Representations of general linear groups.

*(English)*Zbl 0541.20025
London Mathematical Society Lecture Note Series, 94. Cambridge etc.: Cambridge University Press. XII, 147 p. £9.95; $ 19.95 (1984).

In this book the author studies unipotent representations, i.e., representations contained as composition factors in the permutation representation on the cosets of a Borel subgroup of the group \(G_ n=GL(n,q),\) over a field K of characteristic 0 or prime to q. The theorems proved show a similarity with the modular representation theory of the symmetric group \(S_ n\), which might be described as the \(''q=1''\) case; however, the proofs of the theorems themselves do not carry over to that case. The characters of these representations, when \(char K=0,\) were obtained by R. Steinberg [Trans. Am. Math. Soc. 71, 274-282 (1951; Zbl 0045.302)] but the present work is concerned with constructing modules for the representations. In an earlier paper [J. Algebra 74, 443-465 (1982; Zbl 0482.20028)] the author gave a construction for the unipotent representations of \(G_ n\), but the present proof is new, and is simplified by working with an extension field \(\bar K\) of K which contains the pth roots of unity, where q is a power of p.

The first eight chapters are preliminary in nature, and describe, for example, the root subgroups, Chevalley commutator relations, Bruhat decomposition, and the structure of parabolic and certain other subgroups in \(G_ n\). Readers who wish to see these concepts from the theory of algebraic groups worked out in an elementary way in this special case will find these chapters useful, especially as there are many examples.

In Chapter 9 certain idempotents in the group algebra \(\bar KG_ n\) are constructed using some unipotent subgroups of \(G_ n\). These give a way of decomposing (Theorem 9.11) a \(\bar KG_ n\)-module M as a \(\bar KG^*_{n-1}\)-module, where \(G^*_{n-1}\) is a maximal parabolic subgroup with Levi factor isomorphic to \(G_{n-1}\). Next, for a composition \(\lambda\) of n (i.e., a sequence of integers \(\{\lambda_ i\}\) such that \(\sum \lambda_ i=n\); this generalizes a partition of n) the permutation module \(M_{\lambda}\) on the cosets of a parabolic subgroup \(P_{\lambda}\) of type \(\lambda\) is studied. Its structure as \(\bar KG_{n-1}\)-module is given in Corollary 10.16; the author points out the similarity with the case of \(S_ n\), as also the extra difficulty inherent here. In Chapter 11, at first an idempotent \(E_{\lambda}\) is constructed (11.4) using a character of a maximal unipotent subgroup \(U^-\) of \(G_ n\); the definition involves a standard Young tableau \(T_{\lambda}\) and a permutation \(\Pi_{\lambda}\) (11.3) associated with \(\lambda\). Then the author defines \(S_{\lambda}\) to be the \(\bar KG_ n\)-submodule of \(M_{\lambda}\) generated by \(\bar P_{\lambda}\Pi_{\lambda}E_{\lambda}\) (here \(\Pi_{\lambda}\) is regarded as an element of the symmetric group, contained in \(G_ n\), and \(\bar P_{\lambda}\) is the sum of the elements in \(P_{\lambda})\), and \(D_{\lambda}\) to be \(S_{\lambda}/S_{\lambda}\cap S^{\perp}_{\lambda},\) where \(\perp\) is orthogonality with respect to a natural bilinear form on \(M_{\lambda}\) (11.1). Then the following is shown (11.12):

Every submodule of \(M_{\lambda}\) either contains \(S_{\lambda}\) or is contained in \(S^{\perp}_{\lambda}\) (this is the Submodule Theorem), and \(D_{\lambda}\) is a self-dual absolutely irreducible \(\bar KG_ n\)- module occurring with multiplicity 1 as a composition factor in \(S_{\lambda}\) and in \(M_{\lambda}\). A corollary (11.14) is that if \(D_{\mu}\) is a composition factor of \(S_{\lambda}\) then \(\mu {\underline \triangleright}\lambda\) where \({\underline \triangleright}\) is a transitive relation on compositions generalizing the usual ordering of partitions, and that \(D_{\lambda}\cong D_{\mu}\) if and only if \(\mu\) is obtained from \(\lambda\) by rearranging the parts. It is then shown (11.16) that if \(K=Q\) then \(S_{\lambda}=D_{\lambda}\) and the \(S_{\lambda}\) (where \(\lambda\) varies over the partitions of n) give modules for the distinct unipotent representations of \(G_ n\) over Q.

