Computational Aspects of the Isomorphism Problem

The aim of this article is to lead the reader on a journey through the representation theory of finite groups of Lie type and Hecke algebras. We will present some basic results obtained in recent years, explain the ideas behind them, and give lots of examples; proofs are usually omitted but we provide explicit references to an extensive bibliography.

[1]  Gordon James,et al.  The Decomposition Matrices of GLn(q) for n ≤ 10 , 1990 .

[2]  Steve Linton,et al.  On vector enumeration , 1993 .

[3]  Geck Meinolf.  Generalized gelfand-graev characters for steinberg's triality groups and their applications , 1991 .

[4]  S. Sehgal.  On the isomorphism of group algebras , 1967 .

[5]  I. Grojnowski.  Representations of affine Hecke algebras (and affine quantum GL n ) at roots of unity , 1994 .

[6]  G. James,et al.  Blocks and Idempotents of Hecke Algebras of General Linear Groups , 1987 .

[7]  A Sylowlike theorem for integral group rings of finite solvable groups , 1993 .

[8]  R. Dipper,et al.  Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs , 1998, math/9808096.

[9]  I. Isaacs.  Character Theory of Finite Groups , 1976 .

[10]  Lee Klingler,et al.  Construction of a counterexample to a conjecture of zassenhaus , 1991 .

[11]  Matthew J. Richards.  Some decomposition numbers for Hecke algebras of general linear groups , 1996 .

[12]  G. Lehrer,et al.  On Harish-Chandra Induction and Restriction for Modules of Levi Subgroups , 1994 .

[13]  G. James,et al.  Representations of Hecke algebras of type Bn , 1992 .

[14]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .

[15]  W. Deskins.  Finite Abelian groups with isomorphic group algebras , 1956 .

[16]  A. Caranti,et al.  The Modular Group Algebras of P-Groups of Maximal Class , 1988, Canadian Journal of Mathematics.

[17]  Gerhard Hiss,et al.  Supercuspidal Representations of Finite Reductive Groups , 1996 .

[18]  R. Dipper.  On quotients of Hom-functors and representations of finite general linear groups, I , 1990 .

[19]  J. Carlson.  Periodic Modules over Modular Group Algebras , 1977 .

[20]  Roger W. Carter,et al.  Representation theory of the 0-Hecke algebra , 1986 .

[21]  G. Lehrer,et al.  Induced cuspidal representations and generalised Hecke rings , 1980 .

[22]  M. Geck.  Kazhdan-Lusztig cells and decomposition numbers , 1998 .

[23]  Young Characters on Coxeter Basis Elements of Iwahori-Hecke Algebras and a Murnaghan-Nakayama Formula , 1994 .

[24]  Katsushi Waki.  A Note on the Decomposition Numbers ofSp(4, q) , 1996 .

[25]  K. Roggenkamp,et al.  A counterexample for the isomorphism-problem of polycyclic groups☆ , 1995 .

[26]  Arun Ram,et al.  A Frobenius formula for the characters of the Hecke algebras , 1991 .

[27]  Götz Pfeiffer,et al.  On the Irreducible Characters of Hecke Algebras , 1993 .

[28]  R. Dipper,et al.  Harish‐Chandra Vertices and Steinberg's Tensor Product Theorems for Finite General Linear Groups , 1997 .

[29]  M. Dokuchaev,et al.  Finite Subgroups in Integral Group Rings , 1996, Canadian Journal of Mathematics.

[30]  G. James,et al.  Representations of general linear groups , 1984 .

[31]  J. Du,et al.  Harish-Chandra vertices. , 1993 .

[32]  P. Fong,et al.  The blocks of finite classical groups. , 1989 .

[33]  R. Sandling.  The modular group algebra of a central-elementary-by-abelianp-group , 1989 .

[34]  P. Fong,et al.  Generalized Harish-Chandra theory for unipotent characters of finite classical groups , 1986 .

[35]  G. James,et al.  Hecke Algebras of Type Bn at Roots of Unity , 1995 .

[36]  Gordon James,et al.  Unipotent representations of the finite general linear groups , 1982 .

[37]  L. Scott,et al.  Stratifying Endomorphism Algebras Associated to , 1998 .

[38]  Andrew Mathas,et al.  Canonical bases and the decomposition matrices of Ariki--Koike algebras , 1996, q-alg/9607005.

