Jørn B. Olsson

On Residue Symbols and the Mullineux Conjecture

This paper is concerned with properties of the Mullineux map, which plays a rôle in p-modular representation theory of symmetric groups. We introduce the residue symbol for a p-regular partitions, a variation of the Mullineux symbol, which makes the detection and removal of good nodes (as introduced by Kleshchev) in the partition easy to describe. Applications of this idea include a short proof of the combinatorial conjecture to which the Mullineux conjecture had been reduced by Kleshchev.

Prime Power Degree Representations of the Symmetric and Alternating Groups

In 1998, the second author of this paper raised the problem of classifying the irreducible characters of S n of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of S n of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskiis problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p -power degree > 1 and all of the irreducible characters of G have degrees that are relatively prime to p or are powers of p ? The case of the double covers of the symmetric and alternating groups will be dealt with in a forthcoming paper; in particular, this completes the answer to Zalesskiis problem. The paper is organized as follows. In Section 2, some results on hook lengths in partitions are proved. These results lead to an algorithm which allows us to show that every irreducible representation of S n with prime power degree is labelled by a partition having a large hook. In Section 3, we obtain a new result concerning the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove Theorem 2.4, the main result. To do so, we combine the algorithm above with Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for large n . To complete the proof, we check that the algorithm terminates appropriately for small n (that is, those n [les ] 9.25 · 10 8 ) with the aid of a computer. In the last section we derive the classification of irreducible characters of A n of prime power degree, and we solve Hupperts question for alternating groups.

Remarks on symbols, hooks, and degrees of unipotent characters

Abstract Motivated by certain questions concerning the degrees of unipotent characters in finite classical groups we study combinatorial properties of the so-called symbols, which index these characters. In particular, the theory of hooks, cores, and quotients of partitions is generalized to symbols.

Character correspondences in finite general linear, unitary and symmetric groups

Let r > 0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauers and Robinsons solution of Nakayamas conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r > 2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible characters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height h t z = 0 . Therefore Alperins conjecture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspondences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-subgroup /~-~R such that there is a natural height preserving one-to-one correspondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b).

A theorem on the cores of partitions

If s and t are relatively prime positive integers we show that the s-core of a t-core partition is again a t-core partition. A similar result is proved for bar partitions under the additional assumption that s and t are both odd.

SEPARATING CHARACTERS BY BLOCKS

We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

Residue Symbols and Jantzen-Seitz Partitions

Jantzen?Seitz partitions are thosep-regular partitions ofnwhich labelp-modular irreducible representations of the symmetric groupSnwhich remain irreducible when restricted toSn?1; they have recently also been found to be important for certain exactly solvable models in statistical mechanics. In this article we study their combinatorial properties via a detailed analysis of their residue symbols; in particular thep-cores of Jantzen?Seitz partitions are determined.

On Mullineux symbols

Abstract The Mullineux symbols of a special class of p-regular partitions are classified. As a consequence it is shown that two apparently very difficult conjectures in the modular representation theory of finite symmetric groups are compatible.

Fourier transforms with respect to monomial representations

Let F be a field of characteristic zero, and let f : G --+ F be a function from the finite group G into F. The Fourier transform of f a t the representation p : G ~ GL(m, F) is defined as f ( p ) = ~g~af(g)p(#)~ Mat(m, F), which denotes the ring of all m x m matrices with entries in F. The actual evaluation off(p) can be a tremendous computational task. However, this calculation is required for applications in statistics and elsewhere. The case where p is a permutation or monomial representation is attractive for computation because the sparseness of the matrices p (g) makes direct computation o f f ( p ) relatively easy. On the other hand most applications also require the calculation of the inverse Fourier transform, given by the formula

Spin Representations and Powers of 2

The elementary divisors of the reduced spin 2–decomposition matrix for the double covers of the finite symmetric groups are described. In addition, the maximal power of 2 dividing all spin character values on a fixed conjugacy class corresponding to a cycle type with odd parts only is determined.