Modular Representation Theory of Endomorphism Rings
|
|
- Christal Bradford
- 5 years ago
- Views:
Transcription
1 Modular Representation Theory of Endomorphism Rings Lehrstuhl D für Mathematik RWTH Aachen University Bad Honnef, April 2010
2 Alperin s Weight Conjecture Assumptions: Let G be a finite group and k be an algebraically closed field with characteristic p > 0.
3 Alperin s Weight Conjecture Assumptions: Let G be a finite group and k be an algebraically closed field with characteristic p > 0. Definition and Remark Let X be an indecomposable kg-module. 1 Let H G. Then X is called H-projective, if X (X H ) G. 2 Any minimal subgroup Q of G such that X is Q-projective is called a vertex of X. 3 Vertices are p-subgroups of G. The set of vertices of X is a conjugacy-class of G.
4 Alperin s Weight Conjecture Theorem (Green Correspondence) Let Q G be a p-subgroup and N G (Q) H G. Then there is a one-to-one-correspondence between {X : X indecomposable kg-module with vertex Q} = g {Y : Y indecomposable kh-module with vertex Q} = given via 1 X H = f(x) Z, where each indecomposable direct summand of Z has a vertex in {Q g H : g G \ H}, 2 Y G = g(y ) W, where each indecomposable direct summand of W has a vertex in {Q g Q : g G \ H}. f
5 Alperin s Weight Conjecture Definition A weight for G is a tupel (Q,S), where Q is a p-subgroup of G and S is a simple kn G (Q)-module with vertex Q. In this case, we call S a weight module and the Green correspondent g(s) of S in G a weight Green correspondent (WGC).
6 Alperin s Weight Conjecture Definition A weight for G is a tupel (Q,S), where Q is a p-subgroup of G and S is a simple kn G (Q)-module with vertex Q. In this case, we call S a weight module and the Green correspondent g(s) of S in G a weight Green correspondent (WGC). Weights are understood up to the natural equivalence induced by conjugation of G. (Q,S) is a weight if and only if S is a simple and projective kn G (Q)/Q-module. Any WGC is (isomorphic to) a direct summand of the permutation module k G P, where P Syl p(g).
7 Alperin s Weight Conjecture Conjecture (Alperin s Weight Conjecture, 1987) The number of simple kg-modules (up to isomorphism) is equal to the number of weights for G (up to equivalence).
8 Endomorphism Ring of k G P Let Y := k G P = n i=1 Y i be a decomposition of k G P into indecomposable direct summands and put E := End kg (Y ). For ϕ E, ϕ(y) denotes the image of y Y under ϕ. The Hom-Functor Hom kg (Y, ) : mod-kg mod-e induces a decomposition of E E into the PIMs P i := Hom kg (Y,Y i ) (Fitting Correspondence). Of course, hd(p i ) is a simple E-module, but soc(p i ) is in general not simple.
9 Computational Experiments With the computer algebra programes GAP, MeatAxe:
10 Computational Experiments With the computer algebra programes GAP, MeatAxe: Construct Y = k G P, Y = n i=1 Y i, E E, E E = n i=1 P i. Analyze hd(y i ), soc(y i ), hd(p i ), soc(p i ). Determine character table, the Cartan- and decomposition matrix of E.
11 Computational Experiments With the computer algebra programes GAP, MeatAxe: Construct Y = k G P, Y = n i=1 Y i, E E, E E = n i=1 P i. Analyze hd(y i ), soc(y i ), hd(p i ), soc(p i ). Determine character table, the Cartan- and decomposition matrix of E. Observation For almost all groups analyzed so far: {simple constituents of soc(e E )} = {simple kg-modules} = {weights for G}
12 Computational Experiments With the computer algebra programes GAP, MeatAxe: Construct Y = k G P, Y = n i=1 Y i, E E, E E = n i=1 P i. Analyze hd(y i ), soc(y i ), hd(p i ), soc(p i ). Determine character table, the Cartan- and decomposition matrix of E. Observation For almost all groups analyzed so far: { simple constituents of soc(e E )} / = {simple kg-modules} / = = { weights for G} / Exception: M 11 in characteristic 3.
13 Computational Experiments From this, the following questions and ideas arise: Generalize analysis to more general modules than k G P, e.g. k G Q i for certain p-subgroups Q i of G. Which role does the Hom-functor, and especially its evaluation at a simple kg-module play? Characterize families of groups for which the observation hold.
