Representation Theory of Finite Groups

We shall consider here two generalizations to rings of the concept of induced representation as it occurs in the representation theory of finite groups (6).If A is a ring, S a subring of A, we shall associate with each S-module M an induced pair (I(M), K) consisting of an A -modulo I(M) and an S-homomorphism K: M → I(M).

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-045-6 ◽

1991 ◽

Vol 43 (4) ◽

pp. 792-813 ◽

Cited By ~ 2

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.

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A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1959-007-4 ◽

1959 ◽

Vol 11 ◽

pp. 59-60 ◽

Cited By ~ 4

Author(s):

Hirosi Nagao

Keyword(s):

Finite Group ◽

Irreducible Representation ◽

Representation Theory ◽

Finite Groups ◽

Characteristic Zero ◽

Orthogonality Relations ◽

Schur’S Lemma ◽

Image Position ◽

A Finite Group ◽

Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

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Decomposing the complexity quotient category

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100001572 ◽

1996 ◽

Vol 120 (4) ◽

pp. 589-595

Author(s):

D. J. Benson

Keyword(s):

Representation Theory ◽

Finite Groups ◽

Modular Representation ◽

Recent Effort ◽

Finitely Generated ◽

Modular Representation Theory ◽

Joint Work ◽

Quotient Category ◽

Finitely Generated Modules

In the modular representation theory of finite groups, much recent effort has gone into describing cohomological properties of the category of finitely generated modules. In recent joint work of the author with Jon Carlson and Jeremy Rickard[3], it has become clear that for some purposes the finiteness restriction is undesirable. In particular, in the quotient category of kG-modules by the subcategory of modules of less than maximal complexity, it turns out that finitely generated modules can have infinitely generated summands, and that including these summands in the category repairs the lack of Krull–Schmidt property.

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