Algebra Colloquium

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture affirming that the number of G-conjugacy classes of weights of G coincides with the number of isomorphism classes of simple kG-modules, where k is an algebraically closed field of characteristic p. Thirty years ago, Tetsuro Okuyama already proved that in the class of p-solvable groups this conjecture holds. In this paper, for the p-solvable groups, on the one hand we exhibit a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of isomorphism classes of simple kG-modules M and of G-conjugacy classes of weights (R,Y), up to the choice of a polarization. On the other hand, we determine the relationship between a multiplicity module of M and Y. In an Appendix, we show that the bijection defined by Gabriel Navarro for the groups of odd order coincides with our bijection for a particular choice of the polarization.

Let S=K[x₁,…,x_n] be a polynomial ring. Herzog and Zheng conjectured that the i-th Hilbert coefficient of a finitely generated graded Cohen-Macaulay S-module N generated in degree 0 is bounded by the functions of the minimal and maximal shifts in the minimal graded free resolution of N over S and the 0-th Betti number of N. Also, Römer asked whether under the Cohen-Macaulay assumption the i-th Betti number of S/I, where I ⊂ S is a graded ideal, can be bounded by the functions of the minimal and maximal shifts of S/I. In this paper, we provide elementary proofs to establish Herzog and Zheng's conjecture and the upper bound part of Römer's question for the Stanley-Reisner ring of a matroid complex.

We generalize the Auslander-Reiten isomorphisms to the relative n-Auslander-Reiten isomorphisms.

Suppose that G is a finite group and H is a subgroup of G. We say that H is s-semipermutable in G if HG_p = G_pH for any Sylow p-subgroup G_p of G with (p, |H|) = 1; H is s-semiembedded in G if there is a normal subgroup T of G such that HT is s-semipermutable in G and H ∩ T ≤ H_ssG, where H_ssG is the subgroup of H generated by all those subgroups of H which are s-semipermutable in G. We investigate the influence of s-semiembedded subgroups on the structure of finite groups. Some recent results are generalized and unified.

In this paper, we classify and determine the structure of finite groups whose character degree graphs are empty graphs.

For a group G, by δ(G) we denote the number of conjugate classes of the non-cyclic subgroups of G. In this paper, the groups G with δ(G)=3 are classified.

In this paper we continue to study the lifting of outer actions of groups and introduce the concept of equivalence of two lifting homomorphisms, maximal lifting homomorphisms and maximal split extensions. Some criteria are obtained.

Let k be an algebraically closed field of odd characteristic p, and let D_n be the dihedral group of order 2n such that p|2n. Let D(kD_n) denote the quantum double of the group algebra kD_n. In this paper, we describe the structures of all finite-dimensional indecomposable left D(kD_n)-modules, equivalently, of all finite-dimensional indecomposable Yetter-Drinfeld kD_n-modules, and classify them.

The main work of this article is to give a construction of semilattice graded weak Hopf algebras via S-Hopf quivers, mainly based on the work of Cibils, Rosso, and Montgomery. This provides a class of non-commutative and non-cocommutative pointed semilattice graded weak Hopf algebras: they have the natural bases consisting of paths and the underlying coalgebra structures are path coalgebras. It also provides a new way of verifying old results and testing new ideas on pointed Hopf algebras, such as the decomposition of pointed semilattice graded weak Hopf algebras according to the result of Montgomery.

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed.

In this article, we characterize some rings in terms of the DG-projective, injective and flat complexes. We also give some connections between DG-injective complexes and DG-flat complexes and prove that under some additional conditions, every complex of left R-modules has a DG-flat preenvelope.

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol).

Let G be a finite group with a normal Sylow p-subgroup P. Let be a complete discrete valuation ring with residue field F of characteristic p. Let M be an indecomposable endo-monomial -module. In this paper we prove that M extends to an -module if and only if M is G-stable. A similar and well-known version for endo-permutation modules is due to Dade.

Let be a Lie algebra of Block type with basis {L_α,i| α ∈ ℤ, i ∈ ℤ₊} and relations [L_α,i,L_β,j]= ((α-1)(j+1)-(β-1)(i+1))L_{α+β, i+j}. In the present paper, the derivation algebra and automorphism group of are explicitly described. In particular, it is shown that the outer derivation space is 1-dimensional and the inner automorphism group of is trivial.