14 M 11 in characteristic 3 p = 3, G = M 11, P Syl 3 (G), G = k G P hd soc E E k G P hd soc
15 M 11 in characteristic 3 Generalize kp G to kg P kg Q, where Q = 3. Y := (kp G kg Q )/(Defect 0, multiplicities)
16 M 11 in characteristic 3 Y := (kp G kg Q )/(Defect 0, multiplicities) Y hd soc Y hd soc
17 M 11 in characteristic 3 Y := (kp G kg Q )/(Defect 0, multiplicities) Y hd soc Y hd soc E E Y hd soc E E Y hd soc
18 Hom-Functor F := Hom kg (Y, ) evaluated at simple kg-module - what can happen?
19 Hom-Functor F := Hom kg (Y, ) evaluated at simple kg-module - what can happen? In general: F(S) for a simple kg-module S not simple. For example: A 6 in characteristic 5: F(8) = 1 a 1 b. Even soc(f(s)) may not be simple. For example: M 11 in characteristic 3. Then F(10 3 ) =
20 Hom-Functor Assumptions for this section:
21 Hom-Functor Assumptions for this section: Consider some kg-module Y such that {S : S hd(y ) simple }/ = = {T : T soc(y )}/ =. Let Y = n i=1 Y i be a decomposition into indecomposable direct summands. Assume: Head of Y i is simple for all 1 i n. Y i = Yj i = j. Set E := End kg (Y ).
22 Hom-Functor Assumptions for this section: Remark Consider some kg-module Y such that {S : S hd(y ) simple }/ = = {T : T soc(y )}/ =. Let Y = n i=1 Y i be a decomposition into indecomposable direct summands. Assume: Head of Y i is simple for all 1 i n. Y i = Yj i = j. Set E := End kg (Y ). There is a partial order on {Y i : 1 i n}, given via: Y i Y j, if and only if there is a surjection ϕ : Y i Y j.
23 Hom-Functor Notation: Fix S = hd(y i ) for some 1 i n. Then i ϕ : Y i S may be considered as an element of F(S) = Hom kg (Y,S) E.
24 Hom-Functor Notation: Lemma Fix S = hd(y i ) for some 1 i n. Then i ϕ : Y i S may be considered as an element of F(S) = Hom kg (Y,S) E. (a) Let ψ k F(S) be simple E-submodule. Then ψ is of the form ψ = i ϕ for some 1 i n such that hd(y i ) = S. (b) The following are equivalent. (i) Y i is maximal w.r.t. for some 1 i n. (ii) i ϕ k is a simple E-submodule of F(S).
25 Hom-Functor Theorem Let S be a simple socle constituent of Y and assume (possibly after renumbering) that hd(y k ) = S for all 1 k l for some l n and that hd(y k ) = S for all l + 1 k n. (a) The following are equivalent: (i) The E-submodule soc(f(s)) has exactly r non-isomorphic constituents. (ii) Among {Y k : 1 k l} there are exactly r maximal elements with respect to. (b) Let ψ k E E be simple E-submodule. Then there is 1 i n such that ψ = E i ϕ.
26 Hom-Functor Corollary Assume that for each simple module occurring in the head of Y there is exactly one maximal element with respect to.
27 Hom-Functor Corollary Assume that for each simple module occurring in the head of Y there is exactly one maximal element with respect to. Then soc(f(s)) is a simple E-module for each simple constituent S of hd(y ). Moreover, {S : S hd(y ) simple}/ = {T : T soc(e) simple}/ = S soc(f(s)) is a bijection.
28 Indecomposable Liftable Modules in Cyclic Blocks For which families of groups does observation hold?
29 Indecomposable Liftable Modules in Cyclic Blocks For which families of groups does observation hold? Joint work with Gerhard Hiss.
30 Indecomposable Liftable Modules in Cyclic Blocks For which families of groups does observation hold? Joint work with Gerhard Hiss. Assumptions: Let B be cyclic kg-block with defect group Q. Question 3 : Which indecomposable B-modules are liftable?
31 Indecomposable Liftable Modules in Cyclic Blocks For which families of groups does observation hold? Joint work with Gerhard Hiss. Assumptions: Let B be cyclic kg-block with defect group Q. Question 3 : Which indecomposable B-modules are liftable? Kupisch 1969: Description of indecomposable B-modules in terms of paths on the Brauer tree:
32 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type I: Brauer tree, with multiplicity m = 3 of exceptional vertex: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 χ Λ χ 3
33 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type I: Choose path: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 χ Λ χ 3
34 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type I: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3
35 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type I: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 From this, we get the following indecomposable kg-module: A 1 E 3 E 2
36 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type I: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 A 1 A2 E1 E3 B 1 E 2 B 2 E 3 E 2
37 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type I: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 A 1 A2 E1 E3 B 1 E 2 B 2 E 3 E 2 Note: The module Y is of Type I if and only if, soc(y ) and hd(y ) have no common constituents.