The next main theorem is the Kernel Intersection Theorem (15.19): If \(\lambda\) is any composition of n then \(S_{\lambda}=\cap_{\theta}Ker \theta,\) the intersection being over all the \(\bar KG_ n\)-homomorphisms \(\theta\) which map \(M_{\lambda}\) into some \(M_{\mu}\) with \(\mu \triangleright \lambda\). The proof involves a delicate analysis of semistandard tableaux. Various consequences of this theorem are then derived in Chapter 16, e.g., that \(D_{\lambda}\) is a composition factor of \(S_{\lambda}\) (and of \(M_{\lambda})\) of multiplicity one, and all other composition factors are of the form \(D_{\nu}\), where \(\nu \triangleright \lambda\). Indeed, (16.4) every unipotent representation of \(G_ n\) is isomorphic to some \(D_{\mu}\), where \(\mu\) is a partition of n. There is also a Branching Theorem (16.10) (so-called because of the analogy with the symmetric group) describing the restriction of \(S_{\lambda}\) to \(G^*_{n-1}\). The rest of the book is devoted to the study of the composition factors of the modules \(S_{\lambda}\). The part of the decomposition matrix (for primes not dividing q) of \(G_ n\) corresponding to unipotent representations is worked out for \(n\leq 4\). The calculations show remarkable connections with representations of the groups GL(d,F) over F, where F is a field of prime characteristic. In the least chapter the composition factors of \(S_{\lambda}\) are worked out completely when \(\lambda\) has at most two parts.

This book requires no background other than a knowledge of standard facts on the representation theory of finite groups. It also contains a wealth of examples, and analogies with the theory of the symmetric groups are pointed out when they arise. Important questions (such as a knowledge of the dimensions of the \(D_{\lambda}\) and of the decomposition matrices) remain unsolved both for the symmetric groups and the general linear groups. In this context, the author displays much evidence that the modular representation theory of \(S_ n\) is in his words ”the tip \((q=1)\) of a very big iceberg”.

The first eight chapters are preliminary in nature, and describe, for example, the root subgroups, Chevalley commutator relations, Bruhat decomposition, and the structure of parabolic and certain other subgroups in \(G_ n\). Readers who wish to see these concepts from the theory of algebraic groups worked out in an elementary way in this special case will find these chapters useful, especially as there are many examples.

In Chapter 9 certain idempotents in the group algebra \(\bar KG_ n\) are constructed using some unipotent subgroups of \(G_ n\). These give a way of decomposing (Theorem 9.11) a \(\bar KG_ n\)-module M as a \(\bar KG^*_{n-1}\)-module, where \(G^*_{n-1}\) is a maximal parabolic subgroup with Levi factor isomorphic to \(G_{n-1}\). Next, for a composition \(\lambda\) of n (i.e., a sequence of integers \(\{\lambda_ i\}\) such that \(\sum \lambda_ i=n\); this generalizes a partition of n) the permutation module \(M_{\lambda}\) on the cosets of a parabolic subgroup \(P_{\lambda}\) of type \(\lambda\) is studied. Its structure as \(\bar KG_{n-1}\)-module is given in Corollary 10.16; the author points out the similarity with the case of \(S_ n\), as also the extra difficulty inherent here. In Chapter 11, at first an idempotent \(E_{\lambda}\) is constructed (11.4) using a character of a maximal unipotent subgroup \(U^-\) of \(G_ n\); the definition involves a standard Young tableau \(T_{\lambda}\) and a permutation \(\Pi_{\lambda}\) (11.3) associated with \(\lambda\). Then the author defines \(S_{\lambda}\) to be the \(\bar KG_ n\)-submodule of \(M_{\lambda}\) generated by \(\bar P_{\lambda}\Pi_{\lambda}E_{\lambda}\) (here \(\Pi_{\lambda}\) is regarded as an element of the symmetric group, contained in \(G_ n\), and \(\bar P_{\lambda}\) is the sum of the elements in \(P_{\lambda})\), and \(D_{\lambda}\) to be \(S_{\lambda}/S_{\lambda}\cap S^{\perp}_{\lambda},\) where \(\perp\) is orthogonality with respect to a natural bilinear form on \(M_{\lambda}\) (11.1). Then the following is shown (11.12):

