Centralizer and normalizer

About: Centralizer and normalizer is a research topic. Over the lifetime, 2752 publications have been published within this topic receiving 27760 citations. The topic is also known as: centralizer subgroup.

Hecke algebras of type An and subfactors.

Abstract: In his paper [J-l] V. Jones introduced an index, which 'measures' the size of a subfactor in a II1 factor. The main result of that paper is that the index of a subfactor has to be either greater or equal than 4 or it has to be equal to 4cosZ(x//) for some l~N, I>3 and that there exist subfactors for all these index values. Similarly as for subgroups, the index alone does not characterize the subfactor up to conjugacy by automorphisms. The fact that there are only countably many possible index values < 4 seems to be related to another invariant. Subfactors with index less than 4 always have trivial centralizers, (or relative commutants), i.e. the only elements of the factor which commute with every element of the subfactor are multiples of the identity. On the other hand, the examples given in I-J-l] for subfactors with index greater than 4 all have nontrivial centralizers. Furthermore, all known examples of subfactors with trivial relative commutants have as index an algebraic integer. At the current state of knowledge, it is still unknown whether there are only countably many values possible for the index of subfactors with trivial centralizers. Note however, that the set of all possible index values of a subfactor with trivial centralizer in an arbitrary II~ factor has to be a closed subset of R (see [HW]). Our original motivation for this paper was to study how subfactors of the hyperfinite II1 factor can be constructed via AF algebras. We provide a method of computing the index and we give an upper bound for the size of the centralizer of the constructed subfactor. Our general results will then be applied to the series of complex Hecke algebras H,(q), n~N of type A,_I. Their standard generators gx, g2, -.., gn1 satisfy the same relations as a set of simple reflections of the symmetric group S, except that the reflection property g~ = 1 is replaced by g ~ = ( q 1 ) g i + q . It is well-known that H,(q) is isomorphic to C S , if q is not a root of unity. If the parameter is a root of unity, Hn(q) may no longer bc sernisimple and its structure is not known in general. This is, however, the most interesting case as far as subfactors are concerned. We define representations p of Ha(q) such that p(H,(q)) is semisimple for all n~N. Together with

The Subconstituent Algebra of an Association Scheme, (Part I)

Abstract: We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the P- and Q-polynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple \Bbb {C}-algebra T e T(x) whose structure reflects the combinatorial structure of Y. We call T the subconstituent algebra of Y with respect to x. Roughly speaking, T is a combinatorial analog of the centralizer algebra of the stabilizer of x in the automorphism group of Y. In general, the structure of T is not determined by the intersection numbers of Y, but these parameters do give some information. Indeed, we find a relation among the generators of T for each vanishing intersection number or Krein parameter. We identify a class of irreducible T-moduIes whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say Y is thin if every irreducible T(y)-module is thin for every vertex y of Y. We compute the possible thin, irreducible T-modules when Y is P- and Q-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If Y is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”. We give a combinatorial characterization of the thin P- and Q-polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible T-modules actually occur. We close with some conjectures and open problems.