38 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type II: Brauer tree, with multiplicity m = 3 of exceptional vertex: A 2 A 1 χ 0 E χ 1 1 E 2 E 3 B 1 B 2 χ Λ χ 3
39 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type II: Choose path: A 2 A 1 χ 0 χ E 1 1 E 2 E 3 B 1 B 2 χ Λ χ 3
40 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type II: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 χ E 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 From this, we get the following indecomposable kg-module:
41 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type II: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 χ E 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 From this, we get the following indecomposable kg-module: E 2 B 2 A 1 E 2
42 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type II: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 χ E 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 From this, we get the following indecomposable kg-module: E 2 A 1 E 3 E2 E3 B2 E 2
43 Indecomposable Liftable Modules in Cyclic Blocks Paths of Type II: Choose marking on path and multiplicity t = 2 m = 3: A 2 A 1 χ 0 χ E 1 1 E 2 E 3 B 1 B 2 t=2 χ Λ χ 3 From this, we get the following indecomposable kg-module: E 2 A 1 E 3 E2 E3 B2 Note: The module Y is of Type II if and only if, soc(y ) and hd(y ) have common constituents. E 2
44 Indecomposable Liftable Modules in Cyclic Blocks Theorem Let τ be a path of Type I on the Brauer tree of B. Then the indecomposable module X constructed from τ is liftable if and only if τ is as in the following figure and one of the following cases occurs: χ 0 E χ 0 1 E 1 (a) χ 1 is not the exceptional vertex. Then the character of a lift of X equals χ 1.
45 Indecomposable Liftable Modules in Cyclic Blocks Theorem Let τ be a path of Type I on the Brauer tree of B. Then the indecomposable module X constructed from τ is liftable if and only if τ is as in the following figure and one of the following cases occurs: χ 0 E χ 0 1 E 1 (a) χ 1 is not the exceptional vertex. Then the character of a lift of X equals χ 1. (b) χ 1 is the exceptional vertex, which has multiplicity m. Each of E 0 and E 1 occurs t times as composition factor of X for some 1 t m. The character of any lift of X is a sum of t distinct exceptional characters.
46 Indecomposable Liftable Modules in Cyclic Blocks χ 0 E 0 χ 1 E 1 Note: If X of Type I, then X is uniserial in both cases. The head of X is either isomorphic to E 0 or E 1. If the head of X is isomorphic to E 0, the successor of E 1 around χ 1 equals E 0, and if the head of X is isomorphic to E 1, the successor of E 0 around χ 1 equals E 1.
47 Liftable Indecomposable Modules in Cyclic Blocks Theorem Let τ be a path of Type II on the Brauer tree. Then the indecomposable module X constructed from τ is liftable if and only if τ satisfies one the following three cases: Case 1: The vertex χ 0 is a leaf of the Brauer tree. χ 0 E 0 χ 1 χ s E s t χλ
48 Liftable Indecomposable Modules in Cyclic Blocks Theorem Let τ be a path of Type II on the Brauer tree. Then the indecomposable module X constructed from τ is liftable if and only if τ satisfies one the following three cases: Case 1: The vertex χ 0 is a leaf of the Brauer tree. χ 0 E 0 χ 1 χ s E s t χλ Case 2: Either E u0 is in the head or in the socle of X. If E u0 is in the head of X, the successor of E 0 around χ 0 is E u0. If E u0 is in the socle of X, the successor of E u0 around χ 0 is E 0. χ u0 χ E 0 u0 E 0 χ 1 χ s E s t χλ
49 Liftable Indecomposable Modules in Cyclic Blocks Theorem Let τ be a path of Type II on the Brauer tree. Then the indecomposable module X constructed from τ is liftable if and only if τ satisfies one the following three cases: Case 3: The successor of E u0 around χ 0 is E 0 and E u0 is in the socle of X. χ u0 E u0 χ 0 E 0 χ 1 χ s E s t χλ χ d0 E d0
50 Liftable Indecomposable Modules in Cyclic Blocks Theorem Let τ be a path of Type II on the Brauer tree. Then the indecomposable module X constructed from τ is liftable if and only if τ satisfies one the following three cases: Case 3: The successor of E u0 around χ 0 is E 0 and E u0 is in the socle of X. χ u0 E u0 χ 0 E 0 χ 1 χ s E s t χλ χ d0 E d0 The character of any lift of X equals χ 0 + χ 1 + χ s + µ, where µ is a sum of t distinct exceptional characters.
51 Thank you for listening!
Endomorphism rings of permutation modules
Endomorphism rings of permutation modules Natalie Naehrig Lehrstuhl D für Mathematik RWTH Aachen University naehrig@math.rwth-aachen.de December 9, 2009 Abstract Let k be an algebraically closed field
More informationON THE SOCLE OF AN ENDOMORPHISM ALGEBRA
ON THE SOCLE OF AN ENDOMORPHISM ALGEBRA GERHARD HISS, STEFFEN KÖNIG, NATALIE NAEHRIG Abstract. The socle of an endomorphism algebra of a finite dimensional module of a finite dimensional algebra is described.
