Quotients Of Hecke Algebras Research Articles

Finite-dimensional quotients of Hecke algebras

Let W be a complex reflection group. We prove that there is the maximal finite dimensional quotient of the Hecke algebra H_q(W) of W and that the dimension of this quotient coincides with |W|. This is a weak version of a Brou\'e-Malle-Rouquier conjecture from 1998. The proof is based on categories O for Rational Cherednik algebras.

Fusion symmetric spaces and subfactors

We construct analogs of the embedding of orthogonal and symplectic groups into unitary groups in the context of fusion categories. At least some of the resulting module categories also appear in boundary conformal field theory. We determine when these categories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors. This paper is a sequel of our previous paper [W4], where we introduced a q-deformation of Brauer’s centralizer algebra for orthogonal and symplectic groups; this algebra had already appeared more or less before in [Mo], see also discussion in [W4]. It is motivated by finding a deformation of orthogonal or symplectic subgroups of a unitary group which is compatible with the standard quantum deformation of the big group. This has been done before on the level of coideal subalgebras of Hopf algebras by Letzter. However, our categorical approach also allows us to extend this to the level of fusion tensor categories, where we find finite analogs of symmetric spaces related to the already mentioned groups. Moreover, we can establish C∗ structures, necessary for the construction of subfactors, in this categorical setting; this is not so obvious to see in the setting of co-ideal algebras. It is well-known how one can use a subgroup H of a (for simplicity here) finite group G to construct a module category of the representation category Rep G of G. This module category also appears in the context of subfactors of II1 von Neumann factors as follows: Let R be the hyperfinite II1 factor, and let N = RG ⊂M = RH be the fixed points under outer actions of G and H. Then the category of N − N bimodules is equivalent to Rep G, and the module category is given via the M−N bimodules of the inclusion N ⊂ M; its simple objects are labeled by the irreducible representations of H. In particular, an important invariant called the principal graph of the subfactor is determined by the restriction rules for representations from G to H. Important examples of subfactors were constructed from fusion categories whose Grothendieck semirings are quotients of the ones of semisimple Lie groups. So a natural question to ask is whether one can perform a similar construction in this context. More precisely, can we find restriction rules for type A fusion categories which describe a subfactor as before, and which will approach in the classical limit the usual restriction rules from U(N) to O(N). We answer this question in the positive in this paper via a fairly elementary construction. We show that certain semisimple quotients of the q-Brauer algebras have a C∗ structure and contain C∗-quotients of Hecke algebras of type A. The subfactor is then obtained as the closure of inductive limits of such algebras. Due to its close connection to Lie groups, we can

ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type En(for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.

Bridges with pillars: a graphical calculus of knot algebra

The paper comprises a graphical calculus which is designed to deal with the Coxeter-Dynkin series of type E and some generalizations. Temperley-Lieb algebras of type E are defined as quotients of Hecke algebras and the module structure of the algebra associated to E 6 is determined. The graphical calculus is a refinement of the calculus for the ordinary Temperley-Lieb algebra: a planar strip is decomposed by the arcs of a diagram into domains and the domains are used to incorporate additional information into the figure.

Structure of a Hecke algebra quotient

Let W W be a Coxeter group with Coxeter graph γ \gamma . Let H \mathcal {H} be the associated Hecke algebra. We define a certain ideal I \mathcal {I} in H \mathcal {H} and study the quotient algebra H ¯ = H / I \bar {\mathcal {H}} = \mathcal {H}/\mathcal {I} . We show that when γ \gamma is one of the infinite series of graphs of type E E , the quotient is semi-simple. We examine the cell structures of these algebras and construct their irreducible representations. We discuss the case where γ \gamma is of type B B , F F , or H H .

A Hecke Algebra quotient and some combinatorial applications

A Hecke Algebra quotient and some combinatorial applications

A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \bar H of H and show that it has a basis parametrized by a certain subset Wc of the Coxeter group W. Specifically, Wc consists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite Wc and compute the cardinality of Wc when W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for \omega \in W_c) of an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \bar H.

Hecke algebra solutions to the reflection equation

We construct solutions to Sklyanin's reflection equation in the case in which the bulk Yang-Baxter solution is of Hecke algebra type. Each solution constitutes an extension of the Hecke algebra together with a spectral parameter dependent boundary operator, K( theta ). We solve for the defining relations of the extension, and for the spectral parameter dependence of the boundary operator. The problem of finding concrete matrix solutions to the reflection equation is thus reduced to that of constructing finite dimensional representations of the Hecke algebra extension. This problem is solved in the generic case by establishing isomorphism with an affine Hecke algebra quotient whose representation theory is known.

Relative Commutants of Hecke Algebra Subfactors

real region and root of unity region display intrinsically different behavior, the most notable example of which is the fact that in the discrete (root of unity) case the relative commutant, which consists of elements of the factor which commute with all of the subfactor, contains only scalar multiples of the identity, while in the continuous (real) case there is a nontrivial relative commutant. This commutant is not there in the algebraic closure of the standard generators, but makes its appearance only in the weak closure. For this reason it is difficult to show the existence of the nontrivial relative commutant [PP86]. Jones' factor forms a representation of the braid group, and the associated Markov trace gives the Jones knot polynomial. Natural generalizations arise by replacing the factor with a quotient of a Hecke algebra. The quotient generates a H,i factor with a Markov trace for the same values of the parameter A, and has a natural subfactor with analogous properties in the discrete and continuous region. Again, no relative commutant is algebraically generated by the standard generators, but it is reasonable to think that there is a relative commutant in the closure in the continuous case (where the closure is taken in the weak topology given by the G.N.S. construction with the Markov trace). That this is true is the principal result of this paper. By imbedding the Hecke algebra quotient inside a much larger algebra, we may express the relative commutant as concrete elements, as well as phrase the question in terms that don't depend on the language of subfactors. We shall think of the Hecke algebra quotient as being a subalgebra of the infinite tensor product of Mr, the algebra of r by r complex matrices, with the Markov trace given by the restriction of a Powers state (the weak closure of the quotient is a von Neumann subalgebra of the weak closure of the tensor product algebra under the G.N.S. construction with this state). The relative commutant consists of the diagonal subalgebra of the first copy of Mr in the tensor product. More generally, the weak closure of the quotient is the fixed point algebra of the modular group

Reaction-diffusion processes as physical realizations of Hecke algebras

We show that the master equation governing the dynamics of simple diffusion and certain chemical reaction processes in one dimension gives time evolution operators (Hamiltonians) which are realizations of Hecke algebras. In the case of simple diffusion one obtains, after similarity transformations, reducible hermitian representations while in the other cases they are non-hermitian and correspond to supersymmetric quotients of Hecke algebras.

Hecke Algebra Representations of Braid Groups and Link Polynomials

By studying representations of the braid group satisfying a certain quadratic relation we obtain a polynomial invariant in two variables for oriented links. It is expressed using a trace, discovered by Ocneanu, on the Hecke algebras of type A. A certain specialization of the polynomial, whose discovery predated and inspired the two-variable one, is seen to come in two inequivalent ways, from a Hecke algebra quotient and a linear functional on it which has already been used in statistical mechanics. The two-variable polynomial was first discovered by Freyd-Yetter, Lickorish-Millet, Ocneanu, Hoste, and Przytycki-Traczyk.