A cluster tilting module for a representation-infinite block of a group algebra

It is shown that (1) every almost self-injective group algebra is self-injective and (2) if the group algebra KG is continuous, then G is a locally finite group. Furthermore, it follows that the following assertions are equivalent: a CS group algebra KG is continuous; KG is principally self-injective; the group G is locally finite.

We classify, in terms of the structure of the finite group G, all group algebras KG for which all right ideals are right annihilators of principal left ideals. This means in the language of coding theory that we classify code-checkable group algebras KG which have been considered so far only for abelian groups G. Optimality of checkable codes and asymptotic results are discussed.

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

Let K be any field of characteristic zero. We show that there are at least four ideals in the group algebra KG of every simple locally finite group G of 1-type, thus providing the final step in solving an old question of I. Kaplansky's for locally finite groups. We also determine the ideal lattice in KG for those 1-type groups G which are a direct limit of finite direct products of alternating groups.

On modules with cyclic vertices in the Auslander-Reiten quiver

Lie properties of symmetric elements in group rings

We define classes of automorphically rigid modular group algebras KG over a finite field K and a finite group G.

Let K be a field of characteristic p ≠ 2 and let G be any group. A characterization of group algebras KG satisfying the Lie identity [[x,y],[u,v],[z,t]] = 0 for all x,y,u,v,z,t ∈ KG is obtained.

Using representation theoretical methods we investigate self-dual group codes and their extensions in characteristic 2. We prove that the existence of a self-dual extended group code heavily depends on a particular structure of the group algebra KG which can be checked by an easy-to-handle criteria in elementary number theory. Surprisingly, in the binary case such a code is doubly even if the converse of Gleason's theorem holds true, i.e., the length of the code is divisible by 8. Furthermore, we give a short representation theoretical proof of an earlier result of Sloane and Thompson which states that a binary self-dual group code is never doubly even if the Sylow 2-subgroups of G are cyclic. It turns out that exactly in the case of a cyclic or Klein four group as Sylow 2-subgroup doubly even group codes do not exist.

In this paper, the hyperoctahedral group algebra \(\mathscr{F}[\overrightarrow{S_{n}}]\) over a splitting field \(\mathscr{F}\) of wreath product \(\overrightarrow{S_{n}}\) with \(\text{char}(\mathscr{F})\nmid|\overrightarrow{S_{n}}|\), is considered and the unique idempotents corresponding to the four linear characters of the group \(\overrightarrow{S_{n}}\) are explored. Also, by establishing the minimum weights and dimensions, all group codes generated by the linear idempotents in the aforementioned group algebra are completely characterized for every \(n\). The nonlinear idempotents corresponding to nonlinear characters of \(\overrightarrow{S_{3}}\) are also obtained and various group codes in \(\mathscr{F}[\overrightarrow{S_{3}}]\) generated by linear and nonlinear idempotents are examined.

Let F be any field of prime characteristic p and let q be a power of p. We assume that F contains the finite field of order q. A q-polynomial L over F is an element of the polynomial ring F[x] with the property that all exponents are a power of q. We assume that the coefficient of the x term of L is nonzero. We investigate the Galois group G of L over F, under the assumption that L(x)/x is irreducible in F[x]. It is well known that if L has q-degree n, its roots are an n-dimensional vector space over the field of order q and G acts linearly on this space.Our main theorem is the following. We consider a monic q-polynomial L whose q-degree is a prime, r, say, with L(x)/x irreducible over F. Assuming that the coefficient of the x term of L is (−1)r, we show that the Galois group of L over F must be the special linear group SL(r,q) of degree r over the field of order q when q>2. We also prove a projective version for the projective special linear group PSL(r,q), and we present the analysis when q=2.

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical$p$-Kazhdan–Lusztig polynomials.

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. Kawai’s result is a tool to calculate the cohomology variety of a module in a block B of a finite group algebra kG restricted to subgroups of a defect group P, provided that P is abelian. Kawai’s result coincides with a Theorem of Avrunin and Scott specialized to modules in the principal block and their restrictions to p-subgroups. J. Rickard raised the question whether Kawai’s result can be extended to modules in blocks with arbitrary defect groups. We show that this is indeed the case for modules whose corresponding module over some almost source algebra is fusion stable. We show that this fusion stability hypothesis is automatically satisfied for principal blocks and blocks with abelian defect groups.

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne–Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne–Lusztig varieties, and show that the “unramified” varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for SL 2 ( F q 〚 ϖ 〛 / ( ϖ 2 ) ) , with odd q , the extended Deligne–Lusztig varieties do indeed afford all the irreducible representations.