More informationBroué s abelian defect group conjecture holds for the sporadic simple Conway group Co 3
Revised Version of 11 September 2011 Broué s abelian defect group conjecture holds for the sporadic simple Conway group Co 3 Shigeo Koshitani a,, Jürgen Müller b, Felix Noeske c a Department of Mathematics,
More informationTHE 2-MODULAR DECOMPOSITION MATRICES OF THE SYMMETRIC GROUPS S 15, S 16, AND S 17
THE 2-MODULAR DECOMPOSITION MATRICES OF THE SYMMETRIC GROUPS S 15, S 16, AND S 17 Abstract. In this paper the 2-modular decomposition matrices of the symmetric groups S 15, S 16, and S 17 are determined
More informationTHE BRAUER CHARACTERS OF THE SPORADIC SIMPLE HARADA-NORTON GROUP AND ITS AUTOMORPHISM GROUP IN CHARACTERISTICS 2 AND 3
THE BRAUER CHARACTERS OF THE SPORADIC SIMPLE HARADA-NORTON GROUP AND ITS AUTOMORPHISM GROUP IN CHARACTERISTICS 2 AND 3 GERHARD HISS, JÜRGEN MÜLLER, FELIX NOESKE, AND JON THACKRAY Dedicated to the memory
More informationTHE 7-MODULAR DECOMPOSITION MATRICES OF THE SPORADIC O NAN GROUP
THE 7-MODULAR DECOMPOSITION MATRICES OF THE SPORADIC O NAN GROUP ANNE HENKE, GERHARD HISS, AND JÜRGEN MÜLLER Abstract. The determination of the modular character tables of the sporadic O Nan group, its
More informationTESTING MODULES OF GROUPS OF EVEN ORDER FOR SIMPLICITY
TESTING MODULES OF GROUPS OF EVEN ORDER FOR SIMPLICITY GERHARD O. MICHLER AND ØYVIND SOLBERG Abstract. In this paper we exhibit an intimate relationship between the simplicity of an F G-module V of a finite
More informationThe Major Problems in Group Representation Theory
The Major Problems in Group Representation Theory David A. Craven 18th November 2009 In group representation theory, there are many unsolved conjectures, most of which try to understand the involved relationship
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More information`-modular Representations of Finite Reductive Groups
`-modular Representations of Finite Reductive Groups Bhama Srinivasan University of Illinois at Chicago AIM, June 2007 Bhama Srinivasan (University of Illinois at Chicago) Modular Representations AIM,
More informationHERMITIAN FUNCTION FIELDS, CLASSICAL UNITALS, AND REPRESENTATIONS OF 3-DIMENSIONAL UNITARY GROUPS
HERMITIAN FUNCTION FIELDS, CLASSICAL UNITALS, AND REPRESENTATIONS OF 3-DIMENSIONAL UNITARY GROUPS GERHARD HISS Abstract. We determine the elementary divisors, and hence the rank over an arbitrary field,
More informationRAPHAËL ROUQUIER. k( )
GLUING p-permutation MODULES 1. Introduction We give a local construction of the stable category of p-permutation modules : a p- permutation kg-module gives rise, via the Brauer functor, to a family of
More informationModular representation theory
Modular representation theory 1 Denitions for the study group Denition 1.1. Let A be a ring and let F A be the category of all left A-modules. The Grothendieck group of F A is the abelian group dened by
More informationRECOLLECTION THE BRAUER CHARACTER TABLE
RECOLLECTION REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE LECTURE III: REPRESENTATIONS IN NON-DEFINING CHARACTERISTICS AIM Classify all irreducible representations of all finite simple groups and related
More informationNAVARRO VERTICES AND NORMAL SUBGROUPS IN GROUPS OF ODD ORDER
NAVARRO VERTICES AND NORMAL SUBGROUPS IN GROUPS OF ODD ORDER JAMES P. COSSEY Abstract. Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro
More informationTHREE CASES AN EXAMPLE: THE ALTERNATING GROUP A 5
THREE CASES REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE LECTURE II: DELIGNE-LUSZTIG THEORY AND SOME APPLICATIONS Gerhard Hiss Lehrstuhl D für Mathematik RWTH Aachen University Summer School Finite Simple
More informationOn p-monomial Modules over Local Domains
On p-monomial Modules over Local Domains Robert Boltje and Adam Glesser Department of Mathematics University of California Santa Cruz, CA 95064 U.S.A. boltje@math.ucsc.edu and aglesser@math.ucsc.edu December
More informationFOULKES MODULES AND DECOMPOSITION NUMBERS OF THE SYMMETRIC GROUP
FOULKES MODULES AND DECOMPOSITION NUMBERS OF THE SYMMETRIC GROUP EUGENIO GIANNELLI AND MARK WILDON Abstract. The decomposition matrix of a finite group in prime characteristic p records the multiplicities
More informationALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS
ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample
More informationThe Ring of Monomial Representations
Mathematical Institute Friedrich Schiller University Jena, Germany Arithmetic of Group Rings and Related Objects Aachen, March 22-26, 2010 References 1 L. Barker, Fibred permutation sets and the idempotents
More informationNew York Journal of Mathematics. Cohomology of Modules in the Principal Block of a Finite Group
New York Journal of Mathematics New York J. Math. 1 (1995) 196 205. Cohomology of Modules in the Principal Block of a Finite Group D. J. Benson Abstract. In this paper, we prove the conjectures made in
More informationSOLUTIONS FOR FINITE GROUP REPRESENTATIONS BY PETER WEBB. Chapter 8
SOLUTIONS FOR FINITE GROUP REPRESENTATIONS BY PETER WEBB CİHAN BAHRAN I changed the notation in some of the questions. Chapter 8 1. Prove that if G is any finite group then the only idempotents in the
More informationarxiv: v1 [math.rt] 12 Jul 2018
ON THE LIFTING OF THE DADE GROUP CAROLINE LASSUEUR AND JACQUES THÉVENAZ arxiv:1807.04487v1 [math.rt] 12 Jul 2018 Abstract. For the group of endo-permutation modules of a finite p-group, there is a surjective
More informationDECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY C.M. RINGEL)
DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY CM RINGEL) C BOWMAN, SR DOTY, AND S MARTIN Abstract We give an algorithm for working out the indecomposable
More informationChain complexes for Alperin s weight conjecture and Dade s ordinary conjecture in the abelian defect group case
Chain complexes for Alperin s weight conjecture and Dade s ordinary conjecture in the abelian defect group case Robert Boltje Department of Mathematics University of California Santa Cruz, CA 95064 U.S.A.
