Modular representation theory: New trends and methods.

*(English)*Zbl 0564.20004
Lecture Notes in Mathematics. 1081. Berlin etc.: Springer-Verlag. XI, 231 p. DM 31.50; $ 11.10 (1984).

This is a very timely book, to be welcomed by those interested in modular representation theory. It deals with the totality of finitely generated kG-modules, where k is a field of characteristic p and G a finite group.

The first principal worker in this area was J. A. Green who introduced the Green ring A(G), vertices and sources and the transfer theorems; A(G) is the \({\mathbb{C}}\)-algebra with \({\mathbb{C}}\)-basis the indecomposable kG- module classes, multiplication coming from tensor products.

The initial emphasis is upon finding structure in A(G). The notion of relative projectiveness gives rise to ideal direct summands spanned by relative projective modules. This is refined further by the decomposition of semisimple finite dimensional subalgebras A(G,triv) spanned modules with trivial source and A(G,cyc) spanned by classes with cyclic vertex (this latter is an ideal direct summand). Semisimplicity is proved by finding species of such subalgebras A, i.e. \({\mathbb{C}}\)-algebra homomorphisms \(s: A\to {\mathbb{C}}\); for the projective ideal A(G(1)) spanned by indecomposable projective kG-module classes, the species s coincide with usual Brauer characters, one species for each p-regular conjugacy class of G. Each species has an origin (a hypoelementary group) and a vertex (a p-subgroup) determined up to conjugacy in G. \(\psi\)-operations are used to construct powers \(s^ n\) of species; \(\psi^ n\) is a ring endomorphism of A(G) and the value \((s^ n,y)\) of \(s^ n\) on y in A(G) is given by \((s^ n,y)=(s,\psi^ n(y))\); for a Brauer species s above given by character values taken at the p-regular element x of G, \(s^ n\) is the Brauer species evaluated on \(x^ n\). \(\lambda\)-operators are defined in terms of the \(\psi^ n\) and make A(G) into a \(\lambda\)-ring.

Special attention is paid to permutation representations on sets S. The notion of a module V being S-projective handles relative projectiveness with respect to a set of subgroups (sub-conjugacy closed) of G; the set comes from the stabilizers of the transitive components of S. The glue of a short exact sequence \(0\to V''\to V\to V'\to 0\) is V-V’-V” considered as an element of A(G). The S-projectives span an ideal direct summand A(G,S) with complement \(A_ 0(G,S)\) spanned by the glues of S-split short exact sequences; the proof of this involves tensor induction. Let \(E(S)=E_ G(S)=End_{kG}(kS)\) be the endomorphism ring of the permutation module kS. A k-basis of E(S) is given by ”suborbit maps” corresponding to G-orbits in \(S\times S\); each suborbit map has a defect group; each primitive idempotent in E(S) has a defect group; The Brauer homomorphism establishes a one-to-one correspondence between primitive idempotents of E(S) of defect group D and equivalence classes of those of \(E_ N(S^ D)\), where \(N=N_ G(D)\) and \(S^ D\) is the set of points of S fixed by D. Brauer’s classical main theorem is then recovered by \(S=G\) acted upon by \(G\times G\), upon suitable interpretation.

The Auslander-Reiten theory of almost split exact sequences is next applied to (the Artin algebra) kG. If V is indecomposable, the atom \(\tau\) (V) is defined to be soc(V) when V is projective and otherwise to be the glue of an almost split exact sequence associated to V. Two closely related inner products \(<U,V>\) and (U,V) are considered on A(G) each defined from \(Hom_{kG}(U,V)\). Both products are nonsingular with dual bases being given in terms of the atoms \(\tau\) (V). Any module may be regarded as a formal sum of atoms, namely its composition factors and the glues holding it together. When information is projected onto certain finite dimensional direct summands A of A(G), A spanned by indecomposable classes V, there arise two tables, the atom table and the representation table, these being the values of the species of A on the atoms \(\tau\) (V) and the indecomposables V.

Carlson’s idea of associating a variety \(X_ G(V)\) to a module V is introduced. This is a subvariety of \(X_ G=spec H^{ev}(G,k)\), the spectrum of the even cohomology ring of G. This last is a commutative noetherian graded ring and \(H^*(G,V)\) is a graded module over it; \(X_ G(V)\) comes from the annihilator Ann(V) of \(H^*(G,V)\) in \(H^{ev}(G,k)\). The ”dimension” or complexity \(C_ X(V)\) is the order of the pole at \(t=1\) of the rational function \(\eta_ V(t)\) describing the dimensions of the terms of a minimal projective resolution of V. It is also shown that the residue of \(\eta_ V(t)\) at \(t=1\) is a positive integer \(\eta\) (V).

