REPRESENTATION THEORY CONCEPTS TECHNEAQUES AND TRENDS MATH SEMINAR UQU REPRESENTATION THEORY CONCEPTS TECHNEAQUES AND TRENDS Ahmed A. Khammash E-mail: aakhammash@uqu.edu.sa MATHEMATICS AND ITS APPLICATIONS IMAM UNIV MARCH 2011

MATH SEMINAR UQU OUTLINE (1) REPRESENTATIONS AND MODULES (2) INDECOMPOSABLE MODULES (3) INDUCED MODULES & HECKE ALGEBRAS (4) DECOMPOSITION: THE POWER OF THE ENDOMORPHISMS (5) BRAUER-FITTING CORRESPONDENCE (6) PSEAUDOBLOCKS (7) CONNECTION WITH TENSOR ALGEBRAS

HISTORY AND APPLICATIONS MATH SEMINAR UQU HISTORY AND APPLICATIONS Representation theory of groups was initiated in 1896 in a question raised by F. Frobenius (1849-1917) to R. Dedekind (1831-1916) concerning the former's observation on the group determinant [CC, §3]. Nowdays representation theory of groups and its tools have become so diversified with wide range of applications we mension some of them.  Applications to the structure of groups ( Burnside (p,q)-theorem) [CC]  Solid state and particle physics ( See [HG])  Molecular Vibration (See [JL] and [GJ2])  Algebraic Cryptography ( See [NK])  Coding Theory ( See for example [TZ] and [WW])  Chemistry (Spectroscopy) ( See for example [BJ]) ….. etc.

[CR] REPRESNTATIONS AND MODULES MATH SEMINAR UQU [CR] REPRESNTATIONS AND MODULES A representation of a group over a field is an algebra homo where is a -space, is the group algebra of over and is the algebra of -linear maps is the degree of the representation. If , choose a basis for we get a matrix rep (afforded by ). The matrix rep defines a -action on an n-dimensional -space extended linearly to define a -module. Each step can be reversed and so STUDYING REPRESENTATIONS OF OVER STUDYING -MODULES

IRREDUCIBLE MODULES  EXTENSIONS MATH SEMINAR UQU IRREDUCIBLE MODULES  EXTENSIONS If is a -module affording matrix representation and then is a -submodule of (Not. if is closed under the -action (or invariant under ) If then we may choose a suitable basis for to get the following block triangulated form for the matrix rep is an extension of by the quotient module is irreducible (simple) if it has no proper submodule.

MATH SEMINAR UQU From Jordan-Holder theorem it follows that has a comp. series and , have the following shapes where is irreducible rep afforded by . So irreducible matrix representations [simple -modules] are the building blocks for all representations [ -modules].

P1. Determine all simple -modules MATH SEMINAR UQU SO THE PROBLEMS IN STUDYING REPRESENTATIONS CAN BE SUMMARIZED IN THE FOLLOWING P1. Determine all simple -modules P2. Determine all extensions [ short exact sequence -modules] of two simple -modules and .

P1 Dealing with this problem depends on MATH SEMINAR UQU P1 Dealing with this problem depends on If , then character theory [II] is the tool for P1, since the character of defined by Completely determines so P1 reduces to the determination of irreducible characters of . An early result in this direction is the work of J. A. Green [JG1] in which he determined irreducible complex characters for the finite general linear groups (2) If & then every - module is completely reducible (direct sum of simples) . But if then the character as defined in (1) does not characterize the representation, instead we have Brauer (modular) characters which are defined on p-regular conj. classes of whose number is the same as number irreducible modular characters.

Alperin (Weight) Conjecture (1986) Dade Conjecture Broue's Conjecture MATH SEMINAR UQU THE MODULAR CHARACTERS are partitioned into blocks (2 sided ideal of ) associated with certain invariants such as defects and defect groups. There are several (not settled) conjectures dealing with modular characters in a block by localizing to p-subgroups. The most famous conjectures are : McKay Conjecture (1971) Alperin (Weight) Conjecture (1986) Dade Conjecture Broue's Conjecture See [CPS] for an overview and realizations of those conjectures. The book of Alperin [JA] provides a comprehensive treatment for local representation theory.

Concerning the second problem P2 MATH SEMINAR UQU Concerning the second problem P2 If or then every -module is projective and so the extension problem is trivial in this case. If then P2 is connected with cohomology of . The extension and torsion functors Ext and Tor as well as the derived functors are some of the tools for this problem [DB]. Another relevant subject to this problem is the cohomology ring whose structure and some properties were studied by Evens [IE]. The book of J. Carlson [JC] provides many worked examples of cohomology rings.