More informationFINITE GROUPS OF LIE TYPE AND THEIR
FINITE GROUPS OF LIE TYPE AND THEIR REPRESENTATIONS LECTURE IV Gerhard Hiss Lehrstuhl D für Mathematik RWTH Aachen University Groups St Andrews 2009 in Bath University of Bath, August 1 15, 2009 CONTENTS
More informationA note on the basic Morita equivalences
SCIENCE CHINA Mathematics. ARTICLES. March 204 Vol.57 No.3: 483 490 doi: 0.007/s425-03-4763- A note on the basic Morita equivalences HU XueQin School of Mathematical Sciences, Capital Normal University,
More informationFIRST COHOMOLOGY GROUPS OF CHEVALLEY GROUPS IN CROSS CHARACTERISTIC
FIRST COHOMOLOGY GROUPS OF CHEVALLEY GROUPS IN CROSS CHARACTERISTIC ROBERT M. GURALNICK AND PHAM HUU TIEP Abstract. Let G be a simple Chevalley group defined over F q. We show that if r does not divide
More informationSOURCE ALGEBRAS AND SOURCE MODULES J. L. Alperin, Markus Linckelmann, Raphael Rouquier May 1998 The aim of this note is to give short proofs in module
SURCE ALGEBRAS AND SURCE MDULES J. L. Alperin, Markus Linckelmann, Raphael Rouquier May 1998 The aim of this note is to give short proofs in module theoretic terms of two fundamental results in block theory,
More informationLandau s Theorem for π-blocks of π-separable groups
Landau s Theorem for π-blocks of π-separable groups Benjamin Sambale October 13, 2018 Abstract Slattery has generalized Brauer s theory of p-blocks of finite groups to π-blocks of π-separable groups where
More informationVertex subgroups and vertex pairs in solvable groups
c 2010 Contemporary Mathematics Volume 524, 2010 Vertex subgroups and vertex pairs in solvable groups James P. Cossey Abstract. Green introduced the notion of a vertex of an indecomposable module M of
More informationDecomposition numbers for generic Iwahori-Hecke algebras of non-crystallographic type
Decomposition numbers for generic Iwahori-Hecke algebras of non-crystallographic type Jürgen Müller IWR der Universität Heidelberg Im Neuenheimer Feld 368 D 69120 Heidelberg Abstract In this note we compute
More informationON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS
Submitted exclusively to the London Mathematical Society DOI: 0./S0000000000000000 ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS JOHN N. BRAY and ROBERT A. WILSON Abstract In the Kourovka Notebook,
More informationTHE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS
THE DERIVED CATEGORY OF BLOCKS WITH CYCLIC DEFECT GROUPS 1. Introduction Let O be a complete discrete valuation ring with algebraically closed residue field k of characteristic p > 0 and field of fractions
More informationMODULAR SPECIES AND PRIME IDEALS FOR THE RING OF MONOMIAL REPRESENTATIONS OF A FINITE GROUP #
Communications in Algebra, 33: 3667 3677, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870500243312 MODULAR SPECIES AND PRIME IDEALS FOR THE RING OF MONOMIAL
More informationSELF-EQUIVALENCES OF THE DERIVED CATEGORY OF BRAUER TREE ALGEBRAS WITH EXCEPTIONAL VERTEX
An. Şt. Univ. Ovidius Constanţa Vol. 9(1), 2001, 139 148 SELF-EQUIVALENCES OF THE DERIVED CATEGORY OF BRAUER TREE ALGEBRAS WITH EXCEPTIONAL VERTEX Alexander Zimmermann Abstract Let k be a field and A be
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GROUP ACTIONS ON POLYNOMIAL AND POWER SERIES RINGS Peter Symonds Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 GROUP ACTIONS ON POLYNOMIAL
More informationNOTES ON SPLITTING FIELDS
NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationREPRESENTATION THEORY, LECTURE 0. BASICS
REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite
More informationON LOWER BOUNDS FOR THE DIMENSIONS OF PROJECTIVE MODULES FOR FINITE SIMPLE GROUPS. A.E. Zalesski
ON LOWER BOUNDS FOR THE DIMENSIONS OF PROJECTIVE MODULES FOR FINITE SIMPLE GROUPS A.E. Zalesski Group rings and Young-Baxter equations Spa, Belgium, June 2017 1 Introduction Let G be a nite group and p