The properties of \(X_ G(V)\) are determined by its restrictions \(X_ E(V\downarrow E)\), as E runs through nonconjugate elementary subgroups E of G. When G is elementary, \(X_ G(V)\) can be described in terms of the group algebra kG. The elementary subgroups E of G give rise to the Quillen stratification of \(X_ G(V)\). If X is a homogeneous subset of \(X_ G\), then A(G,X) spanned by those V with \(X_ G(V)\subseteq X\) is an ideal of A(G). If V is indecomposable, then the corresponding projective variety \(\bar X_ G(V)\) is topologically connected. Many other properties of these \(X_ G(V)\) are given.

In the final sector of the text, almost split exact sequences are used to construct a locally finite graph with the indecomposables as vertices. Projectives are disregarded and the result is a stable representation quiver. The connected components of this quiver give a refinement of the classification of modules V in as much as if U and V are in the same connected component of the quiver, then \(X_ G(U)=X_ G(V).\)

The Riedtmann structure theorem shows that a connected stable representation quiver is uniquely expressible as a quotient of the universal covering quiver of a tree, by an ”admissible” group of automorphisms. By means of the invariant \(\eta\) (V) above, this tree class is shown to be either a Dynkin diagram (finite or infinite) or a Euclidean diagram. Following Webb, each of these is investigated in turn.

In addition to exercises being provided at the end of the sections, there is an appendix of 48 pages of calculations on finite groups over relevant characteristics. These include the indecomposables in the finite and tame cases, the species and invariants associated to them, the even cohomology ring, the atom and representation tables, the quivers and the tree class. The groups include \(C_ 2\), \(C_ 2\times C_ 2\), \(D_{2^ n}\), \(Q_ 8\), \(C_ p\), groups of orders \(p^ 2\) and \(p^ 3\), alternating groups up to \(A_ 7\) and their associated groups, \(L_ 3(2)\) and \(M_{11}.\)

There are two criticisms of the book. Firstly, it is tough to read; proofs are at research paper level and several sections have appeared as research papers (The reviewer has no complaint about sections on cohomology of groups, spectral sequences and Bockstein and Steenrod Operations; indeed the pictures painted by the author are clearer than those obtained from many standard texts.) Secondly, in such a broad and almost definitive survey of the area, more detail should have been given to attribution of results and proofs to their originators; it is true that general acknowledgements are given in places and the bibliography is extensive, but the author, having such a detailed knowledge of the field, appeared to be in a good position to give a critical appraisal of the work in the field over the last twenty years.

As the author says in the introduction, the book has remarkably little overlap with material currently in textbook form. Further he has pinpointed throughout the text many unsolved interesting questions. He has certainly succeeded in his hope to impress upon the reader that the three topics ”representation rings”, ”almost split exact sequences” and ”complexity and cohomology varieties” are closely connected and in his hope to encourage further investigation of their interplay.

The first principal worker in this area was J. A. Green who introduced the Green ring A(G), vertices and sources and the transfer theorems; A(G) is the \({\mathbb{C}}\)-algebra with \({\mathbb{C}}\)-basis the indecomposable kG- module classes, multiplication coming from tensor products.

The initial emphasis is upon finding structure in A(G). The notion of relative projectiveness gives rise to ideal direct summands spanned by relative projective modules. This is refined further by the decomposition of semisimple finite dimensional subalgebras A(G,triv) spanned modules with trivial source and A(G,cyc) spanned by classes with cyclic vertex (this latter is an ideal direct summand). Semisimplicity is proved by finding species of such subalgebras A, i.e. \({\mathbb{C}}\)-algebra homomorphisms \(s: A\to {\mathbb{C}}\); for the projective ideal A(G(1)) spanned by indecomposable projective kG-module classes, the species s coincide with usual Brauer characters, one species for each p-regular conjugacy class of G. Each species has an origin (a hypoelementary group) and a vertex (a p-subgroup) determined up to conjugacy in G. \(\psi\)-operations are used to construct powers \(s^ n\) of species; \(\psi^ n\) is a ring endomorphism of A(G) and the value \((s^ n,y)\) of \(s^ n\) on y in A(G) is given by \((s^ n,y)=(s,\psi^ n(y))\); for a Brauer species s above given by character values taken at the p-regular element x of G, \(s^ n\) is the Brauer species evaluated on \(x^ n\). \(\lambda\)-operators are defined in terms of the \(\psi^ n\) and make A(G) into a \(\lambda\)-ring.

Special attention is paid to permutation representations on sets S. The notion of a module V being S-projective handles relative projectiveness with respect to a set of subgroups (sub-conjugacy closed) of G; the set comes from the stabilizers of the transitive components of S. The glue of a short exact sequence \(0\to V''\to V\to V'\to 0\) is V-V’-V” considered as an element of A(G). The S-projectives span an ideal direct summand A(G,S) with complement \(A_ 0(G,S)\) spanned by the glues of S-split short exact sequences; the proof of this involves tensor induction. Let \(E(S)=E_ G(S)=End_{kG}(kS)\) be the endomorphism ring of the permutation module kS. A k-basis of E(S) is given by ”suborbit maps” corresponding to G-orbits in \(S\times S\); each suborbit map has a defect group; each primitive idempotent in E(S) has a defect group; The Brauer homomorphism establishes a one-to-one correspondence between primitive idempotents of E(S) of defect group D and equivalence classes of those of \(E_ N(S^ D)\), where \(N=N_ G(D)\) and \(S^ D\) is the set of points of S fixed by D. Brauer’s classical main theorem is then recovered by \(S=G\) acted upon by \(G\times G\), upon suitable interpretation.

