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The star diagram of a diagram [λ] was first obtained by Robinson (5) who denned and studied it, thus giving a unique characterization of the hook structure of [λ]. Staal (6) continued Robinson's work and obtained some important results. Recently, Littlewood (2) published a paper on the modular representations of the symmetric group in which he defined ‘congruent partitions’, ‘p-quotients’, etc., relating to the diagram [λ]. Littlewood's ideas, though superficially different from those of Robinson and Staal, are in essence equivalent to them.

It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.

We first consider the construction of a norm on a direct sum of normed linear spaces and call a norm absolute if it depends only on the norms of the component spaces. Several clharacterizations are given of absolute norms. Absolute norms are then used to construct norms on tensor products of normed linear spaces and on tensor products of operators on normed linear spaces. 1. Introduction. In this paper, we consider the construction of norms on composite linear spaces formed from direct sums and tensor products of normed linear spaces and we consider properties of norms of operators on these spaces. The notion of an absolute norm is introduced as a natural generalization of the relatively familiar idea of an absolute vector norm on the space Cn of ordered n-tuples of complex numbers. Such norms on Cn correspond to the "coordinatewise symmetric" gauge functions as described by Ostrowski (3), and it is shown that our absolute norms on composite spaces correspond in a one-to-one fashion with the absolute vector norms on Cn. We are particularly interested in operator norms for which, in an appropriate sense to be detailed later

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

A set of generators is exhibited for the centre of the integral group ring of the symmetric group on n letters. As an application, a new proof is obtained of Nakayama’s criterion for two representations of the group to belong to the same p -block.

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ

The purpose of this paper is to look at some results in the representation theory of the symmetric groups, both old and recent, from a modern point of view. In the first two sections we construct the irreducible representations of the symmetric groups as left ideals in the group ring. The irreducible modular representations were first constructed by James [ 131. We prove inequivalence by a method more natural than the original. The approach may be compared with that in [6]. A comprehensive survey of recent developments has been published by their originator [ 151. In Section 3 we look at the skew representations. Their dimension may be established by the same method as that used to obtain the dimenssions of the Specht modules in [lo]. A further study of skew representations may be found in [ 161. The induction algebra of symmetric groups is defined in Section 4. It is proved to be a polynomial algebra, a result also obtained in [9]. The algebra is used to give a new proof of Chung’s conjecture that the number of modular irreducible representations in a block is independent of the core. In the last two sections we examine Taulbee’s construction [7] of a class of projective, indecomposable characters. The construction uses the process of r-induction (which has been used to calculate decomposition numbers of

The ordinary irreducible characters X, $ ... 3: X, ;? 0, X, + Aa + ... + A, = n. Similarly, for a given prime number p, the modular irreducible Brauer characters 4” and the modular principal indecomposable characters 7” may be labeled by the p-regular partitions p of n, that is, partitions in which no part is repeated p times or more. This is simply due to the well-known fact (see, e.g., [8, p. 1141) that the number of p-regular conjugacy classes of S, is equal to the number of p-regular partitions of IZ. The lack of a good labeling of characters in the modular theory has always contrasted strongly with the very satisfactory state of affairs in the classical ordinary theory. In this paper we introduce relatively simple combinatorial notions in order to prove the existence of “good” labeling in the modular theory. We use the p-graph of a partition (first introduced by Littlewood [5]) in order to associate with each p-regular partition p a set 1(p) of partitions of n, by means of a special process of stripping and reassembly of the nodes. Littlewood’s p-graph

LetA be an abelian group; we denote byd(A) the homological dimension ofA as a module over its ringE of endomorphisms. In the present note we exhibit a groupA for whichgl dimE=∞ andd(A)≤1, we show that there is a class of torsion-free groupsA of rank>2 for whichd(A)=∞ and we construct an indecomposable torsion-free group of rankr which isE-isomorphic toE.

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.