A natural Morita equivalence reduction for some blocks of G-crossed product O-lattices and applications to the Clifford theory of finite group modular representation theory

2019 ◽  
Vol 48 (4) ◽  
pp. 1726-1743
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Morita Equivalence ◽  
Modular Representation ◽  
Crossed Product ◽  
Modular Representation Theory ◽  
Clifford Theory

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

scholarly journals On Brauer’s height 0 conjecture

1988 ◽  
Vol 109 ◽  
pp. 109-116 ◽  
Author(s):  
T.R. Berger ◽  
R. Knörr
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Defect Group ◽  
Modular Representation Theory

R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the “Height zero conjecture”. It may be stated as follows: CONJECTURE. Let B be a p-block of the finite group G. All irreducible ordinary characters of G belonging to B are of height 0 if and only if a defect group of B is abelian.

Analogs of the Green and Puig correspondences for twisted group algebras

2016 ◽  
Vol 19 (1) ◽  
pp. 1-24
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Group Algebras ◽  
Modular Representation Theory ◽  
Twisted Group ◽  
Standard Derivation ◽  
Green Correspondence

AbstractIn the modular representation theory of finite groups, we show that the standard derivation of the Green correspondence lifts to a derivation of a Green correspondence for twisted group algebras (Theorem 1.3). Then, from these results we derive a lift of the Puig correspondences for twisted group algebras (Theorem 1.6).Clearly twisted group algebras arise naturally in finite group modular representation theory. We conclude with some suggestions for applications in this mathematical area.

A block extension of categorical results in the Green correspondence context

2014 ◽  
Vol 17 (6) ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Short Proof ◽  
Modular Representation ◽  
Academic Press ◽  
Modular Representation Theory ◽  
Representations Of Finite Groups ◽  
Algebra 2 ◽  
Green Correspondence

AbstractIn [J. Pure Appl. Algebra 2 (1972), 371–393, Theorem 4.1], J. A. Green shows that the Green Correspondence in Finite Group Modular Representation Theory is a consequence of an equivalence between two quotient categories of appropriate subcategories in the Green Correspondence context. In [Adv. Math. 104 (1994), 297–314, Theorems 3.5, 3.6 and 3.7], M. Auslander and M. Kleiner prove a similar result. M. Linckelmann suggested that the quotient categories in these results are the same. Utilizing extensions of [The Representation Theory of Finite Groups, North-Holland, Amsterdam, 1982, III, Theorem 7.8] or [Representations of Finite Groups, Academic Press, San Diego, 1988, Chapter 5, Corollary 3.11], we extend these results to blocks of finite groups. In order to state and prove our results and to remain relatively self-contained, we follow the procedures of [Adv. Math. 104 (1994), 297–314] in the Green Correspondent context. This is presented in Section 1. In Section 2 we present our main results. In Section 3 we give a very short proof of a theorem of H. Fitting for 𝒪-algebras that is essential in the proof of basic results of J. A. Green, [J. Pure Appl. Algebra 2 (1972), 371–393, Lemma 3.9 and Theorem 3.10].

Relative characters for H-projective RG-lattices

1988 ◽  
Vol 104 (2) ◽  
pp. 207-213 ◽  
Author(s):  
Peter Symonds
Keyword(s):  
Representation Theory ◽  
Direct Summand ◽  
Modular Representation ◽  
Dedekind Domain ◽  
Modular Representation Theory ◽  
Trivial Group ◽  
Ordinary Character ◽  
Projective Lattices

If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

On the Modular Representations of the Symmetric Group

1954 ◽  
Vol 6 ◽  
pp. 486-497 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
Modular Representation ◽  
Modular Representations ◽  
Modular Representation Theory

The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).

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