Characters of \(GL(n,{\mathbb{F}}_ q)\) and Hopf algebras.

*(English)*Zbl 0551.20022Let \(C_ n=C_ n(q)\) be the vector space of complex valued class functions on GL(n,q) and \(C=C(q)=\oplus_{n\geq 0}C_ n(q)\). The multiplication of class functions defined by J. A. Green by means of induction from parabolic subgroups, and the adjoint operation of comultiplication make C into a graded Hopf algebra. Let \(P=\oplus_{n\geq 0}P_ n\) be the subspace of primitive elements in C, which is the symmetric algebra S(P). The authors define two decompositions of C into a tensor product of Hopf-subalgebras. The first decomposition is closely connected with the conjugacy classes and the second one with the irreducible characters of GL(n,q).

As a consequence, one gets two bases \(B_ C\) and \(B_ R\) of P with the following properties. The elements of \(B_ C\) have simple expressions as linear combinations of the characteristic functions of conjugacy classes. On the other hand, the irreducible characters of GL(n,q) have simple expressions as polynomials in the elements of \(B_ R\), in particular \(B_ R\) contains all cuspidal characters. Thus the evaluation of the irreducible characters of GL(n,q) essentially reduces to the search of the transition matrix from \(B_ R\) to \(B_ C.\)

The main result of the paper is a formula for this matrix. Let \(\Phi_ n\) (resp. \(\Theta_ n)\) denote the set of orbits of \(Gal({\mathbb{F}}_{q^ n}/{\mathbb{F}}_ q)\) on the multiplicative group \({\mathbb{F}}^{\times}_{q^ n}\) (resp. on the set of characters \({\mathbb{F}}^{\times}_{q^ n}\to {\mathbb{C}}^{\times})\). Then the elements of \(B_ C\cap P_ n\) (resp. \(B_ R\cap P_ n)\) are indexed by \(\Phi_ n\) (resp. \(\Theta_ n)\) and the transition matrix from \(B_ R\cap P_ n\) to \(B_ C\cap P_ n\) is, up to sign, the matrix of a natural pairing between \(\Phi_ n\) and \(\Theta_ n\). To prove the theorem, some of the known results are generalized and translated into the language of Hopf algebras and makes them natural and transparent.

As a consequence, one gets two bases \(B_ C\) and \(B_ R\) of P with the following properties. The elements of \(B_ C\) have simple expressions as linear combinations of the characteristic functions of conjugacy classes. On the other hand, the irreducible characters of GL(n,q) have simple expressions as polynomials in the elements of \(B_ R\), in particular \(B_ R\) contains all cuspidal characters. Thus the evaluation of the irreducible characters of GL(n,q) essentially reduces to the search of the transition matrix from \(B_ R\) to \(B_ C.\)

The main result of the paper is a formula for this matrix. Let \(\Phi_ n\) (resp. \(\Theta_ n)\) denote the set of orbits of \(Gal({\mathbb{F}}_{q^ n}/{\mathbb{F}}_ q)\) on the multiplicative group \({\mathbb{F}}^{\times}_{q^ n}\) (resp. on the set of characters \({\mathbb{F}}^{\times}_{q^ n}\to {\mathbb{C}}^{\times})\). Then the elements of \(B_ C\cap P_ n\) (resp. \(B_ R\cap P_ n)\) are indexed by \(\Phi_ n\) (resp. \(\Theta_ n)\) and the transition matrix from \(B_ R\cap P_ n\) to \(B_ C\cap P_ n\) is, up to sign, the matrix of a natural pairing between \(\Phi_ n\) and \(\Theta_ n\). To prove the theorem, some of the known results are generalized and translated into the language of Hopf algebras and makes them natural and transparent.

Reviewer: E.Abe

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

20C30 | Representations of finite symmetric groups |