The Modular Representation Ring of a Cyclic 2-Group

1975 ◽  
Vol s2-11 (1) ◽  
pp. 91-92 ◽  
Author(s):  
Jon F. Carlson
Keyword(s):  
Modular Representation ◽  
Representation Ring

Some computations in the modular representation ring of a finite group

1971 ◽  
Vol 69 (1) ◽  
pp. 163-166 ◽  
Author(s):  
John Santa Pietro
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Modular Representation ◽  
Multiplicative Structure ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Representation Ring ◽  
Ring Isomorphism ◽  
A Finite Group

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

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.

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

Sign in / Sign up

Export Citation Format

Share Document