INDECOMPOSABLE MODULES MATH SEMINAR UQU INDECOMPOSABLE MODULES A -module has a direct sum decomposition if every can be written uniquely as .. is indecomposable if it has no nontrivial such decomposition. If is f.g. then Krull-Schmidt theorem (see [CR], I6.12) ensures the existence of a direct sum -decomposition Hence the indecomposable modules are the building units for the category of -modules and so studying representations of groups is equivalent to the investigation of indec. modules. It follows then that the problems in representation theory (beside the extension problem) can be reduced to the following P3. Classify all indecomposable -modules

MATH SEMINAR UQU GREEN ALGEBRA The Green algebra is the -algebra generated by all indec. -modules with operations defined by direct sum and tensor product. A typical connection between the structure of and the structure of the group is the following D. HIGMAN [DG]: is f.g. if and only if is cyclic.  For a recent treatment on the structure of the Green ring see for example [LH]. Algebras with a finite number of indecomposable modules are called algebras of finite representation type. One of the main tools for classifying algebras with respect to this property is the Auslander-Reiten theory. Almost split sequences and representation of quivers are the main subjects in this field. For a general overview on Auslander-Reiten theory see [AR].

INDUCED MODULES AND HECKE ALGEBRAS MATH SEMINAR UQU INDUCED MODULES AND HECKE ALGEBRAS INDUCED MODULES: HECKE ALGEBRAS : Case ; , permutation - module and the endomorphism algebra is the Hecke algebra . MOTIVATIONS: (1) ; is a finite group of Lie type and is its Sylow p-subgroup. Then the permutation module provides a parameterization for the simple modules in the natural characteristic (see [FR] [JG3],[MC]) and investigating the structure of the Hecke algebra is an effective method for analyzing (§3). (2) , ; row stabilizer of a Young tableaux. Then the permutation module plays a central role in the rep theory of [GJ1]. Specht and Young modules parametrize the simple modules & appear as submodules and direct summands of .

DECOMPOSITION: THE POWER OF THE ENDOMORPHISM ALGEBRA ’MATH SEMINAR UQU DECOMPOSITION: THE POWER OF THE ENDOMORPHISM ALGEBRA Suppose that . Define the -maps It is clear that they satisfy the following identities THEOREM: if and only if for each -maps as above satisfying the identities (1)-(3). 

IDEMPOTENTS If A is a f. d. algebra over . Then is idempotent if . MATH SEMINAR UQU IDEMPOTENTS If A is a f. d. algebra over . Then is idempotent if . Constructing idempotent in is an essential step towards decomposing the module according to the following: THEOREM3: Let be an idempotent, then (1) is idempotent in and . (2) algebra isomorphism.  POLYNOMIAL ALGEBRAS AND IDEMPOTENTS Since A is finite dimensional, if there must be a positive integer n such that the set is linearly dependant. If n is the least with this property, then we have and so is the minimal polynomial for .

MTH SEMINAR UQU THEOREM Let be a finite dimensional k-algebra with and let . Suppose that is the minimal polynomial of , where are non-constant in k[X] with no common divisor. By Euclid’s algorithm : . Let . . Then both , are non-zero and is an orthogonal idempotent decomposition in .

Brauer-FITTING CORRESPONDENCE MATH SEMINAR UQU Brauer-FITTING CORRESPONDENCE The Brauer-Fitting correspondence relates the structure of an A-module Y (A is f.d. algebra) with the representations of the endomorphism algebra . It is known in principle that Y has direct sum decomposition For each indecomposable summand there corresponds an irreducible representation where giving the (Brauer-Fitting) correspondence: (*) V ↔ S For application of BFC to the properties and the structure of See [AK1], [AK2] We are interested in the relating this correspondence with block theory

Connection With Block Theory MATH SEMINAR UQU Connection With Block Theory For an algebra if , write if belong to the same block of . THEOREM1 (([AK3], Theorem1]: Suppose that and are two pair of indecomposable modules related by the BF correspondence. Then The converse of theorem1 is not in general true consider the following example EXAMPLE : at characteristic 2 has 2 blocks with 4 simples on the other hand the induced module ; has the following decomposition

Hence E(Y) (which is of dimension 6) has 4 Linear representations MATH SEMINAR UQU Hence E(Y) (which is of dimension 6) has 4 Linear representations Distributed into 3 blocks Hence although So we seek a parameterization of the indec summands compatible with the (Brauer) block distribution of the endomorphism algebra