Every submodule of \(M_{\lambda}\) either contains \(S_{\lambda}\) or is contained in \(S^{\perp}_{\lambda}\) (this is the Submodule Theorem), and \(D_{\lambda}\) is a self-dual absolutely irreducible \(\bar KG_ n\)- module occurring with multiplicity 1 as a composition factor in \(S_{\lambda}\) and in \(M_{\lambda}\). A corollary (11.14) is that if \(D_{\mu}\) is a composition factor of \(S_{\lambda}\) then \(\mu {\underline \triangleright}\lambda\) where \({\underline \triangleright}\) is a transitive relation on compositions generalizing the usual ordering of partitions, and that \(D_{\lambda}\cong D_{\mu}\) if and only if \(\mu\) is obtained from \(\lambda\) by rearranging the parts. It is then shown (11.16) that if \(K=Q\) then \(S_{\lambda}=D_{\lambda}\) and the \(S_{\lambda}\) (where \(\lambda\) varies over the partitions of n) give modules for the distinct unipotent representations of \(G_ n\) over Q.

The next main theorem is the Kernel Intersection Theorem (15.19): If \(\lambda\) is any composition of n then \(S_{\lambda}=\cap_{\theta}Ker \theta,\) the intersection being over all the \(\bar KG_ n\)-homomorphisms \(\theta\) which map \(M_{\lambda}\) into some \(M_{\mu}\) with \(\mu \triangleright \lambda\). The proof involves a delicate analysis of semistandard tableaux. Various consequences of this theorem are then derived in Chapter 16, e.g., that \(D_{\lambda}\) is a composition factor of \(S_{\lambda}\) (and of \(M_{\lambda})\) of multiplicity one, and all other composition factors are of the form \(D_{\nu}\), where \(\nu \triangleright \lambda\). Indeed, (16.4) every unipotent representation of \(G_ n\) is isomorphic to some \(D_{\mu}\), where \(\mu\) is a partition of n. There is also a Branching Theorem (16.10) (so-called because of the analogy with the symmetric group) describing the restriction of \(S_{\lambda}\) to \(G^*_{n-1}\). The rest of the book is devoted to the study of the composition factors of the modules \(S_{\lambda}\). The part of the decomposition matrix (for primes not dividing q) of \(G_ n\) corresponding to unipotent representations is worked out for \(n\leq 4\). The calculations show remarkable connections with representations of the groups GL(d,F) over F, where F is a field of prime characteristic. In the least chapter the composition factors of \(S_{\lambda}\) are worked out completely when \(\lambda\) has at most two parts.

This book requires no background other than a knowledge of standard facts on the representation theory of finite groups. It also contains a wealth of examples, and analogies with the theory of the symmetric groups are pointed out when they arise. Important questions (such as a knowledge of the dimensions of the \(D_{\lambda}\) and of the decomposition matrices) remain unsolved both for the symmetric groups and the general linear groups. In this context, the author displays much evidence that the modular representation theory of \(S_ n\) is in his words ”the tip \((q=1)\) of a very big iceberg”.

Reviewer: B.Srinivasan

##### MSC:

20G05 | Representation theory for linear algebraic groups |

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

20C30 | Representations of finite symmetric groups |

20C15 | Ordinary representations and characters |