The Auslander-Reiten theory of almost split exact sequences is next applied to (the Artin algebra) kG. If V is indecomposable, the atom \(\tau\) (V) is defined to be soc(V) when V is projective and otherwise to be the glue of an almost split exact sequence associated to V. Two closely related inner products \(<U,V>\) and (U,V) are considered on A(G) each defined from \(Hom_{kG}(U,V)\). Both products are nonsingular with dual bases being given in terms of the atoms \(\tau\) (V). Any module may be regarded as a formal sum of atoms, namely its composition factors and the glues holding it together. When information is projected onto certain finite dimensional direct summands A of A(G), A spanned by indecomposable classes V, there arise two tables, the atom table and the representation table, these being the values of the species of A on the atoms \(\tau\) (V) and the indecomposables V.

Carlson’s idea of associating a variety \(X_ G(V)\) to a module V is introduced. This is a subvariety of \(X_ G=spec H^{ev}(G,k)\), the spectrum of the even cohomology ring of G. This last is a commutative noetherian graded ring and \(H^*(G,V)\) is a graded module over it; \(X_ G(V)\) comes from the annihilator Ann(V) of \(H^*(G,V)\) in \(H^{ev}(G,k)\). The ”dimension” or complexity \(C_ X(V)\) is the order of the pole at \(t=1\) of the rational function \(\eta_ V(t)\) describing the dimensions of the terms of a minimal projective resolution of V. It is also shown that the residue of \(\eta_ V(t)\) at \(t=1\) is a positive integer \(\eta\) (V).

The properties of \(X_ G(V)\) are determined by its restrictions \(X_ E(V\downarrow E)\), as E runs through nonconjugate elementary subgroups E of G. When G is elementary, \(X_ G(V)\) can be described in terms of the group algebra kG. The elementary subgroups E of G give rise to the Quillen stratification of \(X_ G(V)\). If X is a homogeneous subset of \(X_ G\), then A(G,X) spanned by those V with \(X_ G(V)\subseteq X\) is an ideal of A(G). If V is indecomposable, then the corresponding projective variety \(\bar X_ G(V)\) is topologically connected. Many other properties of these \(X_ G(V)\) are given.

In the final sector of the text, almost split exact sequences are used to construct a locally finite graph with the indecomposables as vertices. Projectives are disregarded and the result is a stable representation quiver. The connected components of this quiver give a refinement of the classification of modules V in as much as if U and V are in the same connected component of the quiver, then \(X_ G(U)=X_ G(V).\)

The Riedtmann structure theorem shows that a connected stable representation quiver is uniquely expressible as a quotient of the universal covering quiver of a tree, by an ”admissible” group of automorphisms. By means of the invariant \(\eta\) (V) above, this tree class is shown to be either a Dynkin diagram (finite or infinite) or a Euclidean diagram. Following Webb, each of these is investigated in turn.

In addition to exercises being provided at the end of the sections, there is an appendix of 48 pages of calculations on finite groups over relevant characteristics. These include the indecomposables in the finite and tame cases, the species and invariants associated to them, the even cohomology ring, the atom and representation tables, the quivers and the tree class. The groups include \(C_ 2\), \(C_ 2\times C_ 2\), \(D_{2^ n}\), \(Q_ 8\), \(C_ p\), groups of orders \(p^ 2\) and \(p^ 3\), alternating groups up to \(A_ 7\) and their associated groups, \(L_ 3(2)\) and \(M_{11}.\)

There are two criticisms of the book. Firstly, it is tough to read; proofs are at research paper level and several sections have appeared as research papers (The reviewer has no complaint about sections on cohomology of groups, spectral sequences and Bockstein and Steenrod Operations; indeed the pictures painted by the author are clearer than those obtained from many standard texts.) Secondly, in such a broad and almost definitive survey of the area, more detail should have been given to attribution of results and proofs to their originators; it is true that general acknowledgements are given in places and the bibliography is extensive, but the author, having such a detailed knowledge of the field, appeared to be in a good position to give a critical appraisal of the work in the field over the last twenty years.

As the author says in the introduction, the book has remarkably little overlap with material currently in textbook form. Further he has pinpointed throughout the text many unsolved interesting questions. He has certainly succeeded in his hope to impress upon the reader that the three topics ”representation rings”, ”almost split exact sequences” and ”complexity and cohomology varieties” are closely connected and in his hope to encourage further investigation of their interplay.

Reviewer: S.B.Conlon

##### MSC:

20C20 | Modular representations and characters |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

16S34 | Group rings |

16Gxx | Representation theory of associative rings and algebras |