The absolute order of a permutation representation of a Coxeter group

A permutation representation of a Coxeter group W naturally defines an absolute order. This family of partial orders (which includes the absolute order on W) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on W are investigated when W is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of W is defined.     

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

Double affine Hecke algebra in logarithmic conformal field theory

We construct a representation of the double affine Hecke algebra. The symmetrization of this representation coincides with the center of the quantum group $ U_mathfrak{q} mathcal{S}ell (2) $ U_mathfrak{q} mathcal{S}ell (2) and, by Kazhdan-Lusztig duality with the Verlinde algebra of the (1,p)-model of the logarithmic conformal field theory.     

Structure of a Hecke algebra quotient

Let be a Coxeter group with Coxeter graph . Let be the associated Hecke algebra. We define a certain ideal in and study the quotient algebra . We show that when is one of the infinite series of graphs of type , the quotient is semi-simple. We examine the cell structures of these algebras and construct their irreducible representations. We discuss the case where is of type , , or .

     

Inner products on the Hecke algebra of the braid group

We claim that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. With respect to these structures, the set of positive, respectively negative permutation braids becomes an orthonormal basis. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings.     

A representation of the chamber set Hecke algebra of an affine building

We define a representation of the chamber set Hecke algebra of an affine building. This representation allows to extend to every affine buildings of rank 3 the Kato’s condition for the bijectivity of the Poisson transform.     

The Coxeter polynomial for a one point extension algebra

Let Λ be a finite-dimensional algebra over an algebraically closed field k of finite global dimension. Let M be a finitely generated Λ-module and let $\Gamma =\Lambda [M]$ be the one point extension algebra. We show how to compute the Coxeter polynomial for Γ from the Coxeter polynomial of Λ and homological invariants of M.     

On the center of a Hecke algebra

The center of a group algebra of a finite group and the center of a Hecke algebra for this group are related closely.     

The sorting order on a Coxeter group

Let (W,S) be an arbitrary Coxeter system. For each word ω in the generators we define a partial order—called the ω-sorting order—on the set of group elements Wω⊆W that occur as subwords of ω. We show that the ω-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the ω-sorting order is a “maximal lattice” in the sense that the addition of any collection of Bruhat covers results in a nonlattice.Along the way we define a class of structures called supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.