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We classify Sylow p -subgroups of finite groups whose principal p -blocks have precisely six ordinary irreducible characters. [ABSTRACT FROM AUTHOR]

We give a generic description of the non-defining primes l , for which the Steinberg character lies in the principal l -block of a finite reductive group. [ABSTRACT FROM AUTHOR]

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group S n. In particular, we show that for any parameter c ∈ C , the category of Harish-Chandra H c -bimodules admits a fully faithful embedding into the category O c , and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime dividing | G | . In this paper, we study the structure of G, which satisfies the condition that any two irreducible characters in its principal p-block have distinct degrees. We provide a reduction of this question to simple groups and determine the structure of G for p = 2. Furthermore, we describe the structure of G in which any two characters of the principal p-block of G have coprime degrees. [ABSTRACT FROM AUTHOR]

We describe finite groups whose principal block contains only characters of prime power degree. [ABSTRACT FROM AUTHOR]

We show that the principal p -block of a finite group of order divisible by p has at least 2 p − 1 height-zero characters. Along the way, we describe the p -local structure of finite groups whose principal p -blocks have at most five height-zero ordinary irreducible characters. [ABSTRACT FROM AUTHOR]

We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures. [ABSTRACT FROM AUTHOR]

Given two primes p and q, we study degrees and rationality of irreducible characters in the principal p-block of S n and A n , the symmetric and alternating groups. In particular, we show that such a block always admits an irreducible character of degree divisible by q. This extends and generalizes a recent result of Giannelli–Malle–Vallejo. [ABSTRACT FROM AUTHOR]

Let p ≥ 5 be a prime and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct p'-character degrees in the principal p-block of G. This generalizes a theorem of Isaacs-Smith as well as a recent result of three of the present authors. [ABSTRACT FROM AUTHOR]

Let p , q be different primes and suppose that the principal p - and the principal q -block of a finite group have only one irreducible complex character in common, namely the trivial one. We conjecture that this condition implies the existence of a nilpotent Hall { p , q } -subgroup and prove that a minimal counter-example must be an almost simple group where pq divides the order of its simple nonabelian normal subgroup. As an immediate consequence we obtain that the conjecture holds true for p -solvable or q -solvable groups. Furthermore, we prove the conjecture in case 2 ∈ { p , q } using the classification theorem of finite simple groups. Finally, we consider the situation that the intersection of an arbitrary p -block with an arbitrary q -block contains only one irreducible character. [ABSTRACT FROM AUTHOR]

In this note, it is shown that a finite group G is solvable if for each odd prime divisor p of | G | , | Irr ( B 0 ( G ) 2 ) ∩ Irr ( B 0 ( G ) p ) | ≤ 2 , where Irr ( B 0 ( G ) p ) is the set of complex irreducible characters of the principal p -block B 0 ( G ) p of G . Also, the structure of such groups is investigated. Examples show that the bound 2 is best possible. [ABSTRACT FROM AUTHOR]

Let G be a finite group having a normal p -subgroup N that contains its centralizer C G ( N ) , and let R be a p -adic ring. It is shown that any finite p -group of units of augmentation one in RG which normalizes N is conjugate to a subgroup of G by a unit of RG , and if it centralizes N it is even contained in N . [ABSTRACT FROM AUTHOR]

Let G be a finite group and k a field of characteristic p > 0. In this paper, we obtain several equivalent conditions to determine whether the principal block B of a finite p-solvable group G is p-radical, which means that B has the property that e ( k ) is semisimple as a kG-module, where P is a Sylow p-subgroup of G, k is the trivial kP-module, ( k ) is the induced module, and e is the block idempotent of B . We also give the complete classification of a finite p-solvable group G which has not more than three simple B -modules where B is p-radical. [ABSTRACT FROM AUTHOR]

We determine all finite groups whose first Cartan invariants over a field of characteristic two are two, by using the classification of finite simple groups. [ABSTRACT FROM AUTHOR]

We determine all finite groupsGsuch that the Loewy length (socle length) of the projective coverP(kG) of the trivial kG-modulekGis four, wherekis a field of characteristicp > 0 andkGis the group algebra ofGoverk, by using previous results and also the classification of finite simple groups. As a by-product we prove also that ifp = 2 then all finite groups G such that the Loewy lengths of the principal block algebras ofkGare four, are determined. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As an application we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module. [Copyright &y& Elsevier]

Abstract: We present a simple method for determining Scott modules of finite groups using just the Brauer tree of the principal block and the values of ordinary characters at non-identity p-elements. Previously, determining Scott modules often involved long calculations. The method is applicable to the case where finite groups have cyclic Sylow p-subgroups. The structure of a Scott module is closely related to the length of the shortest path from the vertex labeled by the trivial ordinary character to the exceptional vertex in the Brauer tree. We classify the structures of the Scott modules and the ordinary characters they afford, by the length of the associated path in the Brauer tree. [Copyright &y& Elsevier]

We study integrality of the Frobenius-Perron eigenvalues of the Cartan matrices for the principal blocks of some finite groups of Lie type with noncyclic abelian Sylow p-subgroups. [ABSTRACT FROM AUTHOR]

Abstract: Let A be the principal 3-block of a finite group G with an abelian Sylow 3-subgroup P. Let be the Cartan matrix of A, and we denote by the unique largest eigenvalue of . The value is called the Frobenius–Perron eigenvalue of . We shall prove that is a rational number if and only if A and the principal 3-block of are Morita equivalent. This generalizes earlier Wada''s theorem in 2007, where he proves it only for the case that the order of P is nine, while we prove it for the case that P is an arbitrary finite abelian 3-group. The result presented here uses the classification of finite simple groups. [Copyright &y& Elsevier]

We determine the finite simple groups of Lie type of characteristic p, for which the Steinberg character lies in the principal ℓ-block for every prime ℓ ≠ p dividing the order of the group. [ABSTRACT FROM AUTHOR]

M. Cabanes and J. Rickard showed in [3] that the Alvis–Curtis character duality of a finite group of Lie type is induced in non defining characteristic ℓ by a derived equivalence given by tensoring with a bounded complex X, and they further conjecture that this derived equivalence should actually be a homotopy equivalence. Following a suggestion of R. Kessar, we show here for the special case of principal blocks of general linear groups with abelian Sylow-ℓ-subgroups that this is true, by an explicit verification relating the complex X to the Coxeter complex of the corresponding Weyl group. [ABSTRACT FROM AUTHOR]