More informationCharacters and quasi-permutation representations
International Mathematical Forum, 1, 2006, no. 37, 1819-1824 Characters and quasi-permutation representations of finite p-groups with few non-normal subgroups Houshang Behravesh and Sebar Ghadery Department
More informationOn p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree. Christine Bessenrodt
On p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree Christine Bessenrodt Institut für Algebra, Zahlentheorie und Diskrete Mathematik Leibniz Universität
More informationOn the GL(V ) module structure of K(n) (BV )
On the GL(V ) module structure of K(n) (BV ) By IAN J. LEARY Max-Planck Institut für Mathematik, 53225 Bonn, Germany and BJÖRN SCHUSTER Centre de Recerca Matemàtica, E 08193 Bellaterra (Barcelona), Spain
More informationMath 210B: Algebra, Homework 4
Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,
More informationEXTERIOR AND SYMMETRIC POWERS OF MODULES FOR CYCLIC 2-GROUPS
EXTERIOR AND SYMMETRIC POWERS OF MODULES FOR CYCLIC 2-GROUPS FRANK IMSTEDT AND PETER SYMONDS Abstract. We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group.
More informationSubgroup Complexes and their Lefschetz Modules
Subgroup Complexes and their Lefschetz Modules Silvia Onofrei Department of Mathematics Kansas State University nonabelian finite simple groups alternating groups (n 5) groups of Lie type 26 sporadic groups
More informationPerverse Equivalences and Broué s Conjecture II: The Cyclic Case
Perverse Equivalences and Broué s Conjecture II: The Cyclic Case David A. Craven July 2012 Abstract We study Broué s abelian defect group conjecture for groups of Lie type using the recent theory of perverse
More information. Algebraic tori and a computational inverse Galois problem. David Roe. Jan 26, 2016
Algebraic tori and a computational inverse Galois problem David Roe Department of Mathematics University of Pittsburgh Jan 26, 2016 Outline 1 Algebraic Tori 2 Finite Subgroups of GL n (Z) 3 Tori over R
More informationVERTICES OF SPECHT MODULES AND BLOCKS OF THE SYMMETRIC GROUP
VERTICES OF SPECHT MODULES AND BLOCKS OF THE SYMMETRIC GROUP MARK WILDON Abstract. This paper studies the vertices, in the sense defined by J. A. Green, of Specht modules for symmetric groups. The main
More informationREPRESENTATION THEORY. WEEK 4
REPRESENTATION THEORY. WEEK 4 VERA SERANOVA 1. uced modules Let B A be rings and M be a B-module. Then one can construct induced module A B M = A B M as the quotient of a free abelian group with generators
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationarxiv: v1 [math.gr] 4 Mar 2014
K-stable splendid Rickard complexes Yuanyang Zhou Department of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, P.R. China Email: zhouyy74@163.com arxiv:1403.0661v1 [math.gr]
More informationGENERALIZED BRAUER TREE ORDERS
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 2 GENERALIZED BRAUER TREE ORDERS BY K. W. R O G G E N K A M P (STUTTGART) Introduction. p-adic blocks of integral group rings with cyclic defect
More informationRepresentations of quivers
Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice
More informationOn permutation modules and decomposition numbers for symmetric groups
On permutation modules and decomposition numbers for symmetric groups arxiv:1404.4578v1 [math.rt] 17 Apr 2014 Eugenio Giannelli Abstract We study the indecomposable summands of the permutation module obtained
More informationMATH 527, Topics in Representation Theory
Archive of documentation for MATH 527, Topics in Representation Theory Bilkent University, Spring 2016, Laurence Barker version: 20 May 2016 Source file: arch527spr16.tex page 2: Course specification.