THE PSEUDOBLOCKS DEFINITION: Define the equiv. relation on the set MATH SEMINAR UQU THE PSEUDOBLOCKS DEFINITION: Define the equiv. relation on the set of the indecomposable summands of the module as follows : iff such that The equivalence classes of are called the pseudoblocks of THEOREM2: ([AK3]) iff EXAMPLE: In the previous example has 3 pseudoblocks

CONNECTION WITH TENSOR ALGEBRAS MATH SEMINAR UQU CONNECTION WITH TENSOR ALGEBRAS B-F correspondence and pseudo-blocks of endomorphism algebras are both compatible with the external tensor product of modules and algebras in the sense of the following two theorems THEOREM3: If are two f.d. algebras and is -module . If having BF correspondences then With BF correspondence

MATH SEMINAR UQU , THEOREM4 [AK4] If , then iff

Qs & Further Investigations MATH SEMINAR UQU Qs & Further Investigations FIND Invariants for the pseaudo blocks parallel to the ones for Brauer blocks such as defect , defect group … etc WHAT are the conditions ( either on , or ) so that # = # ( For example this is true if either or is semisimple ) (3) DETERMINE the pseaudoblocks distribution of the Young modules for Symmetricg groups (4) Connection with Auslander-Reiten quiver.

MATH SEMINAR UQU REFERENCES [CC] C. Curtis, Pioneers Of Representation Theory: Frobenius, Burnsie, Schur, and Brauer, History of Mathematics Vol. 15, AMS & LMS 1999 [HG] H. Georgi, Lie Algebras in Particle Physics, Addison-Wesley, 1995 [JL] G. James and M Liebeck, Representations and characters of groups, Cambridge Math. Textbooks 1993 [GJ1] G. James, The representation theory of the symmetric groups, Springer Lecture Notes 682, Berlin 1978. [GJ2] G. James, "The representation theory for the Buckminsterfullerene," J. Alg. 167(1994)803-820 [JG3] J. A. Green, On a theorem of H. Sawada, J. LMS 18(1978),247-252. [NK] N. Koblitz, Algebraic aspects of cryptography, Algorithms and comp. in math. Vol.3, Springer, 1998 [TZ] J. Tillich and G. Zemor, Optimal cycle codes constructed from Ramanujan graphs, SIAM J. On Disc. Math. 10(1997) , 447-459 [WW] H. Ward and J. Wood, Characters and equivalence of codes, J. Combin. Theory A73, 348-352. [BJ] B. Judd, "Lie groups in Atomic and molecular spectroscopy", SIAM J on Applied Math 25(1973) 186-192

MATH SEMINAR UQU [CPS] M. Collins, B. Parshall and L. Scott, ed. ,Modular Representation Theory of Finite Groups , Walter de Gruyter, New York 2001. [JA] J. Alperin, Local Representation Theory, Cambridge studies in advanced mathematics #11, Cambridge University Press 1986 [LE] L. Evens, The cohomology rings of a finite group, Trans. AMS 101(1961), 224-239. [JC] J. Carlson, Cohomology of finite groups, Kluwer Academic Publishers, 2003 [CR] C. Curtis and I. Riener, Methods of Representation Theory Vol.1 &2, Wiley Interscience, New York 1981 &1985 [LH] L. Haberle, The species and idempotents of the Green algebra of a finite group with a cyclic Sylow subgroup, Journal of Algebra 320 (2008) 3120–3132 [AR] M. Auslander, I. Reiten, Representation theory of Artin algebras II, III, Comm. Algebra 1(1974), 269-310 and Comm. Algebra 3(1975), 239-294. [FR] F. Richen, Modular representations of split BN-pairs, Trans. AMS 140(1969), 435-460 [MC] M. Cabanes , Extension groups for modular Hecke algebras, J. Fac. Sci. Univ. Tokyo Math. Sect. IA Math. 36 (1989 ) , 347-362 [GJ1] G. James, The representation theory of the symmetric groups, Springer Lecture Notes 682, Berlin 1978.

MATH SEMINAR UQU [AK1] A. Khammash, Functors and projective summands of permutation modules , J. Algebra 163(3) (1994), 729-738. [AK2] A. Khammash, Hecke algebras and the socle of projective indecomposable modules , J. Algebra 192 , 294-302 (1997) [AK3] A. Khammash, On the blocks of endomorphism algebras, Intern. Math. Journal, Vol.2, 2002, no.6, 535-542. [AK4] A. Khammash, Brauer-Fitting correspondence On Tensor Algebra , Submitted for publication.

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