[39]  G. Peterson.  Automorphisms of the integral group ring of _ , 1976 .

[40]  M. Geck,et al.  Automorphisms of Generic Iwahori–Hecke Algebras and Integral Group Rings of Finite Coxeter Groups☆☆☆ , 1997 .

[41]  George Lusztig,et al.  On a theorem of Benson and Curtis , 1981 .

[42]  Martin Wursthorn,et al.  Isomorphisms of Modular Group Algebras: An Algorithm and its Application to Groups of Order , 1993, J. Symb. Comput..

[43]  M. Geck.  Basic sets of Brauer characters of finite groups of Lie type, III , 1994 .

[44]  M. Geck.  Irreducible brauer characters of the 3-dimensional special unitary groups in non-defining characteristic ∗ , 1990 .

[45]  Peter Landrock,et al.  Principal 2-blocks of the simple groups of Ree type , 1980 .

[46]  Toshiaki Shoji,et al.  Character sheaves and almost characters of reductive groups, II , 1995 .

[47]  Gordon James,et al.  Representations of Hecke Algebras of General Linear Groups , 1986 .

[48]  Alain Lascoux,et al.  Une conjecture pour le calcul des matrices de décomposition des algèbres de Hecke de type A aux racines de l'unité , 1995 .

[49]  Brian Parshall,et al.  Quantum Linear Groups , 1991 .

[50]  Michel Broué,et al.  Bloc et séries de Lusztig dans un groupe réductif fini. , 1989 .

[51]  Jean-Yves Thibon,et al.  Canonical bases of q-deformed Fock spaces , 1996 .

[52]  G. Hiss,et al.  Modular representation of finite groups of Lie type in non-defining characteristic , 1997 .

[53]  D. Kazhdan,et al.  Representations of Coxeter groups and Hecke algebras , 1979 .

[54]  S. Sehgal,et al.  Isomorphism of modular group algebras , 1972 .

[55]  C. Pallikaros,et al.  Representations of Hecke Algebras of Type Dn , 1994 .

[56]  P. Deligné,et al.  Duality for representations of a reductive group over a finite field , 1982 .

[57]  Meinolf Geck,et al.  The decomposition numbers of the hecke algebra of typeF4 , 1991 .

[58]  K. Roggenkamp,et al.  Projective limits of group rings , 1993 .

[59]  J. Müller,et al.  Decomposition Numbers for Generic Iwahori–Hecke Algebras of Noncrystallographic Type☆ , 1997 .

[60]  Michel Broué,et al.  Représentations unipotentes génériques et blocs des groupes réductifs finis , 1993 .

[61]  M. Cabanes.  Finite Reductive Groups: Related Structures and Representations , 1996 .

[62]  M. Geck,et al.  Cuspidal Unipotent Classes and Cuspidal Brauer Characters , 1996 .

[63]  Meinolf Geck,et al.  The decomposition numbers of the hecke algebra of typeF4 , 1991 .

[64]  G. Hiss,et al.  2-blocks and 2-modular characters of the Chevalley groups ₂() , 1992 .

[65]  R. Green.  Hyperoctahedral Schur Algebras , 1997 .

[66]  R. Lyons,et al.  Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups , 1990 .

[67]  P. Fong,et al.  Brauer trees in classical groups , 1990 .

[68]  Gerhard Hiss,et al.  The Brauer trees of the exceptional Chevalley groups of types F4 and 2E6 , 1998 .

[69]  R. Dipper,et al.  Modular Harish-Chandra theory I , 1992 .

[70]  Gerhard Hiss,et al.  Harish‐Chandra Series of Brauer Characters in a Finite Group with a Split BN‐Pair , 1993 .

[71]  Michel Broué,et al.  Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis , 1992 .

[72]  G. Hiss,et al.  Decomposition numbers of finite classical groups for linear primes. , 1997 .

[73]  Hebing Rui.  On Endomorphism Algebras Arising from Multiparameter Hecke Algebras , 1997 .

[74]  G. James,et al.  $q$-tensor space and $q$-Weyl modules , 1991 .

[75]  Gerhard Hiss,et al.  On the decomposition numbers of G2(q) , 1989 .

[76]  D. L. White,et al.  Decomposition numbers of Sp(4, q) for primes dividing q ± 1 , 1990 .