More informationarxiv:math/ v3 [math.qa] 16 Feb 2003
arxiv:math/0108176v3 [math.qa] 16 Feb 2003 UNO S CONJECTURE ON REPRESENTATION TYPES OF HECKE ALGEBRAS SUSUMU ARIKI Abstract. Based on a recent result of the author and A.Mathas, we prove that Uno s conjecture
More informationFinite Group Representations. for the. Pure Mathematician
Finite Group Representations for the Pure Mathematician by Peter Webb Preface This book started as notes for courses given at the graduate level at the University of Minnesota. It is intended to be used
More informationis an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent
Lecture 4. G-Modules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of G-modules, mostly for finite groups, and a recipe for finding every irreducible G-module of a
More informationProjective modules: Wedderburn rings
Projective modules: Wedderburn rings April 10, 2008 8 Wedderburn rings A Wedderburn ring is an artinian ring which has no nonzero nilpotent left ideals. Note that if R has no left ideals I such that I
More informationTopics in Representation Theory University of Cambridge Part III Mathematics 2009/2010
Topics in Representation Theory University of Cambridge Part III Mathematics 2009/2010 Lectured by Dr C.J.B. Brookes (C.J.B.Brookes at dpmms.cam.ac.uk) Typeset by Aaron Chan(akyc2@cam.ac.uk) Last update:
More informationEndotrivial modules. Nadia Mazza. June Lancaster University
Endotrivial modules Nadia Mazza Lancaster University June 2010 Contents Endotrivial modules : background Analysis An exotic turn Final remarks Set-up p a prime; k = k a field of characteristic p; G a finite
More informationREPRESENTATION THEORY WEEK 9
REPRESENTATION THEORY WEEK 9 1. Jordan-Hölder theorem and indecomposable modules Let M be a module satisfying ascending and descending chain conditions (ACC and DCC). In other words every increasing sequence
More informationThe number of simple modules associated to Sol(q)
The number of simple modules associated to Sol(q) Jason Semeraro Heilbronn Institute University of Leicester University of Bristol 8th August 2017 This is joint work with Justin Lynd. Outline Blocks and
More informationCategory O and its basic properties
Category O and its basic properties Daniil Kalinov 1 Definitions Let g denote a semisimple Lie algebra over C with fixed Cartan and Borel subalgebras h b g. Define n = α>0 g α, n = α
More informationThe real root modules for some quivers.
SS 2006 Selected Topics CMR The real root modules for some quivers Claus Michael Ringel Let Q be a finite quiver with veretx set I and let Λ = kq be its path algebra The quivers we are interested in will
More informationTHE GROUP OF ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS
THE GROUP OF ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS JON F. CARLSON, DAVID J. HEMMER, AND NADIA MAZZA Abstract. We complete a classification of the groups of endotrivial modules for
More informationGeneralized Alexander duality and applications. Osaka Journal of Mathematics. 38(2) P.469-P.485
Title Generalized Alexander duality and applications Author(s) Romer, Tim Citation Osaka Journal of Mathematics. 38(2) P.469-P.485 Issue Date 2001-06 Text Version publisher URL https://doi.org/10.18910/4757
More informationThe Terwilliger Algebras of Group Association Schemes
The Terwilliger Algebras of Group Association Schemes Eiichi Bannai Akihiro Munemasa The Terwilliger algebra of an association scheme was introduced by Paul Terwilliger [7] in order to study P-and Q-polynomial
More informationRepresentations Associated to the Group Matrix
Brigham Young University BYU ScholarsArchive All Theses and Dissertations 014-0-8 Representations Associated to the Group Matrix Joseph Aaron Keller Brigham Young University - Provo Follow this and additional
More information6. Dynkin quivers, Euclidean quivers, wild quivers.
6 Dynkin quivers, Euclidean quivers, wild quivers This last section is more sketchy, its aim is, on the one hand, to provide a short survey concerning the difference between the Dynkin quivers, the Euclidean
More informationThus we get. ρj. Nρj i = δ D(i),j.
1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D :
More informationarxiv: v1 [math.rt] 20 Jul 2016
ON REPRESENTATION-FINITE GENDO-SYMMETRIC BISERIAL ALGEBRAS AARON CHAN AND RENÉ MARCZINZIK arxiv:1607.05965v1 [math.rt] 20 Jul 2016 Abstract. Gendo-symmetric algebras were introduced by Fang and Koenig
More informationAuslander-Reiten theory in functorially finite resolving subcategories
Auslander-Reiten theory in functorially finite resolving subcategories arxiv:1501.01328v1 [math.rt] 6 Jan 2015 A thesis submitted to the University of East Anglia in partial fulfilment of the requirements
More informationLecture 6: Etale Fundamental Group
Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and
More informationarxiv: v2 [math.rt] 19 Sep 2013
Strong fusion control and stable equivalences arxiv:1308.5477v2 [math.rt] 19 Sep 2013 Erwan Biland October 3, 2018 Abstract This article is dedicated to the proof of the following theorem. Let G be a finite
More informationINJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to
More informationarxiv: v3 [math.rt] 31 Jul 2015 Introduction
THE EXT ALGEBRA OF A BRAUER GRAPH ALGEBRA EDWARD L GREEN, SIBYLLE SCHROLL, NICOLE SNASHALL, AND RACHEL TAILLEFER Abstract In this paper we study finite generation of the Ext algebra of a Brauer graph algebra
