\cross$

5 $\mathfrak{g}\mathrm{g}$ $\mathrm{i} $ ( $1)_{\text{}}$ 5. normal defect group block aigebra $PE$ ( $P$ $\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p},e$ defect inertial quotient) twisted group algebra Morita ([11] [13]) 1 $\text{}[10]$ $[5]$ Watanabe blocks 2-cocycle source $\mathrm{i}$ $\mathrm{i}=\mathrm{i} f=f\mathrm{i} $ algebra ( $\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}_{\text{}}f$ $\beta$ $w(b)$ source ) normal defect group $f$ $w(b)$ $\mathrm{i} \mathcal{o}g^{s}\mathrm{i} $ $\mathrm{i}\mathcal{o}g\mathrm{i}$ source algebra saurce algebra interior P- $\mathrm{h}\mathrm{o}\mathrm{m}$ algebra. normal defect group block source algebra [17]Th.44.3 $\mathcal{o}$-basis 6. 3(2) first second statement (Brauer character ) Glauberman sign block restriction Glauberman character $M$ $M$ Glauberman 7. [16][14] primitive interior -algebra $G$ $G$-triple (defeet group source algebra defect multiplicity module ) 1 1 ( [16][14] ) 3 statement $S$ centrafise defect group SS $G$ block algebra G-triplc 3 Glauberman \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}1\mathrm{e}$ $G^{S}- ( primitive interior -algebra) ( SS simple $kg$ -module primitive interior -algebra $G$ ) defect group $P$ normalizer $\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$ interior -algebra Green $G$ ( $N_{G}(P)$ $H$ \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}$ $\text{}\mathrm{h}- $G$-tripleJ ( $P$ $G$ normalizer $H$ normalizer ) [16] Green ( ) Puig $G$ -algebra pointed groups $\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}$ F ( module Green ) block $\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}$ algebra block stable algebra Green Morita

6 87 simple module simpie module $\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{n}$ Brauer block algebras ) block algebra block algebra normal defect group Watanabe block algebras 3 first $\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}_{\text{}}$ ; [1] L. Barker, Induction,restriction and $G$ -algebras, Comm.algebra. 22 (1994), [2] E. C. Dade, Counting characters in blocksii, J. Reine Angew. Math. 448 (1994), $\cdot [3] E. C. \mathrm{d}\mathrm{a}\mathrm{d}\mathrm{e}$, Anew approach to Glauberman s correspondence, J. Algebra 270 (2003), [4] G.Glauberman, Correspondence of characters for relatively prime operator groups, Canad. J. Math. 20 (1968), [5] M. E. Harris, Glauberman-Watanabe corresponding $p$-blocks of finite groups with normal defect groups are Morita equivalent, Trans.of the A. M. S. (2004) [6] M. E. Harris and M. Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite $p$-solvable groups, Trams. of the A. M. S. 354(9) (2002), [7] H. Horimoto, A note on the Glauberman correspondence of p-biocks of finite $\mathrm{p}$-solvable groups, Hokkkaido Math. J. 31 (2002), [8] I.M.Isaacs, Character Theory of Finite Groups, Academic Press (1976).

7 98 [9] S. Koshitani, A remark on Glauberman-Watanabe correspondence of $p$-blocks of finite groups, preprint (2002) [10] S.Koshitani and G.O.M ichler, Glauberman correspondence of p- blocks of finite groups, J. Algebra 243 (2001), [11] B. Kiilshammer, Crossed products and biocks with normal defect groups, Comm.Algebra 13 (1985), [12] L. Puig, Pointed groups and construction of characters, Math. Z. 176(1981), [13] L. Puig, Pointed groups and construction of modules, J. Alg. 116(1988), [I4] L. Puig, On Thevenaz parametrization of interior G-algebras, Math. Z [15] L. Puig, On the LocaI Structure of Morita and Rickard Equivalences $\mathrm{b}$etween Brauer Blocks, Birkhauser (1999). $\mathrm{z}$ [16] Thevenaz, The parametrization of interior algebras, Math. (1993), $212$ [17] Thevenaz, $G$-Algebras and Modular Representation Theory, Oxford University Press (1995). [18] A. Watanabe, The Glauberma$\mathrm{n}$ character correspondence and perfect isometries for blocks of finite groups, J. algebra 216 (1999),

数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2011), B25:

Non-existence of certain Galois rep Titletame inertia weight : A resume (Alg Related Topics 2009) Author(s) OZEKI, Yoshiyasu Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2011), B25: 89-92 Issue Date 2011-04