The modular Hecke algebra and Steinberg representation of finite Chevalley groups

The Steinberg lattice of a finite Chevalley group and its modular reduction

On the irreducibility of the complex specialization of the representation of the Hecke algebra of the complex reflection group $G_7$

We consider a 2-dimensional representation of the Hecke algebra $H(G_7, u)$, where $G_7$ is the complex reflection group and $u$ is the set of indeterminates $u = (x_1,x_2,y_1,y_2,y_3,z_1,z_2,z_3)$. After specializing the indetrminates to non zero complex numbers, we then determine a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of the repre...

Representation Theory of the Nonstandard Hecke Algebra

The nonstandard Hecke algebra Ȟr was defined in [11] to study the Kronecker problem. We study a quotient Ȟr,2 of Ȟr , called the nonstandard Temperley-Lieb algebra, which is simpler than Ȟr. It is defined to be the subalgebra of Hr,2⊗Hr,2 generated by Pi := C ′ i ⊗C ′ i +Ci ⊗Ci, i ∈ [r− 1], where Hr,2 is the Temperley-Lieb algebra and C i and Ci are Kazhdan-Lusztig basis elements. We give a com...

in Algebra . Coxeter groups and Hecke algebras

The finite Coxeter groups are the finite groups generated by reflections on real Euclidean spaces. Examples include dihedral groups, the symmetry groups of regular polytopes (e.g. regular polygons and platonic solids) and the Weyl groups of semisimple complex Lie groups and Lie algebras (such as the special linear group and Lie algebra). General Coxeter groups may be defined as certain (special...

Representation theory of finite groups II: Linear algebra