[77]  K. Roggenkamp.  The isomorphism problem for integral group rings of finite groups , 1992 .

[78]  D. Quillen,et al.  The Spectrum of an Equivariant Cohomology Ring: II , 1971 .

[79]  G. Hiss,et al.  Basic sets of Brauer characters of finite groups of Lie type. , 1991 .

[80]  Alain Lascoux,et al.  Hecke algebras at roots of unity and crystal bases of quantum affine algebras , 1996 .

[81]  Gordon James,et al.  The Irreducible Representations of the Finite General Linear Groups , 1986 .

[82]  L. Scott.  Recent Progress on the Isomorphism Problem , 1987 .

[83]  G. Lusztig,et al.  Induced Unipotent Classes , 1979 .

[84]  S. Donkin.  The q-Schur algebra , 1998 .

[85]  M. Geck.  Basic Sets of Brauer Characters of Finite Groups of Lie Type II , 1993 .

[86]  K. Waki,et al.  Decomposition Numbers ofSp(4, q)☆ , 1998 .

[87]  F. Bleher.  FINITE GROUPS OF LIE TYPE OF SMALL RANK , 1999 .

[88]  G. Hiss,et al.  Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type , 1996 .

[89]  M. Geck.  Brauer trees of hecke algebras , 1992 .

[90]  Everett C. Dade.  Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps , 1971 .

[91]  Tom Halverson,et al.  Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of classical type , 1996 .

[92]  Gerhard Hiss,et al.  The brauer trees of the ree groups , 1991 .

[93]  L. Scott,et al.  Isomorphisms of p-adic group rings , 1987 .

[94]  George Lusztig,et al.  Characters of reductive groups over a finite field , 1984 .

[95]  S. Ariki.  On the decomposition numbers of the Hecke algebra of $G(m, 1, n)$ , 1996 .

[96]  S. Sehgal.  Torsion Units in Group Rings , 1984 .

[97]  Irving Reiner,et al.  Methods of Representation Theory , 1981 .

[98]  Meinolf Geck,et al.  ‘Good’ Elements of Finite Coxeter Groups and Representations of Iwahori–Hecke Algebras , 1997 .

[99]  G. Murphy,et al.  On the representation theory of the symmetric groups and associated Hecke algebras , 1992 .

[100]  The affine q-Schur algebra. , 1997, q-alg/9705015.

[101]  M. Geck.  On the decomposition numbers of the finite unitary groups in non-defining characteristic , 1991 .

[102]  Richard Dipper,et al.  On the decomposition numbers of the finite general linear groups. II , 1985 .

[103]  C. W. Curtis,et al.  On the degrees and rationality of certain characters of finite Chevalley groups , 1972 .

[104]  G. James,et al.  Cyclotomic q–Schur algebras , 1998 .

[105]  M. Salim,et al.  The Modular Group Algebra Problem for Small p-Groups Of Maximal Class , 1996, Canadian Journal of Mathematics.

[106]  Meinolf Geck,et al.  On The Character Values of Iwahori–Hecke Algebras of Exceptional Type , 1994 .

[107]  F. Bleher.  Tensor products and a conjecture of Zassenhaus , 1995 .

[108]  G. James,et al.  Identification of the irreducible modular representations of GLn(q) , 1986 .

[109]  G. Higman,et al.  The Units of Group‐Rings , 1940 .

[110]  G. Hiss,et al.  Cuspidal Unipotent Brauer Characters , 1994 .

[111]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[112]  Jochen Gruber.  Green Vertex Theory, Green Correspondence, and Harish-Chandra Induction , 1996 .

[113]  R. Sandling.  The modular group algebra problem for metacyclic -groups , 1996 .

[114]  L. Scott,et al.  Stratifying Endomorphism Algebras Associated to Hecke Algebras , 1998 .

[115]  M. Salim,et al.  The modular group algebra problem for groups of order p5 , 1996, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[116]  S. Sehgal,et al.  Automorphisms of the integral group rings of some wreath products , 1991 .

[117]  N. Iwahori.  On the structure of a Hecke ring of a Chevalley group over a finite field , 1964 .

[118]  Gordon James,et al.  A q‐Analogue of the Jantzen–Schaper Theorem , 1997 .

[119]  S. Sehgal,et al.  Isomorphism of group rings , 1983 .

[120]  H. H. Andersen,et al.  Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p : independence of p , 1994 .