More informationσ = (P 0 < P 1 <... < P k )
EQUIVARIANT K-THEORY AND ALPERIN S CONJECTURE Jacques THÉVENAZ Institut de Mathématiques, Université de Lausanne, CH-1015 Lausanne, Switzerland. Introduction The purpose of this paper is to show that Alperin
More informationSome Classification of Prehomogeneous Vector Spaces Associated with Dynkin Quivers of Exceptional Type
International Journal of Algebra, Vol. 7, 2013, no. 7, 305-336 HIKARI Ltd, www.m-hikari.com Some Classification of Prehomogeneous Vector Spaces Associated with Dynkin Quivers of Exceptional Type Yukimi
More informationDedicated to Helmut Lenzing for his 60th birthday
C O L L O Q U I U M M A T H E M A T I C U M VOL. 8 999 NO. FULL EMBEDDINGS OF ALMOST SPLIT SEQUENCES OVER SPLIT-BY-NILPOTENT EXTENSIONS BY IBRAHIM A S S E M (SHERBROOKE, QUE.) AND DAN Z A C H A R I A (SYRACUSE,
More informationInduction formula for the Artin conductors of mod l Galois representations. Yuichiro Taguchi
Induction formula for the Artin conductors of mod l Galois representations Yuichiro Taguchi Abstract. A formula is given to describe how the Artin conductor of a mod l Galois representation behaves with
More informationImplications of hyperbolic geometry to operator K-theory of arithmetic groups
Implications of hyperbolic geometry to operator K-theory of arithmetic groups Weizmann Institute of Science - Department of Mathematics LMS EPSRC Durham Symposium Geometry and Arithmetic of Lattices, July
More informationFusion Systems: Group theory, representation theory, and topology. David A. Craven
Fusion Systems: Group theory, representation theory, and topology David A. Craven Michaelmas Term, 2008 Contents Preface iii 1 Fusion in Finite Groups 1 1.1 Control of Fusion.................................
More informationCommutative algebra and representations of finite groups
Commutative algebra and representations of finite groups Srikanth Iyengar University of Nebraska, Lincoln Genoa, 7th June 2012 The goal Describe a bridge between two, seemingly different, worlds: Representation
More informationOn Group Ring Automorphisms
On Group Ring Automorphisms MARTIN HERTWECK 1 and GABRIELE NEBE 2 1 Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany. email: hertweck@mathematik.uni-stuttgart.de
More informationFrank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups:
Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups: Definition: The external direct product is defined to be the following: Let H 1,..., H n be groups. H 1 H 2 H n := {(h 1,...,
More informationA note on the Isomorphism Problem for Monomial Digraphs
A note on the Isomorphism Problem for Monomial Digraphs Aleksandr Kodess Department of Mathematics University of Rhode Island kodess@uri.edu Felix Lazebnik Department of Mathematical Sciences University
More informationProjective and Injective Modules
Projective and Injective Modules Push-outs and Pull-backs. Proposition. Let P be an R-module. The following conditions are equivalent: (1) P is projective. (2) Hom R (P, ) is an exact functor. (3) Every
More informationA NOTE ON SPLITTING FIELDS OF REPRESENTATIONS OF FINITE
A NOTE ON SPLITTING FIELDS OF REPRESENTATIONS OF FINITE GROUPS BY Let @ be a finite group, and let x be the character of an absolutely irreducible representation of @ An algebraic number field K is defined
More informationCharacter values and decomposition matrices of symmetric groups
Character values and decomposition matrices of symmetric groups Mark Wildon Abstract The relationships between the values taken by ordinary characters of symmetric groups are exploited to prove two theorems
More informationOn certain family of B-modules
On certain family of B-modules Piotr Pragacz (IM PAN, Warszawa) joint with Witold Kraśkiewicz with results of Masaki Watanabe Issai Schur s dissertation (Berlin, 1901): classification of irreducible polynomial
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationLECTURE NOTES AMRITANSHU PRASAD
LECTURE NOTES AMRITANSHU PRASAD Let K be a field. 1. Basic definitions Definition 1.1. A K-algebra is a K-vector space together with an associative product A A A which is K-linear, with respect to which
More informationA COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998)
A COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998) A category C is skeletally small if there exists a set of objects in C such that every object
More informationSaturated fusion systems as stable retracts of groups
Massachusetts Institute of Technology Department of Mathematics Saturated fusion systems as stable retracts of groups Sune Precht Reeh MIT Topology Seminar, September 28, 2015 Slide 1/24 Outline 1 Bisets
More informationON LOCALLY SEMI-SIMPLE REPRESENTATIONS OF QUIVERS
ON LOCALLY SEMI-SIMPLE REPRESENTATIONS OF QUIVERS CALIN CHINDRIS AND DANIEL LINE ABSTRACT. In this paper, we solve a problem raised by V. ac in [6] on locally semi-simple quiver representations. Specifically,
More informationVersal deformation rings of modules over Brauer tree algebras
University of Iowa Iowa Research Online Theses and Dissertations Summer 05 Versal deformation rings of modules over Brauer tree algebras Daniel Joseph Wackwitz University of Iowa Copyright 05 Daniel Joseph
More informationSUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More information