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Let G be a simple linear algebraic group over a field F, and V an absolutely irreducible representation of G. We show that under some mild hypotheses there exists an invariant homogeneous polynomial f for the action of G on V defined over F, such that twisted forms of f up to a scalar multiple classify twisted forms of G for which the representation V is defined over F. This result extends the classical case of a quadratic form q and its orthogonal group O ( q ) .

A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields., 21 pages

Let G2(q) be the Chevalley group of type G2 defined over a finite field with q=pn elements, where p is a prime number and n is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of G in characteristic other than p to a maximal subgroup of G2(q) is still irreducible. Similar results are obtained for B22(q) and G22(q).

It is proved that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group G into GL d ( F q n ) for n ⩾ 1 is a rational function. It is also proved that the generating function defined by weighted sums over isomorphism classes of representations of G into GL d ( F q n ) for n ⩾ 1 is meromorphic over both the complex numbers and the p-adic complex numbers.

We compute the 2-modular decomposition numbers of Steinberg's triality groups D 4 3 ( q ) , where q is a power of an odd prime, except for two decomposition numbers in the Steinberg character and classify all absolutely irreducible representations of D 4 3 ( q ) in non-defining characteristic up to a certain degree.

Let G=〈X〉 be an absolutely irreducible subgroup of GL(d,K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d,F).K×, where K× denotes the centre of GL(d,K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(|X|d3+d2log|F|) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of GL(d,K).

In Section 2 of this paper, the maximum number of real elements possible in a covering group of C2(n) is determined, and a description of those covering groups in which this maximum is attained is given. Among these “maximally real” examples is that covering group G of C2(n) which is generated by n involutions. For this particular group, the Schur indices of real-valued irreducible characters of each degree are investigated in Section 4. The main result of this section is a set of recurrence relations describing the number of absolutely irreducible characters of G of a given degree of each of three types (non-real-valued and real-valued of index 1 or 2 over R) in terms of related numbers for the corresponding group on n−1 generators.

Hecke algebras for the complex reflection groups G(r,p,n) (where p⩾1 and p∣r) were introduced in the work of Ariki [J. Algebra 177 (1995) 164–185], Broué and Malle [Astérisque 212 (1993) 119–189]. In this paper we consider modular representation theory for these algebras in the case where r=p. We assume that the field K contains a primitive pth root of unity ε. Our method is to study the restrictions of Specht modules Sλ for Hecke algebras Hp,n of type G(p,1,n) with parameters (q;1,ε,…,εp−1). Suppose that f(q,ε)≠0 in K (see 4.7 for definition of f(q,ε)). For any multipartition λ=(λ(1),…,λ(p)) of n, we prove that Sλ↓Hp,p,n≅Sλ[k]↓Hp,p,n for any 1⩽k⩽p−1, where λ[k]=(λ(k+1),λ(k+2),…,λ(p),λ(1),λ(2),…,λ(k)); and if k is the smallest positive integer such that λ=λ[k] (hence k∣p), we explicitly decompose Sλ↓Hp,p,n into a direct sum of p/k smaller Hp,p,n-submodules with the same dimensions. As a result, when f(q,ε)≠0 in K, we show that Hp,p,n is split over K and get a complete classification of all the absolutely irreducible Hp,p,n-modules. This generalizes earlier work of [C. Pallikaros, J. Algebra 169 (1994) 20–48] and [J. Hu, Manuscripta Math. 108 (2002) 409–430] on Hecke algebras of type Dn (which are included as a special case of our main results).

Let p be a prime. This paper classifies finite connected reductive groups G in characteristic p with the property that all complex character values of G belong to an unramified above p extension of the field of rational numbers. The main application of these results is to the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p>3 has been developed in our paper [Proc. London Math. Soc. (3) 84 (2002) 439]. Together with [Proc. London Math. Soc. (3) 84 (2002) 439], Theorem 1.9 of the paper solves this problem for many finite connected reductive groups in characteristic p>3. Additionally, we show that all complex representations of any finite connected reductive group with no composition factor of type E7(2f), E8(2f), and E8(5f) can be realized over a quadratic extension of an unramified (above p) extension of Q .

A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero.

