All maximal finite (absolutely) irreducible subgroups of GL(S, Z) are determined up to Z-equivalence. Moreover, we present a full set of representatives of the Z-classes of the maximal finite irreducible subgroups of GL(n, Z) for n < 9 by listing generators of the groups, the corresponding quadratic forms fixed by these groups, and the shortest vectors of these forms.

We describe the tensor products of two irreducible linear complex representations of the group G = GL(3, Fq) in terms of induced representations by linear characters of maximal torii and also in terms of classical and generalized Gelfand-Graev representations. Our results include MacDonald’s conjectures for G and at the same time they are extensions to G of finite counterparts to classical resu...

We show that a connected split reductive group G over a field of characteristic 0 is uniquely determined up to isomorphism by specifying a maximal torus T of G , the set of isomorphism classes of irreducible representations of G , and the character homomorphism from the Grothendieck ring of G to that of T .

We study the ring of differential operators D(X) on the basic affine space X = G/U of a complex semisimple group G with maximal unipotent subgroup U . One of the main results shows that the cohomology group H(X,OX) decomposes as a finite direct sum of non-isomorphic simple D(X)modules, each of which is isomorphic to a twist of O(X) by an automorphism of D(X). We also use D(X) to study the prope...

This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We st...

We classify partitions which are of maximal p-weight for all odd primes p. As a consequence, we show that any non-linear irreducible character of the symmetric and alternating groups vanishes on some element of prime order.

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK . We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of...

From an abstract algebraic perspective, an explanation for this can be given as follows: since √ D is irrational, the polynomial t −D is irreducible over Q. Since the ring Q[t] is a PID, the irreducible element t − D generates a maximal ideal (t−D), so that the quotient Q[t]/(t−D) is a field. Moreover, the map Q[ √ D]→ Q[t]/(t −D) which is the identity on Q and sends √ D 7→ t is an isomorphism ...