We study a finite group having a faithful character whose square has a small number of irreducible characters as constituents. Let Irr(G) be the set of absolutely irreducible ordinary characters of a finite group G. For each φ ∈ Irr(G), let φ = φ if φ is real valued and φ = φ + φ otherwise, where φ denotes the complex conjugate of φ. Let RIrr(G) = {φ | φ ∈ Irr(G)}. For χ ∈ RIrr(G), let χ 2 = b · 1 + a · χ + Ψ such that Ψ is a character of G which does not contain χ nor the principal character 1 as a constituent. We study the case when Ψ is a scalar multiple of a sum of the characters in IRrr(G), which are in a single orbit with respect to the action of the Galois group Gal(Q/Q(χ)). Here Q denotes the algebraic closure of Q in C and Q(χ) is the field generated by the values of χ. As an application, we give a classification of Q-polynomial group association schemes.

Let D be a division algebra of degree m over its center F . Herstein has shown that any finite normal subgroup of D * ≔ GL 1 ( D ) is central. Here, as a generalization of this result, it is shown that any finitely generated normal subgroup of D * is central. This also solves a problem raised by Akbari and Mahdavi-Hezavehi ( Proc. Amer. Math. Soc. , to appear) for finite-dimensional division algebras. The structure of maximal multiplicative subgroups of an arbitrary division ring D is then investigated. Given a maximal subgroup M of D * whose center is algebraic over F , it is proved that if M satisfies a multilinear polynomial identity over F , then [ D : F ]

Given the ring of integersRof an algebraic number fieldK, for which natural numbernis there a finite groupG ⊂ GL(n, R) such thatRG, theR-span ofG, coincides withM(n, R), the ring of (n × n)-matrices overR? GivenG ⊂ GL(n, R) we show thatRG = M(n, R) if and only if the Brauer reduction ofGmodulo every prime is absolutely irreducible. In addition, the question above is fully answered ifnis an odd prime.

Let Γ be a finite connected regular graph with vertex set V Γ, and let G be a subgroup of its automorphism group Aut Γ. Then Γ is said to be G-locally primitive if, for each vertex α, the stabilizer G α is primitive on the set of vertices adjacent to α. In this paper we assume that G is an almost simple group with socle soc G = S ; that is, S is a nonabelian simple group and S ⊴ G ≤ Aut S . We study nonbipartite graphs Γ which are G -locally primitive, such that S has trivial centralizer in Aut Γ and S is not semiregular on vertices. We prove that one of the following holds: (i) S ⊴ Aut Γ ≤ Aut( S ), (ii) G Y ≤ Aut Γ with Y almost simple and soc Y ≠ S , or (iii) S belongs to a very restricted family of Lie type simple groups of characteristic p , say, and Aut Γ contains the semidirect product Z d p : G , where Z d p is a known absolutely irreducible G -module. Moreover, in certain circumstances we can guarantee that S ⊴ Aut Γ ≤ Aut( S ). For example, if Γ is a connected ( G , 2)-arc transitive graph with Sz( q ) ≤ G ≤ Aut(Sz( q )) ( q = 2 2 n + 1 ≥ 8) or G = Ree( q ) ( q = 3 2 n + 1 ≥ 27), then G ≤ Aut Γ ≤ Aut( G ).

Let H be a normal subgroup of some finite group G and let V be an irreducible (right) FH-module, where F is an arbitrary field. It is one of the primary objectives of Clifford theory to find necessary and sufhcient conditions so that V can be extended to an FG-module. An obvious necessary condition is that V has to be G-invariant (stable). Sufficiency relies upon various properties of the group extension H * G --w S, where S= G/H, and the module itself. The theory is well developed in case V is absolutely irreducible. Recently Isaacs [12] and Dade [4] proved, generalizing a theorem by Gallagher, that V can be always extended to an FG-module provided H is a Hall subgroup of G (and V is G-invariant). The present paper resulted from an effort to understand this remarkable and somewhat surprising result. Assume in the sequel that V is G-invariant. Let D = End& V)’ be the opposite endomorphism ring of K Basic Clifford theory yields a group extension D’ * 52 -P-P S such that there exists an FG-module extending V if and only if 52 =52,(V) splits over the multiplicative group D’ of the division algebra D (see Theorem 1 below). In general D is noncommutative and so the usual cohomological methods seem to fail. Dade even remarked in [4] sceptically that “none of the nice theorems about extensions of V, such as those of Gallagher, which depend upon calculations in the cohomology groups of S with coefficients in a commutative group, can be proved this way.” In the following situation, however, one can argue as usual